Algebraic coherent confluence and higher-dimensional globular Kleene algebras
aa r X i v : . [ c s . L O ] A ug Algebraic coherent confluence andhigher-dimensional globular Kleene algebras
Cameron Calk Eric Goubault Philippe Malbos Georg Struth
Abstract – W e extend the formalisation of confluence results in Kleene algebras to a formalisa-tion of coherent proofs by confluence. To this end, we introduce the structure of modal higher-dimensional globular Kleene algebra, a higher-dimensional generalisation of modal and concurrentKleene algebra. We give a calculation of a coherent Church-Rosser theorem and Newman’s lemma inhigher-dimensional Kleene algebras. We interpret these results in the context of higher-dimensionalrewriting systems described by polygraphs. Keywords –
Modal Kleene algebras, confluence, coherence, higher-dimensional rewriting.
M.S.C. 2010 – Γ -confluence and filling . . . . . . . . . . . . . . . . . . . . . . . . . 13 n -Dimensional globular Kleene algebras . . . . . . . . . . . . . . . . 193.3 A model of higher-dimensional modal Kleene algebras . . . . . . . . 23 n -Kleene algebra . . . . . . . . 324.4 Application in rewriting . . . . . . . . . . . . . . . . . . . . . . . . . 35 . Introduction Introduction
Rewriting is a model of computation widely used in algebra, computer science, and logic. Computationrules or algebraic laws are described by rewrite relations on symbolic or algebraic expressions. Rewritingtheory is strongly based on diagrammatic intuitions. Indeed, a central theme of rewriting is that ofcompleting certain branching shapes with confluence shapes , thus obtaining confluence diagrams . Tra-ditionally, the rewriting machinery was formalised in terms of algebras of binary relations: confluenceproperties are described by union, composition and iteration operations. A natural generalisation of thisis given by the structure of Kleene algebra, in which proofs of classical confluence results such as theChurch-Rosser theorem or Newman’s lemma have been calculated [5, 28, 29]. Beyond that, Kleene alge-bras and similar structures are well known for their ability to capture complex computational propertiesby simple equational reasoning [7, 20, 30, 32] and unify various semantics of computational interest,including formal languages, binary relations, path algebras and execution traces of automata [17].Rewriting provides constructive procedures for proving some coherence properties in categoricalalgebra as well. Coherence properties in this setting are formulated via a notion of contractibility forhigher-dimensional categories. The rewriting approach consists in proving a coherence property from agiven set of higher-dimensional witnesses for (local) confluence diagrams. It was initiated by Squier [25]in the context of homotopical finiteness conditions in string rewriting, which have more recently beenexpressed in the setting of higher dimensional rewriting [14]. By contrast to the standard diagrammatic orrelational approach, witnesses for confluence proofs are provided, in the sense that traditional confluencediagrams are filled with higher dimensional cells. Such a method has been applied, for instance, in [8, 15]to give constructive proofs for coherence in monoids, and in [12] for coherence theorems in monoidalcategories.In this work we combine these two branches of research on Kleene-algebraic and higher-dimensionalrewriting into a coherent framework. We show how some classical calculational confluence proofsusing Kleene algebra can be extended to coherent confluence proofs. To achieve this, we introducehigher-dimensional Kleene algebras, with many compositions and domain and codomain operations,which generalise both modal Kleene algebras [6] and concurrent Kleene algebras [16]. This structurealgebraically captures the semantics of higher dimensional abstract rewriting. As applications of thisformalism we provide calculational proofs for both the coherent Church-Rosser theorem and the coherentNewman’s lemma. We also relate these generalised results to the point-wise approach given by higher-dimensional rewriting systems described by polygraphs. The main contribution of this work is thereforethe provision of a point-free, algebraic approach to coherence in higher dimensional rewriting whichseems of general interest in categorical algebra.
