Finite Groupoids, Finite Coverings and Symmetries in Finite Structures
aa r X i v : . [ m a t h . C O ] O c t Finite Groupoids, Finite Coverings andSymmetries in Finite Structures
Martin OttoDepartment of MathematicsTechnische Universit¨at DarmstadtOctober 2015
Abstract
We propose a novel construction of finite hypergraphs and relational struc-tures that is based on reduced products with Cayley graphs of groupoids.To this end we construct groupoids whose Cayley graphs have large girthnot just in the usual sense, but with respect to a discounted distancemeasure that contracts arbitrarily long sequences of edges within the samesub-groupoid (coset) and only counts transitions between cosets. Reducedproducts with such groupoids are sufficiently generic to be applicable tovarious constructions that are specified in terms of local glueing operationsand require global finite closure. We here examine hypergraph coveringsand extension tasks that lift local symmetries to global automorphisms. ontents Note:
This paper extends and supersedes earlier expositions of core resultsin [14, 15]. 2
Introduction
Consider a partial specification of some global structure by descriptions of itslocal constituents and of the possible links between these, in terms of allowed andrequired direct overlaps between pairs of local constituents. Such specificationstypically have generic, highly regular, free, infinite realisations in the form oftree-like acyclic objects. We here address the issue of finite realisations , whichshould ideally meet similar combinatorial criteria in terms of genericity andsymmetry. Instead of full acyclicity, which is typically unattainable in finiterealisations, we look for specified degrees of acyclicity.Overlaps can be specified by partial bijections between the local constituents.It turns out that the universal algebraic and combinatorial properties of grou-poids, which can be abstracted from the composition behaviour of partial bi-jections, support a very natural approach to the construction of certain highlysymmetric finite instances of hypergraphs and relational structures that providethe desired finite realisations.We use hypergraphs as abstractions for the decomposition of global struc-tures into local constituents. As a collection of subsets of a given structure,the collection of hyperedges specifies the notion of locality: the local view com-prises one hyperedge at a time. Depending on context, individual hyperedgesmay carry additional local structure, e.g., interpretations of relations stemmingfrom an underlying relational structure.
Realisations.
For the synthesis task, we may think of descriptions of localstructure as given in piece-wise, a priori disjoint patches that are taken to be thetemplates for the local constituents in the global structure; a global realisationis to be constructed from isomorphic copies of these local pieces. The secondingredient is a specification of the desired overlaps between pairs of such localconstituents; it describes their admitted and required intersections. In thisgeneral context, we offer a versatile and generic solution to the finite synthesisproblem posed by partial specifications of hypergraphs (relational structureswith a notion of locality) in terms of hyperedges (isomorphism types of localsubstructures) and overlaps between these. Depending on the nature of thesynthesis task, even the very existence of a finite solution may not be immediate.In other circumstances, if some finite realisation is explicitly given or has beenobtained in a first step, we aim to meet additional global criteria in special,qualified finite realisations. The global criteria under consideration are(i) criteria concerning controlled acyclicity w.r.t. the natural gradation ofhypergraph acyclicity or tree decomposability for finite structures;(ii) criteria of global symmetry in the sense of a rich automorphism group thatextends rather than breaks the symmetries of the given specification.In the most general case, the specification of the overlap pattern to be re-alised comes as a disjoint family of abstract regions ( V s ) s ∈ S together with acollection of partial bijections ( ρ e ) e ∈ E , where each e ∈ E has specified sourceand target sites s, s ′ ∈ S and ρ e is a partial bijection between sites V s and V s ′ .3 s V s ′ •• ••• •• ρ e ρ e K K I I • s • s ′ e V V e H H Figure 1: Links between two sites in overlap specification and in the underlyingincidence pattern.We think of the set S as an index set for the different types of local constituents;and of E as an index set for the types of pairwise overlaps. Formally, E will bethe set of edges in a multi-graph with vertex set S .Figure 1 provides an example of a possible overlap specification between twosites V s and V s ′ along with two different modes of overlap, e and e . Note thatminimal requirements w.r.t. an isomorphic (i.e., bijective) embedding of V s intoa desired realisation makes it necessary that, in this example, the e -overlapand the e -overlap of one and the same copy of V s need to go to different copiesof V s ′ , and similarly every V s ′ -copy will have to overlap with distinct V s -copies.Globally, a realisation of an overlap specification consists of partly overlap-ping components V ˆ s that are each isomorphic to one of the local patches V s viasome local projection π ˆ s : V ˆ s → V s , and such that every copy of V s overlaps withcopies of suitable V s ′ according to the overlap specified in the partial bijections ρ e for those edges e ∈ E that link s to some s ′ . Figure 2 gives a local impressionof how a partial bijection ρ e between sites V s and V s ′ of the overlap specificationis to be realised as an actual overlap of isomorphic copies of these patches V s and V s ′ within a realisation.As discussed in [11], groupoids and inverse semigroups naturally occur inconnection with the description of global structure by means of local coordi-nates; atlases for manifolds provide a typical example in the continuous world.The systems of partial bijections that represent changes between different lo-cal coordinates can be abstracted as either inverse semigroups or as groupoids(ordered groupoids in the terminology of [11]). Our use of groupoids in theconstruction of realisations may, in these terms, be associated with atlases for aglobal realisation whose local views and overlaps between local views are speci-fied in the given overlap pattern. The local sites ( V s ) s ∈ S of the specification formthe coordinate domains of an atlas for the realisation, in which local patches V ˆ s corresponding to V s are isomorphically related to their template V s via local4 s V s ′ V ˆ s V ˆ s ′ ρ e π ˆ s (cid:9) (cid:9) ✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒ π ˆ s ′ (cid:23) (cid:23) ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴ Figure 2: A realisation of some ρ e as an overlap at one locus of a realisation.projections π ˆ s ; these projections serve as the local coordinatisations. Coverings.
More concrete overlap specifications arise from an actual struc-ture composed of local patches. A non-trivial further realisation task in thiscontext asks for a replication of the given overlap pattern in another finitestructure, with extra constraints on the global properties of the new realisation.Intuitively, one can think of a finite process of partial unfolding. Formally, wecast this as a covering problem at the level of hypergraphs. These may be thehypergraphs induced by some notion of locality in other kinds structures; thewhole approach thus naturally extends to such settings. A hypergraph coveringaims to reproduce the overlap pattern between hyperedges of a given hyper-graph in a covering hypergraph while smoothing out the overall behaviour, e.g.,by achieving a higher degree of acyclicity. The conceptual connections withtopological notions of (branched) coverings [5] are apparent, but we keep inmind that here we insist on finiteness so that full acyclicity as in universalcoverings cannot generally be expected.It may be instructive to compare first the situation for graphs rather thanhypergraphs. Here the covering would be required to provide lifts for everyedge – and, by extension, every path – in the base graph, at every vertex inthe covering graph above a given vertex in the base graph. For an unbranchedcovering, the in- and out-degrees of a covering node would also be required tobe the same as for the corresponding vertex in the base graph. And indeedgraphs do allow for unbranched, finite coverings in this sense, which achieve anydesired finite degree of acyclicity (i.e., can avoid cycles of length up to N forany desired threshold N ), as the following result from [12] shows. Proposition 1.1.
Every finite graph admits, for each N ∈ N , a faithful (i.e.,unbranched, degree-preserving) covering by a finite graph of girth greater than N (i.e, without cycles of length up to N ). • ✌✌✌✌✌✌✌✌✌✌✌✌✌✌ • ✶✶✶✶✶✶✶✶✶✶✶✶✶✶ •• ✌✌✌✌✌✌✌✌✌✌✌✌✌✌ • ✶✶✶✶✶✶✶✶✶✶✶✶✶✶ •• ✌✌✌✌✌✌✌✌✌✌✌✌✌✌ • ✶✶✶✶✶✶✶✶✶✶✶✶✶✶ •• ✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌ • ✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶ s s s s (2 ,
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Consider a covering task for the simple hypergraph consisting ofthe facets of the 3-simplex. This hypergraph is associated with the faces of thetetrahedron; or it may be seen as just the complete 3-uniform hypergraph on fourvertices. A representation of the overlap specification between the hyperedges(faces) is provided in Figure 3. Possible local views around a single vertex ofa finite covering hypergraph are indicated in Figure 4. It is clear that evenlocally, cycles cannot be avoided (in any trivial sense of avoiding cycles): thecyclic succession of overlapping copies of the hyperedges s , s and s around6 single shared vertex will have to close back onto itself in any finite covering.This also means that the incidence degrees of hyperedges of a certain type withvertices cannot be preserved in any non-trivial finite covering, or that branchedrather than unbranched coverings will have to be considered.Proposition 1.1 above shows that neither of these perceived obstacles, viz. thelack of a clear local-vs-global distinction and the necessity to consider branchedcoverings, arises in the special case of graphs. The uniform and canonical con-struction of N -acyclic graph coverings according to Proposition 1.1, as givenin [12], is based on a natural product between the given graph and Cayleygroups of large girth. The latter in turn can be obtained as subgroups of thesymmetric groups of the vertex sets of suitably coloured finite acyclic graphsin an elegant combinatorial construction attributed to Biggs [4] in Alon’s sur-vey [1].Some of these ideas were successfully lifted and applied to the constructionof hypergraph coverings in [13]. For the combinatorial groups, the generalisationinvolved led to a uniform construction of Cayley groups that not only have largegirth in the usual sense. Instead, they have large girth even w.r.t. to a reduceddistance measure that measures the length of cycles in terms of the number ofnon-trivial transitions between cosets w.r.t. subgroups generated by differentcollections of generators. For an intuitive idea how this concern arises we mayagain look at the above example of the faces of the tetrahedron. There are twodistinct sources of avoidable short cycles in its finite branched coverings: (a)‘local cycles’ around a single pivot vertex, in the 1-neighbourhoods of a singlevertex, and of length 3 k in a locally k -fold unfolding; (b) ‘non-local cycles’ thatenter and leave the 1-neighbourhoods of several distinct vertices. To account forthe length of a cycle of type (b), the number of individual single-step transitionsbetween faces around one of the visited pivotal vertices is typically irrelevant;what essentially matters is how often we move from one pivot to the next, andthis corresponds to a transition between two subgroups (think of a transitionbetween the stabiliser of one pivot and the next).But nothing as simple as a product between a hypergraph and even oneof these ‘highly acyclic’ Cayley graphs will produce a covering by a finite N -acyclic hypergraph (to be defined properly below). The construction presentedin [13] uses such Cayley groups only as one ingredient to achieve suitable hyper-graph coverings through an intricate local-to-global construction by inductionon the maximal size of hyperedges. More importantly, these further steps in theconstruction from [13] are no longer canonical. In particular, they do not pre-serve symmetries of the given hypergraph; it also remains unclear which kindsof singularities and branching are unavoidable as opposed to artefacts due tonon-canonical choices.We here expand the amalgamation techniques that were explored for thecombinatorial construction of highly acyclic Cayley graphs [13] from groupsto groupoids, and obtain ‘Cayley groupoids’ that are highly acyclic in a similarsense. It turns out that groupoids are a much better fit for the task of construct-ing hypergraph coverings as well as for the construction of finite hypergraphsaccording to other specifications. The new notion of Cayley groupoids allows7 s × { g } V s ′ × { gg e } (cid:9) (cid:9) ρ e ✒✒✒✒✒✒ I I ✒✒✒✒✒✒ Figure 5: Identification between layers in a reduced product.for the construction of finite realisations of overlap specifications by means ofnatural reduced products with these groupoids. It is more canonical and sup-ports realisations, and in particular also coverings, of far greater genericity andsymmetry than previously available. The basic idea of the use of groupoids in reduced products is the following. Think of a groupoid G whose elements aretagged by sort labels s ∈ S , which stand for the different sites V s in the overlapspecification; the groupoid is generated by elements ( g e ) e ∈ E which link sort s to sort s ′ if the partial bijection ρ e of the overlap specification links V s to V s ′ .Then a realisation of the overlap specification is obtained from a direct productthat consists of disjoint copies V s × { g } of V s for every groupoid element g ofsort s through a natural identification of elements in V s × { g } and in V s ′ × { gg e } according to ρ e , as sketched in Figure 5. Main Theorem.
Every abstract finite specification of an overlap pattern be-tween disjoint sets, by means of partial matchings between them, admits a real-isation in a finite hypergraph. Moreover, for any N ∈ N , this realisation can bechosen to be N -acyclic, i.e., such that every induced sub-hypergraph of up to N vertices is acyclic, and to preserve all symmetries of the given specification. See Section 3.1, Theorem 3.22 and Corollary 4.10 for the development anddetails.We address two main applications of this general construction. The first ofthese concerns the construction of coverings, as discussed above, as a specialcase of a realisation task in which the overlap specification is induced by a givenhypergraph (or a notion of locality in some decomposition of a given structure).See Section 3.5 for definitions and details, especially Proposition 3.21.
Theorem 1.
Every finite hypergraph admits, for every N ∈ N , a covering by afinite hypergraph that is N -acyclic, i.e., in which every induced sub-hypergraphof up to N vertices is acyclic. In addition, the covering hypergraph can be chosento preserve all symmetries of the given hypergraph. C of relational structures provided there is an infinite extension of therequired kind within C . Here the classes C under consideration are defined interms of forbidden homomorphisms. Herwig and Lascar use the term extensionproperty for partial automorphisms (EPPA) for this extension task from local toglobal symmetries in finite structures. The most general constructions providedin [8] remain rather intricate, while the original construction for graphs from [10]was greatly simplified in a remarkably transparent new argument in [8]. We herenow derive the strongest form of the Herwig–Lascar EPPA theorem as a natu-ral application of the main theorem. This application uses its full power w.r.t.its natural compatibility with symmetries (for a rich automorphism group) andcontrol of cycles (to guarantee a consistent embedding of the given structureand the omission of forbidden homomorphisms).This application also reflects on the role that pseudo-groups, inverse semi-groups and groupoids play in the algebraic and combinatorial analysis of localsymmetries, cf. [11]. See Section 4.4 for details, especially Corollary 4.18. Theorem 2 (Herwig–Lascar) . Every class C that is defined in terms of finitelymany forbidden homomorphisms has the finite model property for the extensionof partial isomorphisms to automorphisms (EPPA). Organisation of the paper.
Section 2 introduces I -graphs and I -groupoidsand their Cayley graphs for incidence patterns I = ( S, E ), which serve as tem-plates for the sites and links in overlap specifications; it proceeds to deal withthe construction of finite I -groupoids with strong acyclicity properties, basedon amalgamation of I -graphs and the abstraction of groupoid actions from I -graphs. In Section 3 we examine the construction of realisations via reducedproducts of I -graphs, which serve as specifications of the desired overlap pat-tern, with I -groupoids, which provide the backbones for suitable finite unfold-ings. Both the main theorem on realisations and the main theorem on coveringsare proved in that section. Readers interested in coverings, and not in the moregeneral setup used for realisations of abstract overlap specifications, can focuson Sections 3.3, 3.5 and 3.6 without loss of coherence as far as the theme of fi-nite coverings is concerned. Section 4 finally deals with the extension of local toglobal symmetries and presents the Herwig–Lascar result on extension proper-9ies of partial automorphisms as a natural application of the generic realisationsof overlap specifications between copies of the given structure. In this section we develop a method to obtain groupoids from operations oncoloured graphs. The basic idea is similar to the construction of Cayley groupsas subgroups of the symmetric group of the vertex set of a graph. In that con-struction, a subgroup of the full permutation group on the vertex set is generatedby permutations induced by the graph structure, and in particular by the edgecolouring of the graphs in question. This method is useful for the construction ofCayley groups and associated homogeneous graphs of large girth [1, 4]. In thatcase, one considers simple undirected graphs H = ( V, ( R e ) e ∈ E ) with edge colours e ∈ E such that every vertex is incident with at most one edge of each colour.In other words, the R e are partial matchings or the graphs of partial bijectionswithin V . Then e ∈ E induces a permutation g e of the vertex set V , where g e swaps the two vertices in every e -coloured edge. The ( g e ) e ∈ E generate a sub-group of the group of all permutations of V . For suitable H , the Cayley graphinduced by this group with generators ( g e ) e ∈ E can be shown to have large girth(no short cycles, i.e., no short generator sequences that represent the identity).We here expand the underlying technique from groups to groupoids and lift it toa higher level of ‘large girth’. The second aspect is similar to the strengtheningobtained in [13] for groups. The shift in focus from groups to groupoids is newhere. Just as Cayley groups and their Cayley graphs, which are particularlyhomogeneous edge-coloured graphs, are extracted from group actions on givenedge-coloured graphs in [13], we shall here construct groupoids and associatedgroupoidal Cayley graphs, which are edge- and vertex-colored graphs ( I -graphs,in the terminology introduced below), from given I -graphs. The generalisationfrom Cayley groups to the new Cayley groupoids requires conceptual changesand presents some additional technical challenges, but leads to objects that arebetter suited to hypergraph constructions than Cayley groups.
The basic idea for the specification of an overlap pattern was outlined in theintroduction. We now formalise the concept with the notion of an I -graph .The underlying structure I = ( S, E ), on which the notion of an I -graph willdepend, is a multi-graph structure whose vertices s ∈ S label the available sitesand whose edges e ∈ E label overlaps between these sites. This structure I serves as the incidence pattern for the actual overlap specification in I -graphsthat instantiate sites and overlaps by concrete sets V s for s ∈ S and partialbijections ρ e for e ∈ E . Definition 2.1. An incidence pattern is a finite directed multi-graph I = ( S, E )with edge set E = ˙ S s,s ′ ∈ S E [ s, s ′ ], where e ∈ E [ s, s ′ ] is an edge from s to s ′ in10 H • s • s ′ V s V s ′ e / / ρ e ❱❱❱❱ + + ❱❱❱❱ Figure 6: Local view of an I -graph H . I , with an involutive, fixpoint-free edge reversal e e − on E that bijectivelymaps E [ s, s ′ ] to E [ s ′ , s ]. Remark 2.2.
There is an alternative, multi-sorted view, which may seem morenatural from a categorical point of view. According to this view, an incidencepattern would be a structure I = ( S, E, ι , ι , ( ) − ) with two sorts S and E ,where ι , ι : E → S specify the start- and endpoints of edges e ∈ E , so that e ∈ E [ ι ( e ) , ι ( e )] . Here edge reversal corresponds to simply swapping ι and ι .In particular, this view fits better with our intention not to identify e with e − for loops e ∈ E [ s, s ] of I and will be our guide when we shall discuss symmetriesof incidence patterns in Section 4. For notational convenience we stick to the shorthand format I = ( S, E ) butkeep in mind that this notation suppresses the two-sorted picture, the typing ofthe edges, and the operation of edge reversal.
Definition 2.3. An I -graph is a finite directed edge- and vertex-coloured graph H = ( V, ( V s ) s ∈ S , ( R e ) e ∈ E ), whose vertex set V is partitioned into subsets V s andin which, for e ∈ E [ s, s ′ ], the directed edge relation R e ⊆ V s × V s ′ induces apartial bijection ρ e from V s to V s ′ , and such that the R e (the ρ e ) are compatiblewith edge reversal, i.e., R e − = ( R e ) − (or ρ e − = ρ − e ).In the following we use interchangeably the functional terminology of partialbijections ρ e and the relational terminology of partial matchings R e : it will beconvenient to pass freely between the view of an I -graph H as either a colouredgraph or as a family of disjoint sets linked by partial bijections.Edges in R e are also referred to as edges of colour e or just as e -edges. Wemay regard I -graphs as a restricted class of S -partite, E -coloured graphs, wherereflexive e -edges (loops) are allowed if e is a loop in I . We do not identify e with e − even for loops e ∈ E [ s, s ] of I . .1.1 Operation of the free I-structure We discuss the structure of the set of all compositions of the partial bijections ρ e in an I -graph. These partial bijections form an inverse semigroup [11], butour main emphasis is on other aspects. Firstly, the analysis of the compositionstructure of the ρ e underpins the idea of I -graphs as specifications of overlappatterns and thus of our crucial concept of realisations . Secondly, it preparesthe ground for the association with groupoids in Section 2.2.For I = ( S, E ), we let E ∗ stand for the set of all labellings of directed paths(walks) in I . A typical element of E ∗ is of the form w = e . . . e n where n ∈ N is its length and, for suitable s i ∈ S , the edges are such that e i ∈ E [ s i , s i +1 ]for 1 i n . We admit the empty labellings of paths of length 0 at s ∈ S ,and distinguish them by their location s as λ s . The set E ∗ is partitioned intosubsets E ∗ [ s, t ], which, for s, t ∈ S , consist of the labellings of paths from s to t in I , so that in particular λ s ∈ E ∗ [ s, s ]. For w = e . . . e n ∈ E ∗ [ s, t ], we write w − := e − n . . . e − for the converse in E ∗ [ t, s ], which is obtained by reversereading w and replacement of each edge label e by its reversal e − . The set E ∗ carries a partially defined associative concatenation operation( w, w ′ ) ∈ E ∗ [ s, t ] × E ∗ [ t, u ] ww ′ ∈ E ∗ [ s, u ] , which has the empty word λ s ∈ E ∗ [ s, s ] as the neutral element of sort s .One may think of this structure as a groupoidal analogue of the familiar wordmonoids. For further reference, we denote it as the free I -structure I ∗ = ( E ∗ , ( E ∗ [ s, t ]) s,t ∈ S , · , ( λ s ) s ∈ S ) . We note that the converse operation w w − of I ∗ does not provide in-verses: obviously, ee − = λ s for any e ∈ E [ s, s ′ ].Consider an I -graph H = ( V, ( V s ) , ( R e )). The partial bijections ρ e prescribedby the relations R e together with their compositions along paths in E ∗ inducea structure of the same type as I ∗ , in fact a natural homomorphic image of I ∗ as follows. For e ∈ E [ s, s ′ ], let ρ e be the partial bijection between V s and V s ′ induced by R e ⊆ V s × V s ′ . For w ∈ E ∗ [ s, t ], define ρ w as the partial bijection from V s to V t that is the composition of the maps ρ e i along the path w = e . . . e n ; inrelational terminology, the graph of ρ w is the relational composition of the R e i .For w ∈ E ∗ [ s, t ], ρ w : V s −→ V t is a partial bijection, possibly empty. We obtain a homomorphic image of thefree I -structure I ∗ = ( E ∗ , ( E ∗ [ s, t ]) , · , ( λ s )) under the map ρ : I ∗ −→ (cid:8) f : f a partial bijection of V (cid:9) w = e . . . e n ρ w = Q ni =1 ρ e i . For convenience we use the notation E ∗ , which usually stands for the set of all E -words,with a different meaning: firstly, E ∗ here only comprises E -words that arise as labellings ofdirected paths in I ; secondly, we distinguish empty words λ s ∈ E ∗ , one for every s ∈ S . ρ ww ′ = ρ w ′ ◦ ρ w wherever defined, i.e., for w ∈ E ∗ [ s, t ] , w ′ ∈ E ∗ [ t, u ] so that ww ′ ∈ E ∗ [ s, u ].The converse operation w w − maps to the inversion of partial maps ρ w − = ( ρ w ) − . Note that the domain of ( ρ e ) − ◦ ρ e is dom( ρ e ) and may be a proper subsetof V s . So we still do not have groupoidal inverses – this will be different onlywhen we consider complete I -graphs (cf. Definition 2.5) in Section 2.2 below. We isolate an important special class of coherent I -graphs in terms of a par-ticularly simple, viz. ‘untwisted’, composition structure of the ρ e . This conceptinvolves a notion of global path-independence. Definition 2.4. An I -graph H = ( V, ( V s ) , ( R e )) is coherent if every composition ρ w for w ∈ E ∗ [ s, s ] is a restriction of the identity on V s : ρ w ⊆ id V s for all s ∈ S, w ∈ E ∗ [ s, s ] . Note that coherence is a property of path-independence for the trackingof vertices via ρ w along paths in I : ρ w ( v ) = ρ w ( v ) for every pair w , w ∈ E ∗ [ s, t ] and for all v ∈ V s in dom( ρ w ) ∩ dom( ρ w ). To see this, consider thepath w = w − w ∈ E ∗ [ t, t ] and apply the map ρ w , which is responsible fortransport along this loop, to the vertex ρ w ( v ). Coherence may also be seen as anotion of flatness in the sense that the operation of the free I -structure does nottwist the local patches in a non-trivial manner. The I -graph representations ofhypergraphs to be discussed in Section 3.1 provide natural examples of coherent I -graphs. Of course I itself is trivially coherent when considered as an I -graph.Coherence of H implies that overlaps between arbitrary pairs of patches V s and V t , as induced by overlaps along connecting paths in I , are well-defined,independent of the connecting path. We let ρ st ( V s ) ⊆ V t stand for the subset of V t consisting of those v ∈ V t that are in the image of ρ w for some w ∈ E ∗ [ s, t ].Then ρ st ( V s ) ⊆ V t is bijectively related to ρ ts ( V t ) ⊆ V s by the partial bijection ρ st := [(cid:8) ρ w : w ∈ E ∗ [ s, t ] (cid:9) , which is well-defined due to coherence. Complete I -graphs trivialise those complicating features of the compositionstructure of the ρ e that arise from the partial nature of these bijections.13 s • tV s V tw * * w ρ w + + ρ w Figure 7: Coherence of I -graphs. Definition 2.5. An I -graph H is complete if the R e induce full rather thanpartial bijections, i.e., if, for all e ∈ E [ s, s ′ ], dom( ρ e ) = V s and image( ρ e ) = V s ′ .Note that I itself may be regarded as a trivially complete I -graph; the Cayleygraphs of I -groupoids will be further typical examples of complete I -graphs; seeDefinition 2.11 below. A process of completion is required to prepare arbitrarygiven I -graphs for the desired groupoidal operation.If H = ( V, ( V s ) , ( R e )) is an I -graph then the following produces a complete I -graph on the vertex set V × S , with the partition induced by the naturalprojection: H × I := (cid:0) V × S, ( ˜ V s ) , ( ˜ R e ) (cid:1) where, for s ∈ S, ˜ V s = V × { s } . For e ∈ E [ s, s ′ ], s = s ′ , the possibly incomplete R e in H is lifted to H × I according to˜ R e = (cid:8) (( v, s ) , ( v ′ , s ′ )) : ( v, v ′ ) an e -edge in H (cid:9) ∪ (cid:8) (( v ′ , s ) , ( v, s ′ )) : ( v, v ′ ) an e -edge in H (cid:9) ∪ (cid:8) (( v, s ) , ( v, s ′ )) : v not incident with an e -edge in H (cid:9) ;and, for e ∈ E [ s, s ], to˜ R e = (cid:8) (( v, s ) , ( v ′ , s )) : ( v, v ′ ) an e -edge in H (cid:9) ∪ (cid:8) (( v ′ , s ) , ( v, s )) : v/v ′ first/last vertex on a maximal e -path in H (cid:9) . We note that this stipulation does indeed produce a complete I -graph: for e ∈ E [ s, s ′ ], it is clear from the definition of the ˜ R e that ˜ R e ⊆ ˜ V s × ˜ V s ′ and that An e -path is a directed path in R He . V s has an outgoing e -edge and every vertex in ˜ V s ′ an incoming e -edge; ˜ R e also is a bijection as required: in the non-reflexive case, either v ∈ V is incident with an e -edge in H , which means that, for a unique v ′ ∈ V , oneof ( v, v ′ ) or ( v ′ , v ) is an e -edge in H , and in both cases (( v, s ) , ( v ′ , s ′ )) and(( v ′ , s ) , ( v, s ′ )) become e -edges in H × I ; or v is not incident with an e -edge in H , and (( v, s ) , ( v, s ′ )) thus becomes the only outgoing e -edge from ( v, s ) as wellas the only incoming e -edge at ( v, s ′ ). Also ˜ R e − = ( ˜ R e ) − as required. Observation 2.6. If H = ( V, ( V s ) , ( R e )) is a not necessarily complete I -graph,then H × I is a complete I -graph; the embedding σ : V −→ V × Sv ( v, s ) for v ∈ V s embeds H isomorphically as a weak substructure. If I is loop-free, or if H isalready complete w.r.t. to all loops e ∈ E [ s, s ] of I for all s ∈ S , then H embedsinto H × I as an induced substructure.Proof. Note that the natural projection onto the first factor provides the inverseto σ on its image. Then ( v, v ′ ) ∈ R e for e ∈ E [ s, s ′ ] implies that v ∈ V s and v ′ ∈ V s ′ and therefore that ( σ ( v ) , σ ( v ′ )) = (( v, s ) , ( v ′ , s ′ )) is an e -edge of H × I .Conversely, let ( σ ( v ) , σ ( v ′ )) = (( v, s ) , ( v ′ , s ′ )) be an e -edge of H × I . If s = s ′ then v = v ′ (as the V s partition V ) and ( v, v ′ ) must be an e -edge of H . If e ∈ E [ s, s ] is a loop of I , an e -edge (( v, s ) , ( v ′ , s )) for v, v ′ ∈ V s may occur in H × I even though ( v, v ′ ) is not an e -edge of H , but then v and v ′ were missingoutgoing, respectively incoming, e -edges in H .In the following we use, as a completion of H , the relevant connected com-ponent(s) of H × I ; i.e., the components into which H naturally embeds. Definition 2.7.
The completion ¯ H of a not necessarily complete I -graph H =( V, ( V s ) , ( R e )) is the union of the connected components in H × I incident withthe vertex set σ ( V ) = { ( v, s ) : v ∈ V s } .Identifying V with σ ( V ) ⊆ H × I , we regard H as a weak subgraph of ¯ H . Corollary 2.8.
For every I -graph H , the completion ¯ H is a complete I -graph.Completion is compatible with disjoint unions: if H = H ˙ ∪ H is a disjointunion of I -graphs H i , then ¯ H = ¯ H ˙ ∪ ¯ H . If H itself is complete, then ¯ H ≃ H .Proof. The first claim is obvious: by definition of completeness, any union ofconnected components of a complete I -graph is itself complete.For compatibility with disjoint unions observe that the connected componentof the σ -image of H in H × I is contained in the cartesian product of H with S , as edges of H × I project onto edges of H , or onto loops, or complete cyclesin H .For the last claim observe that, for complete H , the vertex set of the iso-morphic embedding σ : H → H × I is closed under the edge relations ˜ R e of H × I : due to completeness of H , every vertex in σ ( V s ) is matched to preciselyone vertex in σ ( V s ′ ) for every e ∈ E [ s, s ′ ]; it follows that no vertex in σ ( V ) canhave additional edges to nodes outside σ ( V ) in H × I .15f α = α − ⊆ E we write I α for the reduct of I to its α -edges. We may regardthe α -reducts of I -graphs (literally: their reducts to just those binary relations R e for e ∈ α ) as I α -graphs as well as I -graphs. Note that every I α -graph is alsoan I -graph but, unless α = E , cannot be a complete I -graph. The α -reduct ofthe I -graph H is denoted H ↾ α . Closures of subsets of I -graphs under α -edges(edges of colours e ∈ α ) will arise in some constructions below. Lemma 2.9.