Abstract coherent reduction
Coherence proofs by rewriting are based on coherent formulations of confluence results such as Church-Rosser’s theorem and Newman’s lemma. We present the coherent extension of the Church-Rosser theoremas an example. Recall that an abstract rewriting system → on a set X is a binary relation on X , and thatthe confluence of such a relation is characterised by the inclusion ∗ ← · ∗ → ⊆ ∗ → · ∗ ← ,2 . Introduction where ∗ → denotes the reflexive, transitive closure of the relation → , the relation ← its converse and · stands for relational composition. The relation → has the Church-Rosser property if the inclusion ∗ ↔ ⊆ ∗ → · ∗ ← holds, where ∗ ↔ = ( ← ∪ → ) ∗ is the reflexive, symmetric and transitive closure of → . The Church-Rossertheorem for → , which states that these two properties are equivalent, can be formulated along similarlines in Kleene algebra using the Kleene star operation, an abstraction of the notion of reflexive, transitiveclosure as in Theorem 4 of [28], recalled in (4.1.3), which states that for any x, y in a Kleene algebra K ,the following equivalence holds: x ∗ · y ∗ ≤ y ∗ · x ∗ ⇔ ( x + y ) ∗ ≤ y ∗ · x ∗ . The Church-Rosser theorem for the relation → is the special case where K is the algebra of binary relationson X , x = ← and y = → . This is justified by the fact that the binary relations over any set X form aKleene algebra with respect to relational composition, relational union, the reflexive transitive closureoperation, the empty relation and the unit (or diagonal) relation.The diagrammatic interpretation of the relation → states that there is an arrow u → v , whenever ( u, v ) belongs to → . When ( u, v ) is an element of ∗ → (resp. ∗ ↔ ), we say that u is related to v by a rewriting sequence (resp. zig-zag sequence ). Diagrammaticaly, the Church-Rosser theorem states that,for any branching ( f, g ) of rewriting sequences, there exists an associated confluence ( f ′ , g ′ ) , if and onlyif, for any zig-zag sequence h , there exists an associated confluence ( h ′ , k ′ ) : uf | | ②②②②② g " " ❉❉❉❉❉ u f ′ ! ! ❈❈ v g ′ ~ ~ ⑤ ⑤ u ′ ⇔ uh ′ ❆❆ o o h / / vk ′ ~ ~ ⑦ ⑦ u ′ A coherent extension of this result can be formulated in the context of higher-dimensional rewriting theory.Roughly, it states that if there exists a set Γ of -dimensional cells (of globular shape) such that everybranching can be completed to a confluence diagram filled by elements of Γ pasted together along their -dimensional borders, then every zig-zag sequence may be completed to a Church-Rosser diagram filledby elements of Γ pasted along their -dimensional borders. Pictorially, these statements are represented,respectively, by: uf | | ③③③③③ g ! ! ❉❉❉❉❉ u f ′ ! ! ❇❇ v g ′ ~ ~ ⑤ ⑤ u ′ α (cid:5) (cid:25) ⇔ uf ′ ❆❆ o o h / / vg ′ ~ ~ ⑦ ⑦ u ′ β (cid:5) (cid:25) where α and β are built from -cells in Γ . This result constitutes one step in the proof of Squier’stheorem for higher-dimensional rewriting systems, which provides a constructive approach to coherenceresults akin to the coherence condition satisfied by associativity and units in monoidal categories: ifcertain diagrams of natural isomorphisms commute, then all the diagrams built from the correspondingnatural isomorphisms are commutative. A key issue is therefore to reduce the infinite requirement “every . Introduction diagram commutes”, to a finite requirement “if a specified finite set of diagrams each commute then everydiagram commute”, [21, 26]. The notion of coherent confluence provides a constructive way of provingsuch coherence results. Organisation and main results of the article
Higher-dimensional rewriting.
In Section 2 we recall notions from higher-dimensional rewriting. Wefirst recall the structure of polygraph , which represents a system of generators and relations for higher-dimensional categories. Polygraphs were introduced by Street and Burroni, [3, 27], and are widelyused as rewriting systems presenting higher-dimensional algebraic structures [11, 23]. Furthermore,polygraphs are used to formulate homotopical properties of rewriting systems through polygraphic reso-lutions, [13, 22], as well as coherence properties for monoids, [8], higher categories, [11], and monoidalcategories, [12]. The latter are inspired by Squier’s approach to proving coherence results for monoidsusing convergent string rewriting systems [25]. Explicitly, an n -polygraph P is a higher dimensionalrewriting system made of globular cells of dimension
0, 1, . . . n , such that, for any ≤ k ≤ n , its set of k -cells P k consists in k -dimensional rewriting rules of globular shape: s k − ( α ) s k − ( α ) " " t k − ( α ) < < t k − ( α ) α (cid:5) (cid:25) A cellular extension of the free n -category P ∗ n (resp. the free ( n, n − ) -category P ⊤ n ) generated by P n is a set of globular ( n + ) -cells that relate n -cells of P ∗ n (resp. P ⊤ n ). Coherent confluence. A branching in an n -polygraph P is a pair ( f, g ) of n -cells of the free n -category P ∗ n which have the same ( n − ) -source. A branching is local when f and g are rewritingsteps , i.e. generating elements for the rewriting system given by P , see Section 2.2.1. A cellular extension Γ of the free ( n, n − ) -category P ⊤ n is a confluence filler of the branching ( f, g ) if there exist n -cells f ′ , g ′ in the free n -category P ∗ n and two ( n + ) -cells u < < f − ②②②②② g ! ! ❉❉❉❉❉ u f ′ ! ! ❇❇❇❇ v > > ( g ′ ) − ⑤⑤⑤⑤ u ′ α (cid:5) (cid:25) uf | | ②②②②② u v g ′ ~ ~ ⑤⑤⑤⑤ g − a a ❉❉❉❉❉ u ′ ( f ′ ) − a a ❇❇❇❇ α ′ (cid:5) (cid:25) in the free ( n + ) -category P ⊤ n [ Γ ] generated by Γ over P ⊤ n . We say that the cellular extension Γ is a (local) confluence filler for P if it is a confluence filler for each of its (local) branchings. In the case ofzig-zag sequences, we say that Γ is a confluence filler of an n -cell f in P ⊤ n if there exist n -cells f ′ and g ′ . Introduction in P ∗ n and an ( n + ) -cell α in the free ( n + ) -category P ⊤ n [ Γ ] of the form u f ′ ●●●●●● f / / vu ′ ( g ′ ) − ; ; ①①①①①①① α (cid:5) (cid:25) The cellular extension Γ is a Church-Rosser filler for an n -polygraph P when it is a confluence filler forevery n -cell in P ⊤ n . Theorem 2.3.4 states that for an n -polygraph P , a cellular extension Γ of P ⊤ n is aconfluence filler for P if, and only if, Γ is a Church-Rosser filler for P . Theorem 2.3.7 states that, when P is terminating , then Γ is a local confluence filler, if, and only if, Γ is a confluence filler for P . Theseare the coherent, higher-dimensional extensions of the Church-Rosser theorem and Newman’s lemma,respectively. In Subsection 2.4, we relate this confluence filler property to the coherent confluenceproperty already defined in higher-dimensional rewriting [10]. Higher-dimensional globular Kleeene algebras.