Let α = α − ⊆ E and consider an I -graph H and its α -reduct K := H ↾ α as well as their completions ¯ H and ¯ K as I -graphs and their α -reducts ¯ H ↾ α and ¯ K ↾ α . Then ¯ H ↾ α is an induced subgraph of ¯ K ↾ α , ¯ H ↾ α ⊆ ¯ K ↾ α, and the vertex set of ¯ H is closed under α -edges within ¯ K .Proof. Recall that the completion ¯ H ⊆ H × I consists of the connected com-ponent of the diagonal embedding σ ( V ) ⊆ V × S into H × I . This connectedcomponent is formed w.r.t. the union of the edge relations ( ˜ R e ) e ∈ E of H × I .Similarly, the completion of K is formed by ¯ K ⊆ K × I , where the connectedcomponent of σ ( V ) ⊆ V × S is w.r.t. the union of the edge relations ( ˜ R ′ e ) e ∈ E of K × I = ( H ↾ α ) × I , for all e ∈ E . Let D := σ ( V ) = { ( v, s ) : s ∈ S, v ∈ V s } , which is the same set of seeds for the completions ¯ H and ¯ K as closures of D under ( ˜ R e ) e ∈ E and ( ˜ R ′ e ) e ∈ E , respectively. For e ∈ α , the edge relations ˜ R e and˜ R ′ e of H × I and of K × I also coincide.For e ∈ E \ α , however, ˜ R e and ˜ R ′ e need not agree. Whenever ( v, v ′ ) is an e -edge in H , for some e ∈ E [ s, s ′ ] \ α , then this e -edge is not present in K ,whence, for s = s ′ ,(( v, s ) , ( v ′ , s ′ )) , (( v ′ , s ) , ( v, s ′ )) ∈ ˜ R e in H × I (double arrows in Fig. 8)(( v, s ) , ( v, s ′ )) , (( v ′ , s ) , ( v ′ , s ′ )) ∈ ˜ R ′ e in K × I (single arrows in Fig. 8)For e ∈ E [ s, s ] \ α , no relevant discrepancies occur, since additional e -edgesfor e ∈ E [ s, s ] have no effect on connectivity in either H × I or K × I .Since the vertices on the diagonal ( v, s ) , ( v ′ , s ′ ) ∈ D = σ ( V ) are verticesof both ¯ H and ¯ K , the union of connected components that gives rise to ¯ H isincluded in the one that gives rise to ¯ K : all four of the vertices ( v, s ), ( v ′ , s ′ ),( v, s ′ ) and ( v ′ , s ) are present in ¯ K , while the off-diagonal pair ( v, s ′ ), ( v ′ , s ) mayor may not be present in ¯ H . We now obtain a groupoid operation on every complete I -graph H generated bythe local bijections ρ e : V s → V s ′ for e ∈ E [ s, s ′ ] as induced by the R e . This stepsupports a groupoidal analogue of the passage from coloured graphs to Cayleygroups. 16 s ′ S v •◦ v ′ ◦• VD ⊆ V × S O O O O ; C ⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧⑧ [ c ❄❄❄❄❄❄❄❄❄❄❄❄ Figure 8: e -edges in H × I (double arrows) and ( H ↾ α ) × I (single arrows) for e ∈ E [ s, s ′ ] \ α , provided that ( v, v ′ ) is an e -edge of H . IH • s • s ′ V s V s ′ e / / ρ e ❱❱❱❱ + + ❱❱❱❱ I G • s • s ′ • g ′ = gg e • gG ∗ s G ∗ s ′ e / / g e / / Figure 9: Local view of an I -graph H , and of an I -groupoid G as a complete I -graph; while ρ e may be partial, g e is a full bijection between G ∗ s = S t G ts and G ∗ s ′ = S t G ts ′ . 17 efinition 2.10. An S -groupoid is a structure G = (cid:0) G, ( G st ) s,t ∈ S , · , (1 s ) s ∈ S (cid:1) whose domain G is partitioned into the sets G st , with designated 1 s ∈ G ss for s ∈ S and a partial binary operation · on G , which is precisely defined on theunion of the sets G st × G tu , where it takes values in G su , such that the followingconditions are satisfied:(i) (associativity) for all g ∈ G st , h ∈ G tu , k ∈ G uv : g · ( h · k ) = ( g · h ) · k .(ii) (neutral elements) for all g ∈ G st : g · t = g = 1 s · g .(iii) (inverses) for every g ∈ G st there is some g − ∈ G ts such that g · g − = 1 s and g − · g = 1 t .We are looking to construct S -groupoids as homomorphic images of the free I -structure I ∗ as discussed in Section 2.1.1. For the local view of an I -groupoidcompare the right-hand side of Figure 9. Definition 2.11.
The S -groupoid G is generated by the family ( g e ) e ∈ E if(i) for every e ∈ E [ s, s ′ ], g e ∈ G ss ′ and g e − = ( g e ) − ;(ii) for every s, t ∈ S , every g ∈ G st is represented by a product Q ni =1 g e i , forsome path e . . . e n ∈ E ∗ [ s, t ].An S -groupoid G that is generated by some family ( g e ) e ∈ E for I = ( S, E ) iscalled an I -groupoid . In other words, an I -groupoid is a groupoid that is a homomorphic image ofthe free I -structure I ∗ , under the map G : I ∗ −→ G w = e . . . e n ∈ E ∗ [ s, t ] w G = Q ni =1 g e i ∈ G st . Note that, if I is connected, then an I -groupoid is also connected in the sensethat any two groupoid elements are linked by a path of generators. Otherwise,for disconnected I , an I -groupoid breaks up into connected components thatform separate groupoids, viz., one I ′ -groupoid for each connected component I ′ of I (these are not I -groupoids).For a subset α = α − ⊆ E that is closed under edge reversal we denote by G α the sub-groupoid generated by ( g e ) e ∈ α within G : G α := G ↾ { w G : w ∈ S st α ∗ st } with generators ( g e ) e ∈ α .According to the above, G α may break up into separate and disjoint I β -groupoidsfor the disjoint connected components I β of I α .Recall from Section 2.1.1 how the free I -structure I ∗ induces an operation onan I -graph H if we associate the partial bijections ρ e of H with the generators e ∈ E of I ∗ . The fixpoint-free edge reversal in I induces a converse operation w w − on I ∗ , which corresponds to inversion of partial bijections, ρ w It will often make sense to identify the generator g e with e itself, and we shall often alsospeak of groupoids generated by the family ( e ) e ∈ E . ρ w ) − = ρ w − . But this converse operation I ∗ does not induce groupoidalinverses w.r.t. to the neutral elements 1 s = id V s : for e ∈ E [ s, s ′ ], the domain of( ρ e ) − ◦ ρ e is dom( ρ e ), which may be a proper subset of V s .It is the crucial distinguishing feature of complete I -graphs, cf. Definition 2.5,that we obtain the desired groupoidal inverse. If H is a complete I -graph, then ρ w − ◦ ρ w = ( ρ w ) − ◦ ρ w = id V s for any w ∈ E ∗ [ s, t ], and the image structureobtained in this manner is an S -groupoid G =: cym( H ): ρ : I ∗ −→ G =: cym( H ) , where cym( H ) = G = (cid:0) G, ( G st ) s,t ∈ S , · , (1 s ) s ∈ S (cid:1) ,G st = { ρ w : w ∈ E ∗ [ s, t ] } . The groupoid operation · is the one imposed by the natural composition ofmembers of corresponding sorts: · : S s,t,u G st × G tu −→ G ( ρ w , ρ w ′ ) ∈ G st × G tu ρ w · ρ w ′ := ρ ww ′ ∈ G su . For s ∈ S , the identity 1 s := id V s is the neutral element of sort G ss , inducedas 1 s = ρ λ by the empty word λ s ∈ E ∗ [ s, s ].It is clear from the discussion above that there is a natural groupoidal inverse − : G −→ Gρ w ∈ G st ( ρ w ) − := ρ w − ∈ G ts as ρ w − is the full inverse ( ρ w ) − : V t → V s of the full bijection ρ w : V s → V t . Definition 2.12.
For a complete I -graph H we let cym( H ) be the groupoidabstracted from H according to the above stipulations. We consider cym( H ) asan I -groupoid generated by ( ρ e ) e ∈ E .It is easy to check that cym( H ) is an I -groupoid with generators ( ρ e ) e ∈ E according to Definition 2.11. We turn to the analogue, for I -groupoids, of thenotion of the Cayley graph. Definition 2.13.
Let G = ( G, ( G st ) , · , (1 s )) be an I -groupoid generated by( g e ) e ∈ E . The Cayley graph of G (w.r.t. these generators) is the complete I -graph ( V, ( V s ) , ( R e )) where V = G , V s = G ∗ s := S t G ts , and R e = { ( g, g · e ) : g ∈ V s } for e ∈ E [ s, s ′ ].One checks that this stipulation indeed specifies a complete I -graph, and inparticular that really R e ⊆ V s × V s ′ for e ∈ E [ s, s ′ ]. Compare Figure 9.19 emma 2.14. The I -groupoid induced by the Cayley graph of G is isomorphicto G .Proof. Consider a generator ρ e of the I -groupoid induced by the Cayley graphof G . For e ∈ E [ s, s ′ ] this is the bijection ρ e : V s = G ∗ s −→ V s ′ = G ∗ s ′ g g · g e , so that ρ e operates as right multiplication by generator g e (exactly where de-fined). Since the ( ρ e ) e ∈ E generate the groupoid induced by the Cayley graph of G , it suffices to show that groupoid products of the ρ e (compositions) and thegroupoid products of the g e in G satisfy the same equations, which is obviousfrom the correspondence just established. E.g., if Q i g e i = 1 s in G , then, forthe corresponding w = e . . . e n , we have that ρ w : V s → V s , where V s = G ∗ s ,maps g to g · Q i g e i = g · s = g for all g ∈ V s = G ∗ s , whence ρ w = id V s asdesired.If we identify I -groupoids with their Cayley graphs (which are complete I -graphs), we thus find that the generic process of obtaining I -groupoids fromcomplete I -graphs trivially reproduces the given I -groupoid when applied tosuch. We extend the passage from I -graphs to I -groupoids to the setting of notnecessarily complete I -graphs by combining it with the completion ¯ H of H in H × I . For the following compare Definition 2.7 for the completion ¯ H of an I -graph H and Definition 2.12 for cym( ¯ H ). Definition 2.15.
For a not necessarily complete I -graph H , let the I -groupoidcym( H ) be the I -groupoid cym( ¯ H ) induced by the completion ¯ H of H . Remark 2.16.
The E -graphs of [13] and their role as Cayley graphs of groupsare a special case of I -graphs, also in their roles as Cayley graphs of groupoids. In fact, an E -graph in the sense of [13] is a special I -graph for an incidencepattern of the form I = ( S, E ) where S is a singleton set and E a collection ofloops. An E -graph then is an I -graph in which every R e is a partial matching.It follows that its completion consists of the symmetrisation of R e augmented byreflexive edges at every vertex outside the domain and range of these machings.For the induced I -groupoids abstracted from (complete) I -graphs consisting ofmatchings, this not only means that they are groups rather than groupoids, butalso that they are generated by involutions, as in this case, g e = g e − . Consider two sub-groupoids G α and G β of an I -groupoid G with generators e ∈ E , where α = α − , β = β − ⊆ E are closed under edge reversal. We write G αβ for G α ∩ β and note that α ∩ β is automatically closed under edge reversal.For g ∈ G ∗ s (a vertex of colour s in the Cayley graph) we may think of theconnected component of g in the reduct of the Cayley graph of G to those R e • α i − α i α i +1 g i g i +1 h i Figure 10: Amalgamation: overlap between cosets.with e ∈ α as the G α -coset at g : g G α = { g · w G : w ∈ S t α ∗ st } ⊆ G. If I α is connected, then g G α , as a weak subgraph of (the Cayley graph of) G ,carries the structure of a complete I α -graph. If I α consists of disjoint connectedcomponents, then g G α really produces the coset w.r.t. G α ′ where α ′ ⊆ α is theedge set of the connected component of s in I α . In any case, this I α -graph isisomorphic to the connected component of 1 s in the Cayley graph of G α .Suppose the I α -graph H α and the I β -graph H β are isomorphic to the Cayleygraphs of sub-groupoids G α and G β , respectively. If v ∈ H α and v ∈ H β are vertices of the same colour s ∈ S , then the connected components w.r.t.edge colours in α ∩ β of v in H α and of v in H β are related by a uniqueisomorphism, unique as an isomorphism between the weak subgraphs formedby the ( α ∩ β )-components. We define the amalgam of ( H α , v ) and ( H β , v )(with reference vertices v and v of the same colour s ) to be the result ofidentifying the vertices in these two connected components in accordance withthis unique isomorphism, and keeping everything else disjoint. It is convenientto speak of (the Cayley graphs of) the sub-groupoids G α as the constituents ofsuch amalgams, but we keep in mind that we treat them as abstract I -graphsand not as embedded into G . Just locally, in the connected components of g and g , i.e. in g G αβ ≃ g G αβ , the structure of the amalgam is that of g G αβ ⊆ G in G for any g ∈ V s = G ∗ s ⊆ G .Let, in this sense, ( G α , g ) ⊕ s ( G α , g )stand for the result of the amalgamation of the Cayley graphs of the two sub-groupoids G α i in the vertices g i ∈ V s ⊆ G α i . Note that ( G α , g ) ⊕ s ( G α , g )is generally not a complete I -graph (or I α i -graph for either i ) but satisfies thecompleteness requirement for edges e ∈ α ∩ α .Let ( G α i , g i , h i , s i ) i N be a sequence of sub-groupoids with distinguishedelements and vertex colours as indicated, and such that for all relevant i ( † ) g i ∈ ( G α i ) ∗ s i ⊆ G α i h i ∈ ( G α i ) s i s i +1 ⊆ G α i g i G α i − α i ∩ g i h i G α i α i +1 = ∅ as cosets in G (within g i G α i ).21or the last condition, compare Figure 10: it stipulates that g i cannot belinked to g i +1 by an α i -shortcut that merges the neighbouring α i − - and α i +1 -cosets within the α i -coset that links g i to g i +1 ; intuitively, such a shortcut wouldallow us to eliminate entirely the step involving the α i -coset.If the above conditions are satisfied, then the pairwise amalgams( G α i , g i h i ) ⊕ s i ( G α i +1 , g i +1 )are individually well-defined and, due to the last requirement in ( † ), do notinterfere. Together they produce a connected I -graph H := L Ni =1 ( G α i , g i , h i , s i ) . We call an amalgam produced in this fashion a chain of sub-groupoids G α i of length N .Condition ( † ) is important to ensure that the resulting structure is againan I -graph. Otherwise, an element of the critical intersection g i G α i − α i ∩ g i h i G α i α i +1 could inherit new e -edges from both G α i − and from G α i +1 , for e ∈ ( α i − ∩ α i +1 ) \ α i . Definition 2.17. A coset cycle of length n in an I -groupoid with generator set E is a sequence ( g i ) i ∈ Z n of groupoid elements g i (cyclically indexed) togetherwith a sequence of generator sets (sets of edge colours) α i = α − i ⊆ E such that h i := g − i · g i +1 ∈ G α i and g i G α i α i − ∩ g i +1 G α i α i +1 = ∅ . Definition 2.18. An I -groupoid is N -acyclic if it does not have coset cycles oflength up to N .We now aim for the construction of N -acyclic I -groupoids to be achievedin Proposition 2.22. The following definition of compatibility captures the ideathat some I -groupoid G is at least as discriminating as the I -groupoid cym( H )induced by the I -graph H . Definition 2.19.
For an I -groupoid G and an I -graph H we say that G is compatible with H if, for every s ∈ S and w ∈ E ∗ [ s, s ], w G = 1 s = ⇒ ρ w = id V s = 1 s in cym( H ).The condition of compatibility is such that the natural homomorphisms forthe free I ∗ onto G and onto cym( H ) induce a homomorphism from G ontocym( H ), as in this commuting diagram: I ∗ () G (cid:15) (cid:15) ρ & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ G hom / / cym( H )22ompatibility of G with H also means that G = cym( G ) = cym( G ˙ ∪ H ) –and in this role, compatibility of sub-groupoids G α with certain H will serve asa guarantee for the preservation of these (sub-)groupoids in construction stepsthat render the overall G more discriminating.Note that, by definition, cym( H ) is compatible with H and ¯ H and, byLemma 2.14, with its own Cayley graph. Remark 2.20. If K and H are any I -graphs, then cym( H ˙ ∪ K ) is compatiblewith K , ¯ K and with the Cayley graph of cym( K ) . The following holds the key to avoiding short coset cycles. Note that onlygenerator sets of even sizes are mentioned since we require closure under edgereversal.
Lemma 2.21.
Let G be an I -groupoid with generators e ∈ E , let k, N ∈ N ,and assume that, for every α = α − ⊆ E with | α | < k , the sub-groupoid G α is compatible with chains of groupoids G αβ i up to length N , for any choice ofsubsets β i = β − i ⊆ E . Then there is a finite I -groupoid G ∗ with the samegenerators s.t.(i) for every α = α − ⊆ E with | α | < k , G ∗ α ≃ G α , and(ii) for all α = α − ⊆ E with | α | k , the sub-groupoid G ∗ α is compatiblewith chains G ∗ αβ i up to length N . It will be important later that compatibility of G ∗ α with chains as in (ii) makessure that G ∗ α cannot have cycles of cosets generated by sets α ∩ β i of length upto N : every such cycle in the Cayley group G ∗ α = cym( G ∗ α ) would have to be acycle also in the Cayley group induced by any such chain, including those chainsobtained as linear unfoldings of the proposed cycle (cf. Proposition 2.22 below). Proof of the lemma.
We construct G ∗ as G ∗ := cym( G ˙ ∪ H ) for the I -graph G ˙ ∪ H consisting of the disjoint union of (the Cayley graph of) G and certainchains of sub-groupoids of G .Specifically, we let H be the disjoint union of all amalgamation chains oflength up to N of the form L mi =1 ( G αβ i , g i , h i , s i )for α = α − , β i = β − i ⊆ E , 1 i m N , where | α | k .By construction and Remark 2.20, G ∗ = cym( G ∪ H ) is compatible withchains G αβ i of the required format; together with (i) this implies (ii), i.e., that G ∗ is compatible with corresponding chains of G ∗ αβ i : either the chain in questionhas only components G ∗ αβ with | α ∩ β | < k so that, by (i), G ∗ αβ ≃ G αβ ; or thereis some component G ∗ αβ with | α ∩ β | = 2 k , which implies that α = β ∩ α , andby ( † ) (page 21) the merged chain is isomorphic to G ∗ α , thus trivialising thecompatibility claim.For (i), it suffices to show that, for | α ′ | < k , G α ′ is compatible with eachconnected component of H . (That G ∗ is compatible with G α ′ is clear since G α ′
23s itself a component of H and hence of G ∪ H ; compatibility of G ∗ with G isobvious for the same reason.)Consider then a component of the form L mi =1 ( G αβ i , g i , h i , s i ). The α ′ -reducts of α ′ -connected components of its completion arise as substructures ofthe completions of merged chains of components of the form G α ′ αβ , accordingto Lemma 2.9. Since | α ′ | < k , the assumptions of the lemma imply compat-ibility of G α ′ with any such component. It follows that G ∗ = cym( G ∪ H ) iscompatible with all G α ′ for | α ′ | < k , and thus G ∗ α ′ ≃ G α ′ for | α ′ | < k . Proposition 2.22.
For every incidence pattern I = ( S, E ) and N ∈ N thereare finite N -acyclic I -groupoids with generators e ∈ E . Moreover, such an I -groupoid can be chosen to be compatible with any given I -graph H .Proof. Start from an arbitrary finite I -groupoid G , or with G := cym( H )in order to enforce compatibility with a given I -graph H . Then inductivelyapply Lemma 2.21 and note that the assumptions of the lemma are trivial for k = 1, because the trivial sub-groupoid generated by ∅ , which just consists ofthe isolated neutral elements 1 s , is compatible with any I -graph. In each stepas stated in the lemma, compatibility with corresponding chains implies that G ∗ cannot have coset cycles of length up to N with cosets generated by setsof the form αβ i were | α | k . For 2 k = | E | , this rules out all coset cycles oflength up to N . Observation 2.23.
For any -acyclic I -groupoid G and any subsets α = α − , β = β − ⊆ E , with associated sub-groupoids G α , G β and G αβ : G α ∩ G β = G αβ . Proof.
Just the inclusion G α ∩ G β ⊆ G αβ needs attention. Let h ∈ G α ∩ G β ,i.e., h = w G = ( w ′ ) G for some w ∈ α ∗ st and w ′ ∈ β ∗ st . Let g ∈ G ∗ s and put g := g · h ∈ G ∗ t . Then g , g with h = g − · g = h ∈ G α and h = g − · g = h − ∈ G β , form a coset 2-cycle with generator sets α := α , α := β , unless thecoset condition g G αβ ∩ g G αβ = ∅ of Definition 2.17 is violated. So there must be some k ∈ g G αβ ∩ g G αβ , whichshows that h = ( g − · k ) · ( g − · k ) − ∈ G αβ as claimed. A hypergraph is a structure A = ( A, S ) where S ⊆ P ( A ) is called the set ofhyperedges of A , A the set of vertices of A . Definition 3.1.
With a hypergraph A = ( A, S ) we associate(i) its
Gaifman graph G ( A ) = ( A, G ( S )) where G ( S ) is the simple undirectededge relation that links a = a ′ in A if a, a ′ ∈ s for some s ∈ S .24ii) its intersection graph I ( A ) = ( S, E ) where E = { ( s, s ′ ) : s = s ′ , s ∩ s ′ = ∅} .Note that the intersection graph I ( A ) captures the overlap pattern of thehyperedges of A . If we regard I ( A ) as an incidence pattern, as we wish to do,its important special properties compared to general incidence patterns are thefollowing: I ( A ) is loop-free and a graph rather than a multi-graph, i.e., each E [ s, s ′ ] = { ( s, s ′ ) } ⊆ E is a singleton set.The hypergraph A itself may be regarded as an I -graph H ( A ) for I = I ( A ).To this end we represent it by the disjoint union of its hyperedges together withthe identifications ρ e induced by the overlaps s ∩ s ′ for e = ( s, s ′ ) ∈ E . SeeSection 3.5 for the precise definition of H ( A ). The important special propertiesof the I -graph representation of A , in comparison to the general notion of I -graphs, are its coherence and a strong transitivity property: if e = ( s, s ′ ) ∈ E [ s, s ′ ] , e ′ = ( s ′ , s ′′ ) ∈ E [ s ′ , s ′′ ] are such that ρ e ′ ◦ ρ e = ∅ , then ρ e ′ ◦ ρ e = ρ e ′′ for e ′′ = ( s, s ′′ ) ∈ E [ s, s ′′ ]. Thus a hypergraph embodies prototypical instances ofincidence patterns, of I -graphs as overlap specifications, and of realisations ofthese overlap specifications (it comes as its own trivial realisation in the senseof Definition 3.3 below).The following criterion of hypergraph acyclicity is the natural and strongestnotion of acyclicity (sometimes called α -acyclicity), cf., e.g., [3, 2]. It is in closecorrespondence with the algorithmically crucial notion of tree-decomposability(viz., existence of a tree-decomposition with hyperedges as bags, cf. discussionafter Proposition 4.14 below) and with natural combinatorial notions of trian-gulation. Definition 3.2.
A finite hypergraph A = ( A, S ) is acyclic if it is conformal and chordal where(i) conformality requires that every clique in the Gaifman graph G ( A ) iscontained in some hyperedge s ∈ S ;(ii) chordality requires that every cycle in the Gaifman graph G ( A ) of lengthgreater than 3 has a chord.For N > A = ( A, S ) is N -acyclic if it is N -conformal and N -chordal where(iii) N -conformality requires that every clique in the Gaifman graph G ( A ) ofsize up to N is contained in some hyperedge s ∈ S ;(iv) N -chordality requires that every cycle in the Gaifman graph G ( A ) oflength greater than 3 and up to N has a chord. N -acyclicity is a natural gradation or quantitative restriction of hypergraphacyclicity, in light of the following. Consider the induced sub-hypergraphs A ↾ A of a hypergraph A = ( A, S ), i.e., the hypergraphs on vertex sets A ⊆ A withhyperedge sets S ↾ A := { s ∩ A : s ∈ S } . Then A is N -acyclic if, and only if,every induced sub-hypergraph A ↾ A for A ⊆ A of size up to N is acyclic. The general case of an arbitrary I -graph over an arbitrary incidence pattern I seems to be a vast abstraction from the special case of an overlap pattern induced25y the actual overlaps between hyperedges in an actual hypergraph. The notionof a realisation concerns this gap and formulates the natural conditions for ahypergraph to realise an abstract overlap specification; for the intuitive ideaof an overlap specification and its realisation in an I -graph see Section 1 andFigures 1 and 2. Definition 3.3.
Let I = ( S, E ) be an incidence pattern, H = ( V, ( V s ) , ( R e )) an I -graph with induced partial bijections ρ w between V s and V t for w ∈ E ∗ [ s, t ].A hypergraph ˆ A = ( ˆ A, ˆ S ) is a realisation of the overlap pattern specified by H if there is a map π : ˆ S → S and a matching family of bijections π ˆ s : ˆ s −→ V s , for ˆ s ∈ ˆ S with π (ˆ s ) = s, such that for all ˆ s, ˆ t ∈ ˆ S s.t. π (ˆ s ) = s , π (ˆ t ) = t , and for every e ∈ E [ s, s ′ ]:(i) there is some ˆ s ′ such that π (ˆ s ′ ) = s ′ and π ˆ s ′ ◦ π − s = ρ e ;(ii) if ˆ s ∩ ˆ t = ∅ , then π ˆ t ◦ π − s = ρ w for some w ∈ E ∗ [ s, t ].ˆ s ∩ ˆ s ′ π ˆ s | | ②②②②②②②②② π ˆ s ′ " " ❋❋❋❋❋❋❋❋❋ V s ρ e / / V s ′ ˆ s ∩ ˆ t π ˆ s } } ④④④④④④④④④ π ˆ t ! ! ❈❈❈❈❈❈❈❈❈ V s ρ w / / V t Some comment on the definition: condition (i) says that all those local over-laps that should be realised according to H are indeed realised at correspondingsites in ˆ A ; condition (ii) says that all overlaps between hyperedges realised in ˆ A are induced by overlaps specified in H in a rather strict sense. In Section 3.5below we shall look at a simpler concept of a covering of a given hypergraph A . Realisations of the overlap pattern H ( A ) abstracted from the given A willbe seen to be special coverings. In this sense the notion of a realisation of anabstract overlap pattern (as specified by an I -graph) extends certain more basicnotions of hypergraph coverings, in which the overlap pattern is specified by aconcrete realisation. Realisations and partial unfoldings.
Regarding an I -graph H as a spec-ification of an overlap pattern to be realised, it makes sense to modify H inmanners that preserve the essence of that overlap specification. A natural ideaof this kind would be to pass to a partial unfolding of H , which preserves the lo-cal links. We use the notion of a covering at the level of the underlying incidencepattern for this purpose.Let ˜ I = ( ˜ S, ˜ E ) and I = ( S, E ) be incidence patterns. A homomorphism from˜ I to I is a map π : ˜ I → I respecting the (two-sorted) multi-graph structure sothat, for ˜ e ∈ ˜ E [˜ s, ˜ s ′ ], π (˜ e ) ∈ E [ π (˜ s ) , π (˜ s ′ )] , as well as the fixpoint-free involutive operations of edge reversal: π (˜ e − ) = ( π (˜ e )) − . efinition 3.4. A surjective homomorphism π : ˜ I → I between incidence pat-terns ˜ I = ( ˜ S, ˜ E ) and I = ( S, E ) is a covering of incidence patterns if it satisfiesthe following lifting property (known as the back -property in back&forth rela-tionships like bisimulation equivalence):( back ): (cid:26) for all s ∈ S, e ∈ E [ s, s ′ ] and ˜ s ∈ π − ( s ) , there exists ˜ s ′ and ˜ e ∈ ˜ E [˜ s, ˜ s ′ ] s.t. π (˜ e ) = e. In the situation of the definition, an I -graph H = (cid:0) V, ( V s ) s ∈ S , ( R e ) e ∈ E (cid:1) induces an ˜ I -graph ˜ H = (cid:0) ˜ V , ( V ˜ s ) ˜ s ∈ ˜ S , ( R ˜ e ) ˜ e ∈ ˜ E (cid:1) on a subset ˜ V of V × ˜ S , where V ˜ s := V π (˜ s ) × { ˜ s } ⊆ ˜ V := S ˜ s ∈ ˜ S V ˜ s ,R ˜ e := (cid:8) (( v, ˜ s ) , ( v ′ , ˜ s ′ )) : ˜ e ∈ ˜ E [˜ s, ˜ s ′ ] , ( v, v ′ ) ∈ R e for e = π (˜ e ) (cid:9) . Lemma 3.5.
Suppose π : ˜ I → I is a covering of the incidence pattern I , ˜ H the ˜ I -graph induced by the I -graph H . Then every realisation of the overlap patternspecified by ˜ H induces a realisation of the overlap pattern specified by H .Proof. Let ˆ A = ( ˆ A, ˆ S ) be a realisation of ˜ H = ( ˜ V , ( V ˜ s ) ˜ s ∈ ˜ S , ( R ˜ e ) ˜ e ∈ ˜ E ) with associ-ated projections ˜ π : ˆ S → ˜ S and ˜ π ˆ s : ˆ s → V ˜ π (ˆ s ) . Combining these projections with π : ˜ S → S and the trivial projection π that maps V ˜ s = V π (˜ s ) ×{ ˜ s } to V π (˜ s ) , weobtain projections π := π ◦ ˜ π : ˆ S → S and π ˆ s := π ◦ ˜ π ˆ s : ˆ s → π ( V ˜ π (ˆ s ) ) = V π (ˆ s ) ,which allow us to regard ˆ A as a realisation of H . Towards the defining condi-tions on realisations, (i) is guaranteed by the back -property for π , while (ii)follows form the homomorphism condition for π and the definition of ˜ H . Inparticular, dom( ρ ˜ w ) = dom( ρ w ) × { ˜ s } for any ˜ w ∈ ˜ E ∗ [˜ s, ˜ t ] with projection w = π ( ˜ w ) ∈ E ∗ [ s, t ] where s = π (˜ s ) and t = π (˜ t ). Direct products.