Section 3 contains the definitions of the variousalgebraic structures we employ. We first recall the notion of modal Kleene algebra [6]. These are Kleenealgebras with forward and backward modal operators defined via domain and codomain operations.Modal Kleene algebras provide an algebraic framework for computational relational models such asabstract rewriting systems, and beyond that for dynamics logics and predicate transformers.In Section 3.2, we introduce a notion of globular higher-dimensional modal Kleene algebra. First, wedefine a -dioid as a bounded distributive lattice, and for n ≥ , an n -dioid as a family ( S, + , 0, ⊙ i , 1 i ) ≤ i Section 4 contains the main results of this article. After recalling theformulation of Church-Rosser’s theorem and Newman’s lemma in modal Kleene algebras in Section 4.1,the notion of fillers in a globular modal n -Kleene algebra K is defined in Section 4.2.1. Given j - . Introduction dimensional elements φ, ψ ∈ K j := d j ( K ) , we say that A ∈ K is an i -confluence filler (resp. i -Church-Rosser filler for ( φ, ψ ) if | A i j ( ψ ∗ i ⊙ i φ ∗ i ) ≥ φ ∗ i ⊙ i ψ ∗ i ( resp. | A i j ( ψ ∗ i ⊙ i φ ∗ i ) ≥ ( ψ + φ ) ∗ i ) . We similarly define a notion of local i -confluence filler . A notion of whiskering in n -Kleene algebras isintroduced in Section 4.2.3 and some of its properties are made explicit. We also define, for φ, ψ ∈ K j and an i -confluence filler A ∈ K of ( φ, ψ ) , the j -dimensional i -whiskering of A : ^ A := ( φ + ψ ) ∗ i ⊙ i A ⊙ i ( φ + ψ ) ∗ i . We then prove two results interpreting the coherent Church-Rosser theorem in the setting of n -Kleene algebras. The first, Proposition 4.2.7, uses an inductive argument external to the n -Kleenestructure. Given ≤ i < j < n , it states that for φ, ψ ∈ K j , any i -confluence filler A of ( φ, ψ ) , and anynatural number k ≥ , there exists an A k ≤ ^ A ∗ j such that r j ( A k ) ≤ ψ ∗ i φ ∗ i and d j ( A k ) ≥ ( φ + ψ ) k i , where ( φ + ψ ) i = i and ( φ + ψ ) k i = ( φ + ψ ) ⊙ i ( φ + ψ ) k i − .The second interpretation of the Church-Rosser theorem, with a proof relying only on the internalfixpoint induction of the Kleene star, constitutes our first main result: Theorem 4.2.8. Let K be a globular n -modal Kleene algebra and ≤ i < j < n . Given φ, ψ ∈ K j and any i -confluence filler A ∈ K of ( φ, ψ ) , we have | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≥ ( φ + ψ ) ∗ i . Thus ^ A ∗ j is an i -Church-Rosser filler for ( φ, ψ ) . In Section 4.3, we introduce notions of termination and well-foundedness in n -Kleene algebras inwhich the domain algebras K i have a Boolean structure for all i ≤ p < n . This leads to our second mainresult, a coherent formulation of Newman’s lemma in such algebras: Theorem 4.3.2. Let ≤ i ≤ p < j < n , and let K be a globular p -Boolean modal Kleenealgebra such that i) ( K i , + , 0, ⊙ i , 1 i , ¬ i ) is a complete Boolean algebra, ii) K j is continuous with respect to i -restriction, i.e. for all ψ, ψ ′ ∈ K j and every family ( p α ) α ∈ I of elements of K i such that sup I ( p α ) exists, we have ψ ⊙ i sup I ( p α ) ⊙ i ψ ′ = sup I ( ψ ⊙ i p α ⊙ i ψ ′ ) . For any ψ ∈ K j i -Noetherian, and φ ∈ K j i -well-founded, if A is a local i -confluence fillerfor ( φ, ψ ) , we have | ^ A ∗ i i j ( ψ ∗ i φ ∗ i ) ≥ φ ∗ i ψ ∗ i . Thus ^ A ∗ j is an i -confluence filler for ( φ, ψ ) . Finally, in Section 4.4, we interpret these results in the context of higher dimensional abstract rewriting,using the higher dimensional path model defined in Section 3.3. . Preliminaries on higher-dimensional rewritingToward an algebraic Squier’s theorem. In this work we provide a formal proofs of the coherent Church-Rosser’s theorem and the coherent Newman’s lemma in higher-dimensional Kleene algebras. These resultsare the main ingredients in the proof of Squier’s coherence theorem, [25] used in constructive proofs ofcoherence in categorical algebra. A main objective remains to formalise this result within our algebraicframework. The first main obstacle towards it is the formalisation of the coherent critical branchinglemma. This requires taking the algebraic and syntactic nature of terms of the rewriting system intoaccount, which is currently an open problem in the formalism of Kleene algebras. The second difficultyis to capture normalisation strategies algebraically in higher-dimensional Kleene algebras. Squier’scoherence theorem is the first step in the construction of cofibrant replacements of algebraic structuresusing convergent presentations. We expect that the material introduced in this article will enable us toobtain an algebraic formalisation of acyclicity, which could in turn provide an algebraic criterion forcofibrance. 2. Preliminaries on higher-dimensional rewriting In this preliminary section, we recall the notions of higher-dimensional rewriting relevant to this article.In its two subsections we recall the definition of polygraphs and their properties as rewriting systemspresenting higher-dimensional categories. In Subsection 2.3 we introduce the notion of confluence filler for a polygraph with respect to a cellular extension. We then formulate and prove the coherent versionsof the Church-Rosser theorem and Neman’s lemma in the point-wise, polygraphic setting. Finally, in itslast subsection, we relate the confluence filler property to the coherent confluence property previouslyintroduced in [10]. Let n be a natural number or ∞ . For a (strict and globular) n -category C , and ≤ k < n we denote by C k the k -category of k -cells of C . As an abuse of notation, we also write C k forthe set of k -cells of C . For a k -cell f of C , and ≤ i < k , we denote by s i ( f ) (resp. t i ( f ) ) the i -source(resp. i -target), and by f its identity ( k + ) -cell of f . The source and target maps s i , t i : C k → C i satisfythe globular relations s i ◦ s i + = s i ◦ t i + , t i ◦ s i + = t i ◦ t i + . (2.1.2)When f and g are i -composable k -cells, that is when t i ( f ) = s i ( g ) , we denote by f ⋆ i g their i -composite.We recall that the composition operations satisfy the exchange relation ( f ⋆ i f ′ ) ⋆ j ( g ⋆ i g ′ ) = ( f ⋆ j g ) ⋆ i ( f ′ ⋆ j g ′ ) , (2.1.3)for any ≤ i < j < n and whenever the compositions are defined. The ( k − ) -composition of k -cells f and g is denoted by juxtaposition fg , and the ( k − ) -source s k − ( f ) and the ( k − ) -target t k − ( f ) ofa k -cell f are denoted by s ( f ) and t ( f ) , respectively. If we denote by f : u ⇒ v a k -cell in C , we denoteby u : p → q the ( k − ) -cells of C and by A : f ⇛ g the ( k + ) -cells of C in order to notationally . Preliminaries on higher-dimensional rewriting distinguish their respective dimensions. These globular cells are depicted as follows: p u ! ! v = = qf (cid:5) (cid:25) g (cid:5) (cid:25) A ❴ % Given a k -cell f , the l -dimensional identity on f for k ≤ l ≤ n isdenoted by ι lk ( f ) and defined by induction, setting ι kk ( f ) := f and ι lk ( f ) := ι l − for k < l ≤ n . In thisway, for ≤ k < l ≤ n , we associate to a k -cell f a unique identity cell ι lk ( f ) of dimension l , called the l -dimensional identity on f .In higher category theory, the use of such iterated identities is of great importance for definingcompositions. Given ≤ i < k < l ≤ n , a k -cell f , and a l -cell g , such that t i ( f ) = s i ( g ) , the i -composite of f and g , is defined by f ⋆ i g = ι lk ( f ) ⋆ i g, and if t i ( g ) = s i ( f ) , we define g ⋆ i f = g ⋆ i ι lk ( f ) .For ≤ i < j ≤ k , an ( i, j ) -whiskering of a k -cell f is a k -cell ι kj ( u ) ⋆ i f ⋆ i ι kj ( v ) , where u and v are j -cells,. To simplify notation, we denote this k -cell by u ⋆ i f ⋆ i v . A ( k − 1, k − ) -whiskering u ⋆ k − f ⋆ k − v of a k -cell f will be called a whiskering of f and is denoted by ufv . ( n, p ) -categories. If C is an n -category, for ≤ i < k ≤ n , a k -cell f of C is i -invertible if thereexists a k -cell g in C , with i -source t i ( f ) and i -target s i ( f ) in C , called the i -inverse of f , which satisfies f ⋆ i g = s i ( f ) and g ⋆ i f = t i ( f ) . The i -inverse of a k -cell is necessarily unique. When i = k − , we say that f : u → v is invertible and wedenote by f − : v → u its ( k − )− inverse, simply called inverse for short. If moreover the ( k − ) -cells u and v are invertible, then there exist k -cells u − ⋆ k − f − ⋆ k − v − : u − → v − , v − ⋆ k − f − ⋆ k − v − : u − → v − in C . For a natural number p ≤ n , or for p = n = ∞ , an ( n, p ) -category is an n -category whose k -cellsare invertible for every k > p . When n < ∞ , this is a p -category enriched in ( n − p ) -groupoids and,when n = ∞ , a p -category enriched in ∞ -groupoids. Let C be an n -category. A -sphere of C is a pairof -cells of C . For ≤ k ≤ n , a k -sphere of C is a pair ( f, g ) of k -cells such that s k − ( f ) = s k − ( g ) and t k − ( f ) = t k − ( g ) . We denote by Sph k ( C ) the set of k -spheres of C .When n < ∞ , the n -category C is aspherical if any n -sphere of C is of the form ( f, f ) , with f in C n .A cellular extension of C is a set Γ equipped with a map ∂ : Γ → Sph n ( C ) . For α ∈ Γ , the boundary ofthe sphere ∂ ( α ) is denoted ( s n ( α ) , t n ( α )) , defining in this way two maps s n , t n : Γ → C n satisfying thefollowing globular relations s n − ◦ s n = s n − ◦ t n and t n − ◦ s n = t n − ◦ t n .8 .2. Rewriting properties of polygraphs The free ( n + ) -category generated by Γ over C is the ( n + ) -category, denoted by C [ Γ ] , whoseunderlying n -category is C and whose ( n + ) -cells are built as formal i -compositions, for ≤ i ≤ n ,of elements of Γ and k -cells of C , seen as ( n + ) -cells with source and target in C n . The free ( n + 1, n ) -category generated by Γ over C is denoted by C ( Γ ) . We refer the reader to [22] for explicitfree constructions on cellular extensions over an n -category. n -polygraph. An n -polygraph P consists of a set P and for every ≤ k < n a cellular extension P k + of the free k -category P ∗ k = P [ P ] . . . [ P k ] . For ≤ k ≤ n , the elements of P k are called the generating k -cells of P . The free n -category P [ P ] . . . [ P n − ][ P n ] ( resp. the free ( n, n − ) -category P [ P ] . . . [ P n − ]( P n ) ) generated by P is denotedby P ∗ n ( resp. P ⊤ n ). We refer to [22] for the details of the free constructions on an n -polygraph. Note thata -polygraph is a set and an -polygraph corresponds to a directed graph, whose set of vertices is P and P is the set of arrows f with source s ( f ) and target t ( f ) . Low-dimensional polygraphs describe abstract rewrit-ing systems. Recall that an abstract rewriting system consists of a set X and a family → = { → i } i ∈ I ofbinary relations on X , i.e. → i ⊆ X × X for all i ∈ I . We refer the reader to [31] for a complete treatmenton abstract rewriting. An abstract rewriting system A = ( X, { → i } i ∈ I ) , can be described by a -polygraph,denoted P ( A ) , whose set of -cells is X , and whose set of -cells consists of u ( x,y,i ) : x → y for any x, y ∈ X and i ∈ I such that ( x, y ) ∈ → i . When I is a singleton, the -cells of the free -category P ( A ) ∗ correspond to the elements of the reflexive and transitive closure ∗ → of the relation → . Moreoverthe -cells of the free ( 1, 0 ) -category P ( A ) ⊤ correspond to the elements of the symmetric closure of therelation → ∗ .A string rewriting system is an abstract rewriting system on a free monoid [2], and can be describedby a -polygraph with a single -cell. Finally, -polygraphs describe term rewriting systems [9] orthree-dimensional rewriting [23]. A rewriting step for an n -polygraph P is an n -cell of the n -category P ∗ n of the form u n − ⋆ n − ( u n − ⋆ n − . . . ⋆ ( u ⋆ ( u ⋆ f ⋆ v ) ⋆ v ) ⋆ . . . ⋆ n − v n − ) ⋆ n − v n − , for a generating n -cell f ∈ P n and i -cells u i , v i with ≤ i < n . We denote by P cn the set of rewritingsteps of P . A rewriting path in P of length k is an ( n − ) -composition f ⋆ n − f ⋆ n − . . . ⋆ n − f k of rewriting steps of P . A zig-zag in P of length k is an ( n − ) -composition f ǫ ⋆ n − f ǫ ⋆ n − . . . ⋆ n − f ǫ k k . Preliminaries on higher-dimensional rewriting of rewriting steps for P , where ǫ , . . . , ǫ k ∈ { − 1, 1 } , and which is reduced with respect the reduction f ⋆ n − f − → .The set of rewriting steps induces an abstract rewriting system on the set of ( n − ) -cells of P ∗ n denoted by → P n , and defined by u → P n u ′ if there exists a rewriting step for P that reduces u to u ′ . Inthis case, we say that u rewrites to u ′ . An ( n − ) -cell u in P n is irreducible with respect to P if there isno rewriting step for P that reduces u . Given a cellular extension Γ of an n -category C , we also denote by Γ c the set of cells of Γ in context , that is the set of ( n + ) -cells of the form f n ⋆ n − . . . ⋆ ( f ⋆ ( f ⋆ α ⋆ g ) ⋆ g ) ⋆ . . . ⋆ n − g n , where f i , g i are i -cells of C for ≤ i ≤ n , and α ∈ Γ . Any ( n + ) -cell in the free ( n + ) -category C [ Γ ] can be written as an n -composition of elements of Γ c using the algebraic laws of higher categories, mostnotably the exchange relation. In particular, this means that the n -cells of P ∗ n correspond to the reflexive,transitive closure of → P n in the sense that given an n -cell f of P ∗ n , we have f = f ⋆ n − f ⋆ n − . . . ⋆ n − f k , where f i ∈ P cn . n -polygraph. Let