We define a natural direct product H × G of an I -graph H = ( V, ( V s ) s ∈ S , ( R e ) e ∈ E ) with an I -groupoid G . The construction may beviewed as a special case of a more general, natural notion of a direct productbetween two I -graphs. In geometrical-combinatorial terms we are interested in H × G because it plays the role of a finite unfolding or covering of H , at leastif G satisfies the compatibility condition in the sense of Definition 2.19.For an I -graph H and I -groupoid G we define the direct product H × G to be the following I -graph on the disjoint union of the products V s × G ∗ s , cf.Figure 11: H × G := (cid:16)S s ∈ S ( V s × G ∗ s ) , (cid:0) V s × G ∗ s (cid:1) s ∈ S , ( R e ) e ∈ E (cid:17) where R e = (cid:8) (( v, g ) , ( ρ e ( v ) , gg e )) : v ∈ V s , g ∈ G ∗ s (cid:9) for e ∈ E [ s, s ′ ].Just like H , this direct product admits an operation of the free I -structurein terms of compositions of partial bijections. These are based on the natural27ifting of e to ρ e (in H ) and further to ρ H × G e (in H × G ) according to ρ H × G e : V s × G ∗ s −→ V s ′ × G ∗ s ′ ( v, g ) ( ρ e ( v ) , gg e ) . This extends to paths w ∈ E ∗ [ s, t ], which are lifted to ρ w and further to ρ H × G w ,cf. Figure 11. Observation 3.6. If G is compatible with H , then the liftings along differentpaths w , w ∈ E ∗ [ s, t ] linking the same groupoid elements agree in their commondomains in H × G : for every g ∈ G ∗ s and all v ∈ dom( ρ w ) ∩ dom( ρ w ) ⊆ V s , w G = w G ⇒ ρ w ( v, g ) = ρ w ( v, g ) in H × G .Proof. Wherever the composition ρ H × G w is defined, it agrees in the first com-ponent with the operation of ρ w on H and on the completion ¯ H , which givesrise to cym( H ). Therefore w G = w G , or equivalently ( w − w ) G = 1 s , impliesby compatibility that ρ w − w = id V s in cym( H ), whence ρ w ( v ) = ρ w ( v ) for v ∈ dom( ρ w ) ∩ dom( ρ w ).In other words, compatibility of G with H guarantees that any w ∈ E ∗ [ s, s ]such that w G = 1 s (i.e., any cycle in G ) induces a partial bijection ρ w : V s × G ∗ s → V s × G ∗ s that is compatible with the identity of V s × G ∗ s : w G = 1 s = ⇒ ρ H × G w ⊆ id V s × G ∗ s . The path-independence expressed in the observation is, however, character-istically weaker than coherence of H × G as an I -graph, because we only comparepaths that link the same groupoid elements. But H × G also carries the struc-ture of an I ( G )-graph, where I ( G ) is the natural incidence pattern associatedwith the Cayley graph of G I ( G ) = ( G, ˜ E ) where ˜ E = S e ∈ E (cid:8) ( g, gg e ) : e ∈ E [ s, s ′ ] , g ∈ G ∗ s (cid:9) . The relationship between I ( G ) and I is that of a covering as in Definition 3.4w.r.t. to the natural projection that maps G ∗ s ⊆ G to s ∈ S . Moreover, H × G isthe I ( G )-graph induced by H in the sense of Lemma 3.5. That lemma thereforetells us that any realisation of H × G will provide a realisation of H . Thisin turn yields an interesting reduction of the general realisation problem tothe realisation problem for coherent I -graphs, since the path-independence ofObservation 3.6 precisely states that H × G is coherent as an I ( G )-graph in thesense of Definition 2.4. This route to realisations is pursued in Proposition 3.12. Observation 3.7. If G is compatible with H , then the direct product H × G is coherent when viewed as an I ( G ) -graph, and every realisation of H × G thusinduces a realisation of H . educed products. The reduced product H ⊗ G between an I -graph andan I -groupoid is simply obtained as the natural quotient of the direct product H × G w.r.t. the equivalence relation ≈ induced by( v, g ) ≈ ( ρ e ( v ) , gg e ) for e ∈ E [ s, s ′ ] , v ∈ V s , g ∈ G ∗ s . Note that, by transitivity, for arbitrary ( v, g ) , ( v ′ , g ′ ) ∈ H × G ,( v, g ) ≈ ( v ′ , g ′ ) iff v ′ = ρ w ( v ) for some w ∈ E ∗ [ s, t ] with g ′ = g · w G . We denote equivalence classes w.r.t. ≈ by square brackets, as in [ v, g ] := { ( v ′ , g ′ ) : ( v ′ , g ′ ) ≈ ( v, g ) } and extend this notation naturally to sets as in[ V s , g ] := [ V s × { g } ] = { [ v, g ] : v ∈ V s } . Definition 3.8.
Let H = ( V, ( V s ) s ∈ S , ( R e ) e ∈ E ) be an I -graph, G an I -groupoid.The reduced product H ⊗ G is defined to be the hypergraph H ⊗ G := ˆ A = ( ˆ A, ˆ S )with vertex set ˆ A = (cid:8) [ v, g ] : ( v, g ) ∈ S s V s × G ∗ s (cid:9) and set of hyperedges ˆ S = (cid:8) [ V s , g ] : s ∈ S, g ∈ G ∗ s (cid:9) , where square brackets denote passage to equivalence classes w.r.t. ≈ as indicatedabove.Note that the hyperedges [ V s , g ] of H ⊗ G are induced by the patches V s of H . As already indicated in Observation 3.6 above, these hyperedges turn outto be bijectively related to those patches in cases of interest. This takes us onestep towards a realisation of the overlap pattern specified by H . Lemma 3.9. If G is compatible with the I -graph H = ( V, ( V s ) , ( E s )) , then thenatural projection π s,g : [ V s , g ] −→ V s [ v, g ] v is well-defined in restriction to each hyperedge [ V s , g ] of H ⊗ G , and relates eachhyperedge [ V s , g ] = { [ v, g ] : v ∈ V s } for g ∈ G ∗ s bijectively to V s .Proof. It suffices to show that [ v, g ] = [ v ′ , g ] implies v = v ′ , which shows that π s,g is well-defined. By compatibility of G with H , w G = 1 implies ρ w ⊆ id V s ,for any w ∈ E ∗ [ s, s ]. (For the last step compare Observation 3.6 about path-independence.)An even higher degree of path-independent transport in H × G is achievedif H itself is a coherent I -graph in the sense of Definition 2.4 and if G is at least2-acyclic in the sense of Definition 2.18.29 s • s ′ V s V s ′ e / / ρ e ❱❱❱❱ + + ❱❱❱❱ • g • gg e V s × { g } V s ′ × { gg e } e / / ρ e ❱❱❱❱ + + ❱❱❱❱ • gG ∗ s • g ′ G ∗ t V s × { g } V t × { g ′ } w * * w ρ w + + ρ w E ∗ [ s,t ] / / Figure 11: Fibres in an I -graph and in its product with an I -groupoid.Recall from the discussion after Definition 2.4 that coherence implies the ex-istence of a unique and well-defined partial bijection ρ st between those elementsof V s and V t that can be linked by any ρ w for w ∈ E ∗ [ s, t ], viz., ρ st = [ { ρ w : w ∈ E ∗ [ s, t ] } . Observation 3.10. If H is coherent and G is compatible with H and -acyclic,then there is, for any g ∈ G ∗ s and g ′ ∈ G ∗ t a unique maximal subset of V s among all subsets dom( ρ w ) ⊆ V s for those w ∈ E ∗ [ s, t ] with g ′ = g · w G . Hence, in the reduced product H ⊗ G , the full intersection between hyperedges [ V s , g ] and [ V t , g ′ ] is realised by the identification via ρ w for a single path w ∈ E ∗ [ s, t ] such that g ′ = g · w G . The second formulation is the key to the importance of this observationtowards the construction of realisations.The observation could also be phrased in terms of the direct product asfollows. For any fixed site V s × { g } in H × G , the maximal overlap of V s × { g } with any other site V t × { g ′ } via some composition of partial bijections ρ H × G e is well-defined. In the reduced product H ⊗ G , this maximal overlap representsthe full intersection between the hyperedges [ V s , g ] and [ V t , g ′ ]. Proof of the observation.
It suffices to show that, if w , w ∈ E ∗ [ s, t ] are suchthat w G = w G , then dom( ρ w ) ∪ dom( ρ w ) ⊆ dom( ρ w ) for some suitable choiceof w ∈ E ∗ [ s, t ] for which also w G = w G i .By coherence of H , the image of any v ∈ dom( ρ w i ) ⊆ V s under any appli-cable composition of partial bijections ρ e is globally well-defined, so that the30oint-wise image of the sets dom( ρ w i ) ⊆ V s at any V u can be addressed as ρ su (dom( ρ w i )) ⊆ V u . As ρ w i maps every element of dom( ρ w i ) along the path w i , this path can only involve generators from α i = [ u,u ′ ∈ S (cid:8) e ∈ E [ u, u ′ ] : ρ su (dom( ρ w i )) ⊆ dom( ρ e ) (cid:9) , for i = 1 ,
2. So w G i ∈ G α i . By 2-acyclicity of G , g := 1 s and g := w G = w G does not form a coset cycle w.r.t. the generator sets α i . As ( g ) − g = w G = w G ∈ G α and ( g ) − g = ( w G ) − ∈ G α , it follows that the coset conditionmust be violated. So there must be some w ∈ ( α ∩ α ) ∗ with w G = w G = w G . But α ∩ α = [ u,u ′ ∈ S (cid:8) e ∈ E [ u, u ′ ] : ρ su (dom( ρ w ) ∪ dom( ρ w )) ⊆ dom( ρ e ) (cid:9) , so that w ∈ ( α ∩ α ) ∗ implies that dom( ρ w i ) ⊆ dom( ρ w )) for i = 1 , Corollary 3.11. If G is compatible with the coherent I -graph H = ( V, ( V s ) , ( E s )) and -acyclic, then the reduced product H ⊗ G with its natural projections is arealisation of the overlap pattern specified in H .Proof. Compatibility of G with H guarantees that the natural projections π ˆ s : [ V s , g ] −→ V s [ v, g ] v are well-defined in restriction to each hyperedge [ V s , g ] of H ⊗ G , and map thishyperedge bijectively onto V s , by Lemma 3.9. For condition (i) in Definition 3.3,it is clear by construction of H ⊗ G that for e ∈ E ∗ [ s, s ′ ] and g ∈ G ∗ s , ˆ s =[ V s , g ] overlaps with ˆ s ′ := [ V s ′ , gg e ] according to ρ e ; this overlap cannot bestrictly larger than | ρ e | , as g e is not in the sub-groupoid generated by E \{ e, e − } , due to 2-acyclicity of G . Condition (ii) of Definition 3.3 is settled byObservation 3.10. Combining the constructions of direct and reduced products with the existenceof suitable groupoids we are ready to prove the first major step towards themain theorem on realisations: the existence of realisations for overlap patternsspecified by arbitrary I -graphs. We shall then see in Section 3.6 below thatthe degree of acyclicity in realisations can be boosted to any desired level N through passage to N -acyclic coverings. That will then take us one step closer, inTheorem 3.22, to the full statement of the main theorem from the introduction.For the full content of the main theorem as stated there, however, compatibilitywith symmetries will have to wait until Section 4, see Corollary 4.10.31 roposition 3.12. For every incidence pattern I and I -graph H , there is afinite hypergraph ˆ A that realises the overlap pattern specified by H .Proof. From a given I -graph H we first obtain its product H × G with an I -groupoid G that is compatible with H (see Proposition 2.22 for existence).Regarding H × G as a coherent ˜ I -graph for ˜( I ) = I ( G ) (cf. Observation 3.7),we obtain a realisation of that ˜ I -graph H × G by a reduced product with a 2-acyclic ˜ I -groupoid that is compatible with H × G , according to Corollary 3.11.As ˜ I = I ( G ) is a covering of I in the sense of Definition 3.4 and H × G the˜ I -graph induced by H , Lemma 3.5 guarantees that the resulting hypergraph isindeed a realisation of H .We remark that the approach to realisations as presented above is differentfrom the one outlined in [14, 15]. That construction first produces some kindof ‘pre-realisations’ H ⊗ G , which satisfy condition (i) for realisations but havemore identifications than allowed by condition (ii). These pre-realisations canthen be modified in a second unfolding step w.r.t. a derived incidence patternto set condition (ii) right. The present approach seems more natural in that itputs realisations rather than coverings and unfoldings at the centre. A hypergraph homomorphism is a map h : A → B between hypergraphs A =( A, S ) and B such that, for every s ∈ S , h ↾ s is a bijection between the hyperedge s and some hyperedge h ( s ) of B . Definition 3.13.
A hypergraph homomorphism h : ˆ A → A between the hyper-graphs ˆ A = ( ˆ A, ˆ S ) and A = ( A, S ) is a hypergraph covering (of A by ˆ A ) if itsatisfies the back -property w.r.t. hyperedges: for every ˆ s ∈ ˆ S , s = h (ˆ s ) ∈ S and s ′ ∈ S there is some ˆ s ′ ∈ ˆ S such that h (ˆ s ′ ) = s ′ and h (ˆ s ∩ ˆ s ′ ) = s ∩ s ′ .As mentioned above, a hypergraph A = ( A, S ) directly translates into anequivalent representation as an I -graph H ( A ), where the intersection graph I = I ( A ) plays the role of the incidence pattern I . Explicitly, H ( A ) = (cid:0) V, ( V s ) s ∈ S , ( R e ) e ∈ E (cid:1) where E = { ( s, s ′ ) : s = s ′ , s ∩ s ′ = ∅} is the edge relation of the intersectiongraph, V is the disjoint union of the hyperedges s ∈ S , V = [ s ∈ S V s where V s = s × { s } , naturally partitioned into the ( V s ) s ∈ S , and with the R e (or ρ e ) that identifyoverlaps in intersections: R e = (cid:8) (( v, s ) , ( v, s ′ )) : v ∈ s ∩ s ′ (cid:9) for e = ( s, s ′ ) ∈ E. We recall from the discussion above that H ( A ) is coherent.32ny realisation ˆ A = ( ˆ A, ˆ S ) of H ( A ) then is a hypergraph covering for A ,albeit one that avoids certain redundancies w.r.t. intersections. Cf. property (ii)in Definition 3.3 for realisations for the following rendering of this condition forcoverings, which we want to call strict coverings. Definition 3.14.
A hypergraph covering h : ˆ A → A is strict if every intersectionˆ s ∩ ˆ t between hyperedges of ˆ A is induced by a sequence of intersections in A inthe sense that h (ˆ s ∩ ˆ t ) × { ˆ s } = dom( ρ w ) in H ( A ), for some path w ∈ E ∗ [ s, t ]from s = h (ˆ s ) to t = h (ˆ t ) in I ( A ). Observation 3.15.
Every realisation of H ( A ) is a (strict) hypergraph coveringw.r.t. the natural projection induced by the projections of the realisation.Proof. Let ˆ A = ( ˆ A, ˆ S ) with projections π : ˆ S → S and π ˆ s : ˆ s → V π (ˆ s ) be arealisation of the overlap pattern specified by H = H ( A ). Writing π for theprojection to the first component in V = S s ∈ S ( s × { s } ), we obtain π : ˆ A → A as the compositions π := S ˆ s π ◦ π ˆ s .π is well-defined since condition (ii) for realisations makes sure that overlaps inˆ A are induced by compositions ρ w in H , which, in H = H ( A ), are trivial incomposition with π : π ◦ ρ w ⊆ id A for any w ∈ E ∗ . It is easy to check that π : ˆ A → A is a hypergraph homomorphism (as π ˆ s bijectively maps ˆ s ∈ ˆ S onto V ˆ s = π (ˆ s ) ×{ π (ˆ s ) } ) and satisfies the back -property(by condition (i) on realisations).As H ( A ) is coherent, every reduced product H ( A ) ⊗ G of H ( A ) with a 2-acyclic groupoid G that is compatible with H ( A ), according to Corollary 3.11provides a realisation, and hence a covering. The resulting reduced product canalso be cast more directly as a natural reduced product with A itself, whichoffers a more intuitive view, and puts fewer constraints on G . The followingessentially unfolds and combines the definitions of H ( A ) (cf. discussion belowDefinition 3.13) and H ⊗ G (cf. Definition 3.8). Definition 3.16.
Let A = ( A, S ) be a hypergraph, G an I -groupoid for I = I ( A ). The reduced product A ⊗ G is the hypergraph ˆ A = ( ˆ A, ˆ S ) whose vertexset ˆ A is the quotient of the disjoint union of G ∗ s -tagged copies of all s ∈ S ,ˆ A := (cid:0)S s ∈ S,g ∈ G ∗ s s × { g } (cid:1) (cid:14) ≈ w.r.t. the equivalence relation induced by identifications( a, g ) ≈ ( a, ge ) for e = ( s, s ′ ) ∈ E a := { ( s, s ′ ) ∈ E : a ∈ s ∩ s ′ } . Natural though the general notion of a hypergraph covering may be, it does not rule out,e.g., partial overlaps between different covers ˆ s, ˆ s ′ of the same s ; the example indicated inFigure 4 shows that some such branching behaviour can be unavoidable. Note that the map ρ w in H ( A ) represents a composition of identities in intersections interms of A itself. a, g ) as [ a, g ], and lifting this notation to setsin the usual manner, the set of hyperedges of A ⊗ G isˆ S := { [ s, g ] : s ∈ S, g ∈ G ∗ s } where [ s, g ] := { [ a, g ] : a ∈ s } ⊆ ˆ A. The covering homomorphism π is the natural projection π : [ a, g ] a .We note that ( a, g ) is identified with ( a, g ′ ) in this quotient if, and only if, g ′ = g · w G for some path w ∈ E ∗ [ s, t ] such that ( a, s ) ∈ dom( ρ w ) (and hence ρ w ( a, s ) = ( a, t )) in H = H ( A ). We may think of the generators e = ( s, s ′ ) ∈ E a as preserving the vertex a in passage from a ∈ s to a ∈ s ′ , so that the g -taggedcopy of s and the gg e -tagged copy of s ′ are glued in their overlap s ∩ s ′ .It is easy to see directly that π : ˆ A → A is indeed a hypergraph covering. Proposition 3.17.
For any hypergraph A and I -groupoid G , where I = I ( A ) ,the reduced product A ⊗ G with the natural projection π : A ⊗ G −→ A [ a, g ] a is a hypergraph covering. If G is -acyclic, then this covering is strict. One also checks that, indeed, A ⊗ G ≃ H ( A ) ⊗ G .Another useful link between realisations and coverings is the following. Lemma 3.18. If π : ˆ A ′ → ˆ A is a strict hypergraph covering of a realisation ˆ A of the overlap pattern specified by the I -graph H , then so is ˆ A ′ , w.r.t. theprojections induced by the natural compositions of those of the realisation ˆ A with π .Proof. Let I = ( S, E ), H = ( V, ( V s ) , ( R e )), ˆ A = ( ˆ A, ˆ S ) covered by ˆ A ′ = ( ˆ A ′ , ˆ S ′ )through π , and let π : ˆ S → S and π ˆ s : ˆ s → V π (ˆ s ) be the projections throughwhich ˆ A realises H . Then π ′ := π ◦ π : ˆ S ′ → S together with π ˆ s ′ := π π (ˆ s ′ ) ◦ π : ˆ s ′ → V π ′ (ˆ s ′ ) serve as the projections required in the realisation of H by ˆ A ′ . We show that the absence of short coset cycles in G , i.e., N -acyclicity in thesense of Definition 2.18, implies corresponding degrees of hypergraph acyclicityin the sense of Definition 3.2 in reduced products A ⊗ G . Remark:
The analysis of cycles and cliques in the following two sections couldbe carried out in the slightly more general setting of realisations of coherent I -graphs in reduced products, rather than the setting of coverings by reducedproducts. We choose the latter for the sake of greater transparency. The differ-ence is just that one would have to work with coherent translations of overlap34egions ρ st ( V s ) ⊆ V t in H ⊗ G instead of the much more intuitive use of liftedpre-images of the actual intersections between hyperedges in A – this preciselyis the advantage of having one realisation of H ( A ) already, albeit the trivial oneby A itself. Chordality in coverings by reduced productsLemma 3.19.
Let A = ( A, S ) be a hypergraph, G an N -acyclic I -groupoid for I := I ( A ) , the intersection graph of A . Then A ⊗ G is N -chordal.Proof. Suppose that ([ a i , g i ]) i ∈ Z n is a chordless cycle in the Gaifman graph of A ⊗ G . W.l.o.g. the representatives ( a i , g i ) are chosen such that, for suitable s i ∈ S , [ s i +1 , g i +1 ] is a hyperedge linking [ a i , g i ] and [ a i +1 , g i +1 ] in A ⊗ G . I.e.,there is a path w from s i to s i +1 in I consisting of edges from α i := E a i = (cid:8) ( s, s ′ ) ∈ E : a i ∈ s ∩ s ′ (cid:9) such that g i +1 = g i · w G . In particular, g − i g i +1 ∈ G α i . We claim that ( g i ) i ∈ Z n is a coset cycle w.r.t. the generator sets ( α i ) i ∈ Z n , in the sense of Definition 2.17.If so, n > N follows, since G is N -acyclic.In connection with ( g i ) i ∈ Z n and ( α i ) i ∈ Z n it essentially just remains to checkthe coset condition g i G α i α i − ∩ g i +1 G α i α i +1 = ∅ . Suppose, for contradiction, that there is some k ∈ g i G α i α i − ∩ g i +1 G α i α i +1 ,and let t ∈ S be such that k ∈ G ∗ t . We show that this situation implies that[ a i − , g i − ] and [ a i +1 , g i +1 ] are linked by a chord induced by the hyperedge [ k, t ]:(a) Since k ∈ g i G α i α i − , there is some path w from s i to t consisting ofedges in α i ∩ α i − such that k = g i · w G ; as there also is a path w from s i − to s i in I consisting of edges from α i − such that g i = g i − · w G , it follows that thereis a path w from s i − to t consisting of edges in α i − such that k = g i − · w G .So [ a i − , g i − ] ∈ [ t, k ].(b) Since k ∈ g i +1 G α i α i +1 , there is some path w from s i +1 to t consistingof edges in α i ∩ α i +1 ⊆ α i +1 such that k = g i +1 · w G ; so [ a i +1 , g i +1 ] ∈ [ t, k ].(a) and (b) together imply that the given cycle was not chordless after all. Conformality in coverings by reduced productsLemma 3.20.
Let A = ( A, S ) be a hypergraph, G an N -acyclic I -groupoid for I := I ( A ) , the intersection graph of A . Then A ⊗ G is N -conformal.Proof. Suppose that X := { [ a i , g i ] : i ∈ n } is a clique of the Gaifman graph of A ⊗ G that is not contained in a hyperedge of A ⊗ G , but such that every subsetof n − A ⊗ G . For i ∈ n , choose ahyperedge [ t i , k i ] such that X i := { [ a j , g j ] : j = i } ⊆ [ t i , k i ]. Let h i := k − i k i +1 and α i := T j = i,i +1 E a j . Note that k − i k i +1 ∈ G α i . We claim that ( k i ) i ∈ Z n withgenerator sets ( α i ) i ∈ Z n forms a coset cycle in G in the sense of Definition 2.17.It follows that n > N , as desired. 35uppose, for contradiction, that k ∈ k i G α i α i − ∩ k i +1 G α i α i +1 for some i . Let t ∈ S be such that k ∈ G ∗ t . We show that X ⊆ [ t, k ] would follow.Since k ∈ k i G α i α i − , and [ a j , g j ] ∈ [ t i , k i ], i.e., [ a j , g j ] = [ a j , k i ], for all j = i ,clearly [ a j , g j ] ∈ [ t, k ] for j = i (note that α i ∩ α i − = T j = i E a j ).It therefore remains to argue that also [ a i , g i ] ∈ [ t, k ]. Note that k ∈ k i +1 G α i α i +1 and that α i ∩ α i +1 = T j = i +1 E a j . In particular, generators in α i ∩ α i +1 preserve a i . Since [ a i , g i ] ∈ [ t i +1 , k i +1 ], we have that [ a i , g i ] = [ a i , k i +1 ],and thus [ a i , g i ] ∈ [ t, k ] follows from the fact that k − i +1 · k ∈ G α i α i +1 .Combining the above, we obtain the following by application of the reducedproduct construction A ⊗ G for suitably acyclic groupoids G , which are availableaccording to Proposition 2.22. Proposition 3.21.
For every N ∈ N , every finite hypergraph admits a strictcovering by a finite hypergraph of the from A ⊗ G that is N -acyclic. As any strict hypergraph covering of any given realisation of an I -graph H induces another realisation of H by Lemma 3.18, realisations can be boosted toany desired degree of acyclicity. We have thus proved the main theorem almostas stated in the introduction, viz., up to the analysis of the global symmetrybehaviour to which the next section will be devoted. Theorem 3.22.
Every abstract finite specification of an overlap pattern, byan I -graph w.r.t. to some incidence pattern I , admits, for every N ∈ N , finiterealisation by N -acyclic finite hypergraphs. We discuss global symmetries, and especially the behaviour of our construc-tions under automorphisms of the input data. Section 4.1 provides the basicdefinitions and indicates that the matter is not entirely trivial, for two reasons.Firstly, the straightforward, essentially relational representation of givenstructures or input data that we chose does not necessarily represent the rel-evant symmetries as automorphisms, because the representations themselvesbreak symmetries. The most apparent example lies at the very root of ourformalisations: incidence patterns I and I -graphs. If we choose a relationalrepresentation of I = ( S, E ) as a trivial special I -graph, with singleton rela-tions R e = { ( s, s ′ ) } for every e ∈ E [ s, s ′ ], then the resulting structure is rigid,simply because we have individually labelled the edges by their names. Ratherthan this format, we need to consider I as a two-sorted structure of the form I = ( S, E, ι / , ( ) − ) discussed in Remark 2.2, which leaves room for non-trivialautomorphisms that capture all the intended ‘symmetries’.Secondly, constructions could, in principle involve choices that break sym-metries of the input data or given structures. In essence, this section largely isto show that the constructions presented so far are all sufficiently natural andgeneric in relation to the input data, so that such choices do not occur (or atleast do not have to occur). This is not trivial in all instances. In Section 4.2 we36rst look at those symmetries in direct and reduced products that stem from thehomogeneity of the Cayley graphs of the groupoid factor in those products; Sec-tion 4.3 concerns compatibility of these constructions with symmetries that arepresent in the input structures. In the end we shall know that all our construc-tions – of realisations and coverings based on reduced products with groupoids –are indeed symmetry preserving and in themselves highly symmetric, providedthe groupoids are, and that our construction of groupoids is compatible withthis requirement.In Section 4.4, we use these features of our constructions to provide anew proof of extension properties for partial automorphisms in the style ofHrushovski, Herwig and Lascar [10, 6, 8], and thus show how our generic recipefor the realisation of overlap patterns can be used to lift local symmetries, asmanifested in overlaps between local substructures or local isomorphisms, toglobal symmetries that manifest themselves as automorphisms. Definition 4.1. A symmetry of an incidence pattern I = ( S, E ) is an auto-morphism of the associated two-sorted structure, i.e., a pair η I = ( η S , η E ) ofbijections η S : S → S and η E : E → E , such that η E ( e ) ∈ E [ η S ( s ) , η S ( s ′ )] iff e ∈ E [ s, s ′ ], and such that η E ( e − ) = ( η E ( e )) − . Definition 4.2. A symmetry of an I -graph H = ( V, ( V s ) , ( R e )) based on anincidence pattern I = ( S, E ) consists of a symmetry η I = ( η S , η E ) of I to-gether with a permutation η V of V such that η V ( V s ) = V η S ( s ) and η V ( R e ) := { ( η V ( v ) , η V ( v ′ )) : ( v, v ′ ) ∈ R e } = R η E ( e ) .In this scenario, we think of the symmetry η H of the I -graph H as the triple η H = ( η V , η S , η E ). Definition 4.3. A symmetry of an I -groupoid G based on an incidence pattern I = ( S, E ) consists of a symmetry η I of I together with a permutation η G of G such that for all e ∈ E [ s, s ′ ], η G maps the generator g e of G to the generator η G ( g e ) = g η E ( e ) , and is compatible with the groupoid structure in the sensethat, for all s ∈ S and g ∈ G st , g ∈ G tu :(i) η G (1 s ) = 1 η S ( s ) ;(ii) η G ( g · g ) = η G ( g ) · η G ( g ).In this scenario we think of the symmetry η G of the I -groupoid G as the triple η G = ( η G , η S , η E ). It follows from the definition that, dropping superscripts fornotational ease, η ( g ) ∈ G η ( s ) η ( t ) for all g ∈ G st and that, for w = e . . . e n ∈ E ∗ [ s, t ], η ( w G ) = ( η ( w )) G where η ( w ) = η ( e ) . . . η ( e n ) ∈ E ∗ [ η ( s ) , η ( t )].It is obvious that the last two definitions are compatible with the passagebetween I -groupoids and their Cayley graphs (cf. especially Definitions 2.13, 2.7and 2.12). We often write just η to denote the different incarnations of η , and use superscripts onlyto highlight different domains where necessary. bservation 4.4. If η G = ( η G , η S , η E ) is a symmetry of the I -groupoid G , and H = ( G, . . . ) its Cayley graph, then η H := ( η V , η S , η E ) for η V = η G is also asymmetry of H . Conversely, any symmetry of an I -graph H naturally lifts to asymmetry of its completion ¯ H and of the I -groupoid G = cym( H ) . Note that the groupoid structure, i.e., the partial operation on G is fully de-termined by the Cayley graph structure (viewed as an I -graph, with E -labellededges) together with the identification of the elements 1 s ∈ G ss for s ∈ S . But,due to its homogeneity, the Cayley graph structure does not even distinguish G ss within G ∗ s : it is clear that the automorphism group of the Cayley graph(even as a relational structure with E -labelled edges) acts transitively on eachset G ∗ s . Most importantly, the groupoid constructions from Section 2 are fully com-patible with symmetries. In particular, passage from H to cym( H ) and theamalgamation constructions that lead to N -acyclic groupoids, Proposition 2.22,are such that every symmetry of I (or of the given I -graph H ) lift and extendto symmetries of the resulting I -groupoid G . The induction underlying Propo-sition 2.22 is based on the number of generators in the sub-groupoids, withan individual induction step according to Lemma 2.21 on the amalgamationof chains of sub-groupoids. All these notions are entirely symmetric w.r.t. anysymmetries of I . We can therefore strengthen the claim of Proposition 2.22 asfollows. Corollary 4.5.
For every incidence pattern I = ( S, E ) , I -graph H and N ∈ N ,there are finite N -acyclic I -groupoids G that extend every symmetry of I and H to a symmetry of G . The following notion of hypergraph automorphisms is the obvious one.
Definition 4.6. An automorphism of a hypergraph A = ( A, S ) is a bijection η : A → A such that for every s ⊆ A : η ( s ) := { η ( a ) : a ∈ s } ∈ S if, and only if, s ∈ S .We also denote the induced bijection on S by η and may also think of anautomorphism of the hypergraph A = ( A, S ) as a pair η A = ( η A , η S ), which thenautomatically induces a symmetry η I = ( η S , η E ) of the associated intersectiongraph I := I ( A ) (regarded as an incidence pattern) as well as a symmetry η H = ( η V , η S , η E ) of the I -graph representation H ( A ) of A . The operation of η : A → A naturally induces all the derived operations occurring in these; e.g.,on the vertex set V = S s ∈ S ( s × { s } ) of H ( A ), η V ( s × { s } ) = ( η S ( s ) × { η S ( s ) } )where η S ( s ) = { η ( a ) : a ∈ s } . In fact, the Cayley graph of G consists of a disjoint union of isomorphic complete I -graphs induced on the subsets G t ∗ = S s G ts for t ∈ S (if I is connected, then these are alsothe connected components); the groupoid structure of G can be retrieved from each one ofthese, via cym. .2 Groupoidal symmetries of reduced products Consider a hypergraph H ⊗ G or A ⊗ G obtained as a reduced product ofeither an I -graph or a hypergraph with an I -groupoid (for I = I ( A ) in thehypergraph case). Any such reduced product has characteristic symmetrieswithin its vertical fibres induced by groupoidal or Cayley symmetries of G . Inparticular, these symmetries are compatible with the natural projections ontoto first factor and trivial w.r.t. I . Lemma 4.7.
The automorphism group of the reduced product H ⊗ G betweenan I -graph H and an I -groupoid G acts transitively on the set of hyperedges { [ V s , g ] : g ∈ G ∗ s } for every s ∈ S , with trivial operation on the V s -component.Similarly, in the direct product H × G , any two patches V s × { g } for g ∈ G ∗ s are bijectively related by some symmetry of H × G that is trivial on the firstcomponent and on I . In the natural covering of a hypergraph A = ( A, S ) byits reduced product π : A ⊗ G → A with an I -groupoid G , where I = I ( A ) , anytwo pre-images of the same hyperedge s ∈ S are related by an automorphism of ˆ A = A ⊗ G that commutes with π .Proof. All claims are an immediate consequence of the homogeneity propertiesof the Cayley graphs of groupoids and the fact that the definitions of the directand reduced products in question do not refer to the groupoid structure of G (with distinguished elements 1 s ), other than via the structure of its Cayleygraph. Besides the vertical symmetries within fibres of coverings or realisations thereis an obvious concern relating to the compatibility of reduced products withautomorphisms of the input data. For these considerations, symmetries thatinvolve non-trivial symmetries of the underlying incidence pattern I are of theessence.The aim is to show that all our constructions of realisations and coverings aresufficiently natural or canonical to allow us to lift symmetries of a hypergraph A to its coverings by A ⊗ G , and of an I -graph H to its realisations obtainedin direct and reduced product constructions. This requires the use of groupoids G that are themselves fully symmetric w.r.t. those symmetries of I that areinduced by the structural symmetries of the given A or H .Recall from the discussion of Definition 4.6 that an automorphism of a hy-pergraph A = ( A, S ) takes the form η A = ( η A , η S ) and canonically inducessymmetries η I = ( η S , η E ) of I = I ( A ) and η H ( A ) = ( η V , η S , η E ) of the I -graphrepresentation H ( A ) of A . If, in addition, the I -groupoid G admits a match-ing symmetry η G = ( η G , η I ) = ( η G , η S , η E ), based on the same η I = ( η S , η E ),then the covering π : A ⊗ G → A carries a corresponding symmetry that is bothan automorphism of the covering hypergraph ˆ A = A ⊗ G and compatible withthe given automorphism of A via π , in the sense that the following diagram39ommutes: A ⊗ G π (cid:15) (cid:15) η ˆ A / / A ⊗ G π (cid:15) (cid:15) A η A / / A At the level of A ⊗ G , the automorphism η operates according to η A ⊗ G : [ a, g ] [ η A ( a ) , η G ( g )] , for g ∈ G ∗ s and a ∈ s .From Corollary 4.5 and Proposition 3.21 we thus further obtain the following. Corollary 4.8.
Any finite hypergraph A admits, for N ∈ N , finite strict N -acyclic coverings π : ˆ A = A ⊗ G → A by reduced products with finite N -acyclic I -groupoids G , for I = I ( A ) , such that these coverings are compatible with theautomorphism group of A in the sense that every automorphism η A of A lifts toan automorphism η ˆ A of ˆ A such that π ◦ η ˆ A = η A ◦ π . Turning to realisations of overlap patterns and their symmetries, it is equallyclear that direct and reduced products H × G and H ⊗ G extend every symmetry η H = ( η V , η S , η E ) of the I -graph H = ( V, ( V s ) , ( R e )) provided the I -groupoid G admits a matching symmetry η G = ( η G , η S , η E ) with the same underlyingsymmetry η I = ( η S , η E ). Corollary 4.9.
Every simultaneous symmetry of the I -graph H and the I -groupoid G gives rise to a symmetry of the direct product H × G and to anautomorphism of the reduced product hypergraph H ⊗ G . We may now combine these observations with those from Section 4.2 to ob-tain realisations that respect all symmetries of the overlap specification. Recallfrom Sections 3.2–3.4 that realisations, in the general case, were obtained in atwo-stage process. H −→ H × G −→ ( H × G ) ⊗ ˜ G = ˆ A ( ∗ )The first stage takes us from the I -graph H to a direct product I -graph H × G with a compatible I -groupoid G ; The second stage uses this direct product asa coherent ˜ I -graph for ˜ I = I ( G ) (Observation 3.6), to form a reduced productwith a suitable ˜ I -groupoid ˜ G , which realises the overlap pattern specified in H (Corollary 3.11). The analysis of the relevant symmetries, therefore, involvesstructural symmetries of the input structure H × G in the second stage, whichalso stem from groupoidal symmetries of G . Corollary 4.10.
For any incidence pattern I = ( S, E ) and I -graph H =( V, ( V s ) , ( R e )) , realisations ˆ A as obtained in Theorem 3.22 can be chosen so thatall symmetries of H lift to automorphisms of ˆ A . Moreover, for any two hyper-edges ˆ s and ˆ s of ˆ A that bijectively project to the same V s for s = π (ˆ s ) = π (ˆ s ) via π ˆ s i : ˆ s i → V s , there is a ‘vertical’ automorphism η = η ˆ A of ˆ A that is com-patible with these projections in the sense that π ˆ s = π ˆ s ◦ η . s η / / π ˆ s (cid:15) (cid:15) ˆ s π ˆ s (cid:15) (cid:15) V s id Vs / / V s Proof.
Consider the two-level construction indicated in ( ∗ ) above. It sufficesto choose the two groupoids in the construction of the realisation sufficientlysymmetric. For the first stage, the I -groupoid G can be chosen to be compatiblewith H and to have symmetries η G = ( η G , η S , η E ) based on the same η I =( η S , η E ) for all symmetries η H = ( η V , η S , η E ) of the I -graph H . Then the I -graph H × G has a symmetry η H × G that simultaneously extends η H and η G , forevery symmetry η H of H . In addition, H × G has all the groupoidal symmetriesof the Cayley graph of G according to Lemma 4.7.For the second stage of the construction, H × G is regarded as an ˜ I -graphwhere˜ I = I ( G ) = ( G, ˜ E ) where ˜ E = S e ∈ E (cid:8) ( g, gg e ) : e ∈ E [ s, s ′ ] , g ∈ G ∗ s (cid:9) . This ˜ I -graph also has the natural extension of every symmetry η H of H asa symmetry: clearly, every symmetry of the Cayley graph of G that is inducedby a symmetry of H extends to a symmetry of H × G as an ˜ I -graph. The˜ I -groupoid ˜ G can now be chosen compatible with H × G , N -acyclic for anydesired level N , and such that it extends every simultaneous symmetry of ˜ I and H × G . It then follows that the realisation ˆ A = ( H × G ) ⊗ ˜ G is compatible withevery symmetry of H .For the additional claim about ‘vertical’ symmetries consider two hyperedgesˆ s = [ V s × { g } , ˜ g ] and ˆ s = [ V s × { g } , ˜ g ] that project to the same V s ; here g i ∈ G ∗ s and ˜ g i ∈ ˜ G ∗ g i (note that ˜ S = G in ˜ I = ( ˜ S, ˜ E ) = ( G, ˜ E )).The Cayley graph of G has a symmetry η that takes g to g and whoseunderlying symmetry of I is trivial so that it fixes s . The corresponding simul-taneous symmetry of H and the Cayley graph of G lifts to H × G and inducesmatching symmetries of ˜ I and ˜ G . (This symmetry η ˜ G of ˜ G will typically notbe trivial with respect to ˜ I as it links η ˜ S ( g ) = g in their roles as elements of˜ S .) A purely groupoidal symmetry η of G , which is trivial w.r.t. ˜ I and H × G again, will finally suffice to align η ˜ G (˜ g ) with ˜ g , so that the composition of η and η maps ˆ s = [ V s × { g } , ˜ g ] to ˆ s = [ V s × { g } , ˜ g ] as required. In its basic form, Herwig’s theorem [6, 8, 7] provides, for some given partialisomorphism p of a given finite relational structure A , an extension B ⊇ A of A such that the given partial isomorphism p of A extends to a full automorphism of B . It generalises a corresponding theorem by Hrushovski [10], which makes thesame assertion about graphs. Elegant combinatorial proofs of both theoremscan be found in the paper by Herwig and Lascar [8], which also generalises41hem further to classes of structures that avoid certain weak substructures (cf.Corollary 4.18 below). The variant in which every partial isomorphism of theoriginal finite structure extends to an automorphism of the extension is moreuseful for some purposes. Only provided that the construction of B ⊇ A iscompatible with (full) automorphisms of A , however, can the general form beobtained directly for the basic form via straightforward iteration.W.l.o.g. we restrict attention to relational structures with a single relation R of some fixed arity r . Theorem 4.11 (Herwig’s Theorem) . For every finite relational structure A =( A, R A ) there is a finite extension B = ( B, R B ) ⊇ A such that every partialisomorphism of A is the restriction of some automorphism of B . Note that the term ‘extension’ as applied here means that A is an inducedsubstructure of B , denoted A ⊆ B , which means that A ⊆ B and R A = R B ∩ A r .A partial isomorphism of A is a partial map on A , p : dom( p ) → image( p ) that isan isomorphism between A ↾ dom( p ) and A ↾ image( p ) (induced substructures).In the context of the above theorem, it is also customary to refer to the ‘localsymmetries’, which are to be extended to global symmetries (automorphisms),as ‘partial automorphisms’.We here reproduce Herwig’s theorem in an argument based on our groupoidalconstructions, which may also offer a starting point for further generalisations.Before that, we prove from scratch the basic version of Herwig’s theorem for asingle partial isomorphism p using an idea that is going to motivate our newapproach to the full version below. Excursion: the basic extension task.
Let A = ( A, R A ) be a finite R -structure, p a partial isomorphism of A . We first provide a canonical infinitesolution to the extension task for p and A . Let A × Z = ( A × Z , R A × Z ) bethe structure obtained as the disjoint union of isomorphic copies of A , indexedby Z . Let ≈ be the equivalence relation over A × Z that identifies ( a, n ) with( p ( a ) , n + 1) for a ∈ dom( p ); we think of ≈ as induced by partial matchings orlocal bijections ρ p,n : dom( p ) × { n } −→ image( p ) × { n + 1 } ( a, n ) ( p ( a ) , n + 1) . Then, for m n , ( a , m ) ≈ ( a , n ) iff a = p n − m ( a ) . The interpretation of R in A ∞ := ( A × Z ) / ≈ is R A ∞ := { [¯ a, m ] : ¯ a ∈ R A , m ∈ Z } , where [¯ a, m ] is shorthand for the tuple of the equivalence classes of the com-ponents ( a i , m ) w.r.t. ≈ . By construction, A is isomorphic to the induced sub-structure represented by the slice A × { } ⊆ A × Z , on which ≈ is trivial:42 a, ≈ ( a ′ , ⇔ a = a ′ . Since p and the ρ p,n are partial isomorphisms, thequotient w.r.t. ≈ does not induce new tuples in the interpretation of R that arerepresented in the slice A × { } .The shift n n − η : ( a, n ) ( a, n −
1) and η : [ a, n ] [ a, n −
1] of A × Z and of ( A × Z ) / ≈ .The automorphism η of ( A × Z ) / ≈ extends the realisation of p in A × { } , sincefor a ∈ dom( p ) ⊆ A : η ([ a, a, −
1] = [ p ( a ) , . So B ∞ := ( A × Z ) / ≈ is an infinite solution to the extension task.It is clear that the domain dom( p k ) of the k -fold composition of the partialmap p is eventually stable, and, for suitable choice of k , also induces the identityon dom( p k ). Fixing such k , we look at the correspondingly defined quotient B := ( A × Z k ) / ≈ , which embeds A isomorphically in the induced substructure represented by theslice A × { } . Moreover, B has the automorphism η : [ a, n ] [ a, n −
1] (nowmodulo 2 k in the second component), which extends p , and thus is a finitesolution to the extension task. As B is not in general compatible with (full)automorphisms of A , the passage from A to B ⊇ A that solves the extensiontask for one partial isomorphisms p of A cannot just be iterated to furthersolve the extension task for another partial isomorphism p ′ without potentiallycompromising the solution for p . It turns out that suitably adapted groupoids,instead of a naive use of cyclic groups, implicitly take care of the interactionbetween simultaneous extension requirements.Let us summarise this argument in light of the approach we want take below,i.e., in terms of realisations of I -graphs that specify a desired overlap pattern.For this let I := ( { } , { e p , e − p } ) be the singleton incidence pattern with the twoorientations of a loop for p . The overlap pattern to be realised is specified inthe I -graph H = ( A, A, ( p, p − )) where A is the universe and only partition set,and p and p − stand for the partial bijections associated with the edges e p and e − p . For k ∈ N , let H × Z k := (cid:0) A × Z k , ( A × { n } ) n ∈ Z k , ( ρ p,n , ρ − p,n ) n ∈ Z k (cid:1) with partial bijections ρ p,n : ( a, n ) ( p ( a ) , n + 1) for a ∈ dom( p ). We maythink of Z k as a covering ˜ I of I in the sense of our discussion in Section 3.2; then H × Z k is the ˜ I -graph associated with the I -graph H . According to Lemma 3.5,every realisation of H × Z k induces a realisation of H . Above, we chose k such that H × Z k is coherent in the sense of Definition 2.4. Coherence of H × Z k further implies that any Z mk for m >
1, viewed as an ˜ I -groupoid inthe obvious manner, is compatible with H × Z k . For m >
2, moreover, Z mk is The period ℓ = 2 k (rather than k ) supports the essential condition that the domains ofthe partial bijections p n and of ( p − ) ℓ − n , which relate the same slices albeit along oppositedirections in the cycle Z ℓ , cannot be incomparable; cf. Observation 3.10 and discussion below. m -acyclic ˜ I -groupoid. Therefore, by Observation 3.10, the reduced product H ⊗ Z k is a realisation of H × Z k , and hence also of H . Coherence, and inparticular the 2-acyclicity of Z k , implies that any relation R A on A for which p is a partial isomorphism lifts consistently to H ⊗ Z k , and that in this mannerevery slice represented by some A × { n } for n ∈ Z k is isomorphic to ( A, R A ).The symmetry of the realisation under cyclic shifts in Z k , finally, shows thatthis structure extends the partial isomorphism p of any individual slice to anautomorphism. In the following we expand on this perspective. A generic construction.
We turn to the extension task for a specified col-lection P of partial isomorphisms of A = ( A, R A ). By a (finite) solution to thisextension task we mean any (finite) extension B ⊇ A which lifts every partialisomorphism p ∈ P to an automorphism of B . With this situation we associatea natural incidence pattern I and I -graph specification of an overlap pattern,whose symmetric realisations will solve the extension task.For I = ( S, E ) we use a 1-element set S = { } and endow it with one forwardand one backward loop for each p ∈ P : E = { e p : p ∈ P } ∪ { e − p : p ∈ P } , where the e p are pairwise distinct edge colours. For H we choose the I -graph H ( A , P ) = ( A, ( R e ) e ∈ E ) where R e = ( { ( a, p ( a )) : a ∈ dom( p ) } for e = e p { ( p ( a ) , a ) : a ∈ dom( p ) } for e = e − p is the graph of p or p − , according to the orientation of e .The idea is that the desired automorphisms will be directly induced by ‘ver-tical shifts’ in the sense of the last claim of Corollary 4.10 that link suitablepre-images of V s in realisations of H ( A , P ). Before we state the first version ofa Herwig–Lascar theorem as Theorem 4.13 below, we introduce some terminol-ogy for useful homogeneity properties of hypergraphs. Definition 4.12.
We call a hypergraph ( B, ˆ S ) homogeneous if its automor-phism group acts transitively on its set of hyperedges: for ˆ s, ˆ s ′ ∈ ˆ S there is anautomorphism η of ( B, ˆ S ) such that η (ˆ s ) = ˆ s ′ . For a subgroup Aut of the fullautomorphism group of ( B, ˆ S ), we say that ( B, ˆ S ) is Aut -homogeneous if anytwo hyperedges in ˆ S are related by an automorphism from that subgroup.Homogeneity of ( B, ˆ S ) in particular implies that ( B, ˆ S ) is uniform in thesense that all its hyperedges have the same cardinality; a hypergraph is called k - uniform if all its hyperedges have size k .The following variant of Herwig’s theorem can also be obtained as a corollaryof Herwig’s theorem as stated in Theorem 4.11 above. Its new proof, however,allows for further variations w.r.t. the nature of the hypergraph ( B, ˆ S ), whichmay for instance be required to be N -acyclic. Among other potential gen-eralisations this reproduces the extension of Herwig’s theorem to the class ofconformal structures and, e.g., of k -clique free graphs, obtained in [9] on thebasis of Herwig’s theorem together with a hypergraph covering result.44 heorem 4.13. For every finite A = ( A, R A ) and every set P of partial isomor-phism of A there is a finite relational structure B = ( B, R B ) and an Aut( B ) -homogeneous | A | -uniform hypergraph ( B, ˆ S ) such that(i) every automorphism of B is an automorphism of ( B, ˆ S ) ;(ii) B ↾ ˆ s ≃ A for all ˆ s ∈ ˆ S ;(iii) B ⊇ B ↾ ˆ s ≃ A solves the extension task for A and P .In particular, if P is the set of all partial isomorphisms of A , then every partialisomorphism of B whose domain and image sets are each contained in hyper-edges of ( B, ˆ S ) is induced by an automorphism of B .Proof. Let ˆ A = ( ˆ A, ˆ S ) be a fully symmetric realisation of the overlap patternspecified in H ( A , P ) in the sense of Corollary 4.10, with projections π ˆ s : ˆ s → A .We expand the universe ˆ A to produce an R -structure by pulling the interpreta-tion R A to ˆ A : B := ( ˆ A, R B ) where R B = [ ˆ s ∈ ˆ S π − s ( R A ) . One checks that this interpretation entails that π ˆ s : B ↾ ˆ s ≃ A for every ˆ s ∈ ˆ S .This interpretation of R in B also turns every π -compatible automorphism ofthe hypergraph ˆ A into an automorphism of the R -structure B . We want toshow that it solves the extension task for all p ∈ P , for every embedding of A into B via any one of the maps π − s .By Corollary 4.10, the automorphism group of the hypergraph ˆ A acts tran-sitively on ˆ S in a manner that is compatible with the π ˆ s . In particular, ˆ A ishomogeneous w.r.t. the subgroup of those hypergraph automorphisms that arecompatible with the projections and therefore w.r.t. the automorphism groupof the relational structure B . For p ∈ P , consider any hyperedge ˆ s of ˆ A and ahyperedge ˆ s ′ that, corresponding to condition (i) for realisations, overlaps withˆ s as prescribed by ρ e = p (on the left-hand side of the diagram):ˆ s id ˆ s ∩ ˆ s ′ / / π ˆ s (cid:15) (cid:15) ˆ s ′ π ˆ s ′ (cid:15) (cid:15) A p / / A ˆ s ′ η / / π ˆ s ′ (cid:15) (cid:15) ˆ s π ˆ s (cid:15) (cid:15) A id A / / A Consider the automorphism η of ˆ A that maps ˆ s ′ onto ˆ s and is compatiblewith π ˆ s ′ and π ˆ s (on the right-hand side of the diagram). The combination ofthe two diagrams shows that this automorphism η maps π − s (dom( p )) = π − s ′ (image( p )) = ˆ s ∩ ˆ s ′ ⊆ ˆ s onto π − s (image( p )) ⊆ ˆ s , and extends the partial map p ˆ s := π − s ◦ p ◦ π ˆ s , which represents p in the embedded copy π − s ( A ) = B ↾ ˆ s of A .45he formally stronger homogeneity assertion about the solution B for theextension task for A and the set of all its partial isomorphisms follows from(i)–(iii) by homogeneity of ( B, ˆ S ) w.r.t. the automorphism group of B .The homogeneity properties of the solutions to the extension task obtainedin our construction have further interesting consequences. In essence, we shallsee that sufficiently acyclic realisations of H ( A , P ) at least on a local scalebehave like ‘free’ solutions. This is an interesting phenomenon, because reallyfree solutions will in general be necessarily infinite. Moreover, this feature ofour solutions offers a new and transparent route to the much stronger extensionproperty of Herwig and Lascar [8] in Corollary 4.18 below. Proposition 4.14.
For any finite R -structure A , any collection P of partialisomorphisms of A and for any N ∈ N , there is a finite extension B ⊇ A thatsatisfies the extension task for A and P and has the additional property that anysubstructure B ⊆ B of size up to N can be homomorphically mapped into anyother (finite or infinite) solution C to the extension task for A and P . The proof is essentially based on the analysis of our solutions to the exten-sion task in the case that we use an N -acyclic realisation ( B, ˆ S ) of the overlapspecification in H ( A , P ). To prepare for this, we draw on the alternative char-acterisation of hypergraph acyclicity in terms of the existence of tree decompo-sitions . We then prove two further claims concerning B in relation to ( B, ˆ S )for N -acyclic ( B, ˆ S ).For our purposes, a tree decomposition of a finite hypergraph ( A, S ) can berepresented by an enumeration of the set S of hyperedges as S = { s , . . . , s m } such that for every 1 ℓ m there is some n ( ℓ ) < ℓ for which s ℓ ∩ [ n<ℓ s n ⊆ s n ( ℓ ) . The important feature here is that S S can be built up in a step-wise (andtree-like) fashion, starting from the root patch s by adding one member s ℓ at atime in such a manner that the overlap of the new addition s ℓ with the existingpart is fully controlled in the overlap with a single patch n ( ℓ ) in the existingpart. It is well known that hypergraph acyclicity as defined in terms of chordalityand conformality in Definition 3.2 is equivalent to the existence of a tree decom-position [2]. Correspondingly, the hypergraph ( B, ˆ S ) is N -acyclic if, and onlyif, every induced sub-hypergraph of size up to N admits a tree decomposition.Using an N -acyclic realisation of H ( A , P ) in our construction above, of asolution to the extension task for A , we therefore obtain the following additionalproperties of the hypergraph ( B, ˆ S ) and the induced relational structure B =( B, R B ). Re-thinking the whole process in reverse, we can see this as a decomposition process thatreduces (
A, S ) to the empty hypergraph by simple retraction steps. laim 4.15. In the terminology of the proof of Theorem 4.13, and for an N -acyclic realisation ( B, ˆ S ) of H ( A , P ) : for every substructure B ⊆ B of size upto N there are hyperedges { ˆ s , . . . , ˆ s m } ⊆ ˆ S such that(i) ( B , { ˆ s ∩ B , . . . , ˆ s m ∩ B } ) admits a tree decomposition;(ii) R B ↾ B ⊆ S n m R B ↾ ˆ s n .Proof of the claim. Let B = B ↾ B , | B | N . Due to N -acyclicity of ( B, ˆ S ),the induced sub-hypergraph ( B , { ˆ s ∩ B : ˆ s ∈ ˆ S } ) admits a tree decompositionrepresented by some enumeration of these induced hyperedges as (ˆ s ℓ ∩ B ) ℓ m .Due to the nature of R B as the union of the relations π − s ( R A ) for ˆ s ∈ ˆ S , whichagree in their overlaps, it is clear that condition (ii) is satisfied. Claim 4.16.
Let B = ( B, R B ) be a homogeneous solution to the extension taskfor A and P , based on the hypergraph ( B, ˆ S ) which is obtained as a realisationof the overlap pattern H ( A , P ) as in the proof of Theorem 4.13.Let C ⊇ A be any other (finite or infinite) solution to the extension task for A and P such that every p ∈ P extends to an automorphism f p ∈ Aut( C ) .Let ˆ s, ˆ s ′ ∈ ˆ S and consider any isomorphic embedding of B ↾ ˆ s ≃ A onto anautomorphic image of A within C of the form π := f ◦ π ˆ s : B ↾ ˆ s ≃ f ( A ) ⊆ C for some f ∈ Aut( C ) . Then there is an isomorphic embedding of B ↾ ˆ s ′ ≃ A onto another automorphic image of A within C of the form: π ′ := f ′ ◦ π ˆ s ′ : B ↾ ˆ s ′ ≃ f ′ ( A ) ⊆ C for some f ′ ∈ Aut( C ) such that π ∪ π ′ : B ↾ ˆ s ∪ B ↾ ˆ s ′ → C is a homomorphismfrom the union of these two induced substructures into C . Proof.
We observe that, according to the properties of a realisation, the overlapˆ s ∩ ˆ s ′ in ( B, ˆ S ) is induced by a path w of partial bijections p and p − for p ∈ P in H . The corresponding composition of partial isomorphisms of A gives riseto a composition of automorphisms f w ∈ Aut( C ). So there is an automorphism f ′ = f w ◦ f ∈ Aut( C ) such that f ′ ( A ) ∩ f ( A ) contains π ( B ↾ (ˆ s ∩ ˆ s ′ )). It followsthat π ′ := f ′ ◦ π ˆ s ′ agrees with π on ˆ s ∩ ˆ s ′ , whence π ∪ π ′ is well-defined on B ↾ ˆ s ∪ B ↾ ˆ s ′ and a homomorphism. Proof of the proposition.
Let B = ( B, R B ) be a homogeneous solution to theextension task for A and P , based on the hypergraph ( B, ˆ S ) which is obtainedas an N -acyclic realisation of the overlap pattern H ( A , P ) as in the proof ofTheorem 4.13. Then any B ⊆ B of size up to N admits a tree decomposition by(ˆ s ∩ B ) ℓ m in the sense of Claim 4.15. By Claim 4.16, C contains automorphicimages ( f ℓ ( A )) ℓ m of A that are related to B ↾ ˆ s ℓ by individual isomorphisms of B ↾ ˆ s ∪ B ↾ ˆ s ′ stands for the union of these two induced substructures, which overlap inthe common substructure B ↾ (ˆ s ∩ ˆ s ′ ); in universal algebraic terms, B ↾ ˆ s ∪ B ↾ ˆ s ′ is the freeamalgam over B ↾ (ˆ s ∩ ˆ s ′ ). π ℓ = f ℓ ◦ π ˆ s ℓ and such that π ℓ and π n ( ℓ ) agree on the overlap in B , sothat [ ℓ m π ℓ ( B ↾ ˆ s ℓ )is a homomorphic image of B under S ℓ π ℓ .From this we also obtain a major strengthening of Theorem 4.11 due to [8],which can be phrased as a finite-model property for the extension task. Definition 4.17.
Let C be a class of R -structures.(a) C has the finite model property for the extension of partial isomorphismsto automorphisms (EPPA) if, for every finite A ∈ C and collection P ofpartial isomorphisms of A such that A has some (possibly infinite) solutionto the extension task for A and P in C , there is a finite solution in C tothis extension task.(b) C is defined in terms of finitely many forbidden homomorphisms if, forsome finite list of finite R -structures C i , it consists of all R -structures A that admit no homomorphisms of the form h : C i hom −→ A .The following is now immediate from Proposition 4.14. Corollary 4.18 (Herwig–Lascar Theorem) . Every class C that is defined interms of finitely many forbidden homomorphisms has the finite model property for the extension of partial isomorphisms to automorphisms (EPPA). Concluding remarks.
The strength of the proposed generic approach to therealisation of overlap patterns of finite coverings has here been exemplified bythe conceptually rather simple proof and analysis of the Herwig–Lascar theorem.The use of reduced producs with (the Cayley graphs of) finite groupoids thatsatisfy strong finitary acyclicity properties has been shown to be a natural toolto obtain suitable realisations and to lend itself also directly to the construc-tion of finite coverings of controlled acyclicity for hypergraphs. The explorationof the relationship of this treatment of finite coverings, which yields canonicalcoverings of interest in the discrete world of finite hypergraphs, with classicalconstructions of branched coverings in the continuous setting remains a goal forfurther investigation. Both, the qualitative aspect of finiteness of coverings (inthe sense of having finite fibres) and the quantitative aspect of N -acyclicity (asa natural finitary approximation to full acyclicity), may point to further corre-spondences that are being explored further in ongoing research. At the moretechnical level, current investigations also aim for an assessment of the branch-ing behaviour of the finite coverings that are obtained by these combinatorialmethods; another issue being investigated concerns meaningful size bounds forvariant constructions that do not necessarily invoke the full genericity of theconstruction of N -acyclic groupoids as provided in Section 2.48 cknowledgement I am grateful to Achim Blumensath and Julian Bitter-lich for insightful comments and remarks on earlier versions of this technicalreport. Thanks to Dugald McPherson for pointing out an earlier, erroneousclaim concerning the simple solution to the basic extension task in the excur-sion in Section 4.4, and to his student Daoud Siniora for several text correc-tions. Research on this topic continues to be partially supported by a grantfrom Deutsche Forschungsgemeinschaft on the topic of hypergraphs of controlledacyclicity (DFG grant OT 147/6-1).
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