P be a rewriting property defined on abstractrewriting systems. A polygraph P has the property P if the abstract rewriting system → P n has theproperty P . In particular, P is terminating if there is no infinite rewriting path for P .A branching in P is an unordered pair ( f, g ) of rewriting paths of P such that s n − ( f ) = s n − ( g ) .Such a branching is local when f and g are rewriting steps. We say that P is confluent (resp. locallyconfluent ) if for any branching (resp. local branching) ( f, g ) , there exist rewriting paths f ′ , g ′ of P with t n − ( f ′ ) = t n − ( g ′ ) such that the compositions f ⋆ n − f ′ and g ⋆ n − g ′ are defined, as illustrated in thefollowing diagram: uf | | ②②②②② g " " ❉❉❉❉❉ u f ′ ! ! ❈❈❈❈ v g ′ ~ ~ ⑤⑤⑤⑤ u ′ The source of a branching ( f, g ) is the common ( n − ) -source of f and g . We say that P is Church-Rosser if for any zig-zag h of P , there exist rewriting paths k, k ′ of P as in the following diagram: u o o h / / k ●●●●●● vk ′ { { ①①①①①①① u ′ Let P be an n -polygraph and Γ a cellular extension of P ⊤ n . .3. Coherent confluence The cellular extension Γ is a confluence filler of a branching ( f, g ) of P if there exist rewriting paths f ′ , g ′ of P as in (2.3.2), and two ( n + ) -cells α, α ′ in the free ( n + ) -category P ⊤ n [ Γ ] of the form α : f − ⋆ n − g → f ′ ⋆ n − ( g ′ ) − and α ′ : g − ⋆ n − f → g ′ ⋆ n − ( f ′ ) − : uf | | ②②②②② g " " ❉❉❉❉❉ u f ′ ! ! ❈❈❈❈ v g ′ ~ ~ ⑤⑤⑤⑤ u ′ u < < f − ②②②②② g " " ❉❉❉❉❉ u f ′ ! ! ❈❈❈❈ v > > ( g ′ ) − ⑤⑤⑤⑤ u ′ α (cid:5) (cid:25) uf | | ②②②②② u v g ′ ~ ~ ⑤⑤⑤⑤ g − b b ❉❉❉❉❉ u ′ ( f ′ ) − a a ❈❈❈❈ α ′ (cid:5) (cid:25) (2.3.2)In this case, α and α ′ are n -compositions of ( n + ) -cells of Γ c as recalled in Remark 2.2.2. We say thatthe cellular extension Γ is a confluence filler for the polygraph P if Γ is a confluence filler for each of itsbranchings.More generally, the cellular extension Γ is a confluence filler of an n -cell f in P ⊤ n if there exist n -cells f ′ and g ′ in P ∗ n and an ( n + ) -cell α in the free ( n + ) -category P ⊤ n [ Γ ] of the form α : f → f ′ ⋆ n − g ′ − : u f ′ ●●●●●● f / / vg ′ { { ①①①①①①① u ′ u f ′ ●●●●●● f / / vu ′ ( g ′ ) − ; ; ①①①①①①① α (cid:5) (cid:25) (2.3.3)The cellular extension Γ is a Church-Rosser filler for an n -polygraph P when it is a confluence filler ofevery n -cell in P ⊤ n . Let P be an n -polygraph. A cellular extension Γ of P ⊤ n is a confluence filler for P if, and only if, Γ is a Church-Rosser filler for P .Proof. First suppose that Γ is a Church-Rosser filler for P . Given a branching ( f, g ) , we have that f − ⋆ n − g and g − ⋆ n − f are elements of P ⊤ n and thus Γ is a confluence filler for these n -cells. This gives us thecells α and α ′ as in (2.3.2), and so Γ is a confluence filler for P .Conversely, suppose that Γ is a confluence filler for P , and let f ∈ P ⊤ n be an n -cell. We prove byinduction on the length of f that Γ is a Church-Rosser filler for P . For f of length or , we clearly havethat f is Γ -confluent, since it suffices to take an identity ( n + ) -cell. Suppose that every n -cell of length k ≥ is Γ -confluent and that f is of length k + . Then f = f ⋆ n − f with f : u → u in P ⊤ n of length k and f is of length in P ∗ n either of the form v → u or u → v . By the induction hypothesis thereexist rewriting paths h and k and an ( n + ) -cell α such that α : f ⇒ hk − . If f : u → v , there existrewriting paths k ′ and f ′′ and an ( n + ) -cell β as depicted in diagram (2.3.5) since Γ is a confluencefiller for P . Thus ( αf ) ⋆ n ( hβ ) is a confluence filler for f . u o o f / / h $ $ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏ u f / / vu ′ k − tttttt tttttt k ′ / / u ′′ f ′′ − : : tttttttttttttt α (cid:5) (cid:25) β (cid:5) (cid:25) (2.3.5) . Preliminaries on higher-dimensional rewriting If f : v → u , the ( n + ) -cell αf − ⋆ n h1 k − ( f ) − = αf − is a confluence filler for f . u o o f / / h $ $ ❏❏❏❏❏❏❏❏❏❏❏❏❏❏ u ( f ) − / / vu ′ k − tttttt tttttt ( f ) − / / u ′′ k − : : tttttttttttttt α (cid:5) (cid:25) k − ( f ) − (cid:5) (cid:25) (2.3.6) Let P be a terminating n -polygraph, and Γ acellular extension of P ⊤ n . Then Γ is a local confluence filler, if, and only if, Γ is a confluence filler for P .Proof. Firstly, observe that if Γ is a confluence filler for P , then it is also a local confluence filler for P since local branchings are also branchings.Now suppose that Γ is a local confluence filler for P . We prove by Noetherian induction that, for every ( n − ) -cell u of P ∗ n , Γ is a confluence filler for every branching of P with source u . For the base case,if u is irreducible for P , then ( u , 1 u ) is the only branching with source u , and it is Γ -confluent, takingthe ( n + ) -cell u .Suppose now the induction hypothesis, namely that u is a reducible ( n − ) -cell of P ∗ n and that Γ is aconfluence filler for every branching with source an ( n − ) -cell u ′ such that u rewrites to u ′ . Let ( f, g ) be a branching of P with source u . If one of f or g is an identity, say f , then Γ is a confluence fillerfor ( f, g ) by considering the ( n + ) -cells g and g − . We may now suppose that the n -cells f and g are not identities, thus we write f = f ⋆ n − f and g = g ⋆ n − g , where g , f are rewriting steps and g , f are n -cells of P ∗ n . Since Γ is a local confluence filler for P , there exist n -cells f ′ , g ′ in P ∗ n , and an ( n + ) -cell α in P ∗ n [ Γ ] as in the diagram (2.3.8). We apply the induction hypothesis to the branching ( f , f ′ ) , which yields n -cells f ′ , h in P ∗ n and an ( n + ) -cell β in P ∗ n [ Γ ] as in the diagram (2.3.8). Finally,we apply the induction hypothesis again to the branching ( g ′ ⋆ n − h, g ) yielding n -cells k and g ′ andan ( n + ) -cell γ in P ∗ n [ Γ ] as in (2.3.8). u < < f − ②②②②②②② g ! ! ❉❉❉❉❉❉❉ u f ′ ❇❇ ! ! ❇❇ > > f − ⑥⑥⑥⑥⑥⑥⑥ v > > ( g ′ ) − g ❅❅❅❅❅❅ u f ′ (cid:31) (cid:31) ❄❄❄❄❄❄ u ′ ? ? h − ⑦⑦⑦⑦ v ? ? ( g ′ ) − ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ u ′ k ❆❆❆❆❆❆ u ′′ α (cid:5) (cid:25) β (cid:5) (cid:25) γ (cid:5) (cid:25) (2.3.8)The n -composition δ = ((( f − ⋆ n − α ) ⋆ n ( β ⋆ n − ( g ′ ) − )) ⋆ n − g ) ⋆ n ( f ′ ⋆ n − γ ) (2.3.9) .4. Γ -confluence and filling is an ( n + ) -cell in P ∗ n [ Γ ] with source f − ⋆ n − g and target f ′ ⋆ n − k ⋆ n − ( g ′ ) − . We can similarly findan ( n + ) -cell δ ′ with source g − ⋆ n − f and with target a confluence. Γ is thus a confluence filler for P ,which proves the result. Readers more familiar with abstract rewriting than higher dimensional rewriting maynotice that the proofs of Theorems 2.3.4 and 2.3.7 are similar to the classical proofs of these results forabstract rewriting systems. Indeed, if we forget the ( n + ) -dimensional coherence cells and look onlyat their n -dimensional borders in (2.3.5), (2.3.6) and (2.3.8), we obtain precisely the diagrams used toprove the analogous -dimensional results for abstract rewriting systems. This shows that the higherdimensional approach is consistent with the abstract case while providing several advantages. Firstly,using explicit witnesses for confluence allows for a constructive formulation of classical results in theform of normalisation strategies . Furthermore, since the higher-dimensional cells may be considered asrewriting systems in their own right, and since the procedures describe the above work in any dimension,higher-dimensional rewriting provides a constructive method for calculating resolutions and cofibrantreplacements of algebraic structures. Γ -confluence and filling Recall from [10] that, given an n -polygraph P and a cellular extension of P ∗ n , we say that P is Γ -confluent (resp. Γ -locally confluent ) if for any branching (resp. local branching) ( f, g ) of P there exist n -cells f ′ , g ′ in the free n -category P ∗ n as in (2.3.2), and an ( n + ) -cell α in the free ( n + 1, n ) -category P ∗ n ( Γ ) of the form α : f ⋆ n − f ′ → g ⋆ n − g ′ . Similarly, we say that P is Γ -Church-Rosser if for any n -cell f of P ⊤ n there exist n -cells f ′ , g ′ in the free n -category P ∗ n as in 2.3.3, and an ( n + ) -cell α in the free ( n + 1, n ) -category P ∗ n ( Γ ) of the form α : f ⋆ n − f ′ → g ′ . Note that when Γ = Sph ( P ∗ n ) , the property of(local) Γ -confluence coincides with the property of (local) confluence of P as defined in (2.2.1), and theproperty of Γ -Church-Rosser coincides with the Church-Rosser property of P .Theorems 2.3.4 and 2.3.7, formulated above in terms of fillers, are expressed in terms of Γ -confluenceas follows: Let P be an n -polygraph, and Γ be a cellularextension of P ∗ n . If P is Γ -confluent, then P is Γ -Church-Rosser.Proof. The proof is similar to that of Theorem 2.3.4, but with the ( n + ) -cells oriented horizontally inthe induction step, as pictured in the following diagram: u o o f / / h ❅❅❅❅❅❅❅❅❅❅ v f ′ / / k ⑥⑥⑥⑥ ~ ~ ⑥⑥⑥⑥ f ′′ (cid:0) (cid:0) ✁✁✁✁✁✁✁✁✁✁ u ′ k ′ / / v ′ α % β % (2.4.2)The composite ( α ⋆ n − k ′ ) ⋆ n ( f ⋆ n − β ) makes the n -cell f Γ -confluent. Let P be a terminating n -polygraph, and Γ a cellularextension of P ∗ n . If P is locally Γ -confluent, then P is Γ -confluent. . Preliminaries on higher-dimensional rewriting Proof. The proof is similar to that of Theorem 2.3.7, but with the following induction diagram: uf | | ②②②②②②② g ! ! ❉❉❉❉❉❉❉ u f ′ ❇❇ ! ! ❇❇ f ~ ~ ⑥⑥⑥⑥⑥⑥⑥ v g ′ ⑥⑥ ~ ~ ⑥⑥ g ❅❅❅❅❅❅ u f ′ (cid:31) (cid:31) ❄❄❄❄❄❄ u ′ h ⑦⑦⑦ (cid:127) (cid:127) ⑦⑦⑦ v g ′ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ u ′ k ❆❆❆❆❆❆ u ′′ α % β % γ % (2.4.4)Then the following n -composition δ = ((( f ⋆ n − β ) ⋆ n ( α ⋆ n − h )) ⋆ n − k ) ⋆ n ( g ⋆ n − γ ) (2.4.5)is an ( n + ) -cell in P ∗ n ( Γ ) with source f ⋆ n − ( f ′ ⋆ n − k ) and target g ⋆ n − g ′ , proving the result.Note that for Γ = Sph ( P ∗ n ) , Theorems 2.4.3 and 2.4.1 correspond to Newman’s lemma and theChurch-Rosser theorem [24], see also [18]. In this section, we have defined two approaches to coherence properties of an n -polygraph P with respect to a cellular extension Γ : i) A "vertical" approach in which the coherence cells, i.e. the ( n + ) -cells generated by Γ , have abranching as n -source and a confluence as n -target. This necessitates having inverses of n -cells,that is Γ is a cellular extension of P ⊤ n . However, in the proofs of Theorems 2.3.4 and 2.3.7, we donot need inverses of ( n + ) -cells. ii) A "horizontal" approach in which coherence cells have rewriting paths for both source and target,and we do not need inverses of n -cells, i.e. we consider cellular extensions of P ∗ n , only inverses of ( n + ) -cells in order to prove Theorems 2.4.1 and 2.4.3.These differences can be summed up by saying that, in the first approach, the proofs take place in P ⊤ n [ Γ ] ,whereas, in the second one, the proofs take place in P ∗ n ( Γ ) .Furthermore, it is worth noting that, in the first approach, we specify two filler cells α and α ′ asdepicted in diagram 2.3.2 for each branching ( f, g ) . This is due to the fact that branchings are unorderedpairs, so we must account for both cases. This equally constitutes the reason we require inverses of ( n + ) -cells in the second approach.In the rest of this article, we will exclusively consider the "vertical" approach to paving diagrams withhigher dimensional cells. The motivation of this choice lies in the fact that with Kleene algebras, we pavediagrams from a relational rather than a polygraphic point of view. We thus follow the direction of the n -cells in branchings and confluences— i.e. "vertically". This is a consequence of the quantification onbranchings and confluences: we quantify universally over branchings and existentially over confluences.In the polygraphic approach, this quantification is hidden by specifying the ( n + ) -cells filling confluencediagrams. . Higher dimensional modal Kleene algebras 3. Higher dimensional modal Kleene algebras In this section we introduce the notion of higher-dimensional globular modal Kleene algebra. In its firstsubsection, we recall the axioms of modal Kleene algebra [6]. We then define n -dimensional dioids andequip these with domain and star operations, thus obtaining modal n -Kleene algebras . Finally, we providea model of this structure in the form of a higher-dimensional path algebra associated to an n -polygraphwith a cellular extension Γ . Recall that a semiring is a tuple ( S, + , 0, · , 1 ) made of a set S and two binary oper-ations + and · such that ( S, + , 0 ) is a commutative monoid, ( S, · , 1 ) is a monoid whose multiplicationoperation · distributes on the left and the right over the addition operation + , and is a left and rightzero for multiplication.A dioid is a semiring S in which addition is idempotent, i.e. for all x ∈ S , we have x + x = x . In thiscase, the relation defined by x ≤ y ⇐⇒ x + y = y, (3.1.2)for all x, y ∈ S , is a partial order on S , with respect to which addition and multiplication are monotone,and is minimal. Where there is no possible confusion, we will often denote multiplication simplyby juxtaposition. A bounded distributive lattice is a dioid ( S, + , 0, · , 1 ) , whose multiplication · iscommutative and idempotent, and x ≤ , for every x ∈ S . Recall from [6] that a domain semiring is a dioid ( S, + , · , 0, 1 ) equipped witha domain operation d : S → S that satisfies the following five axioms for all x, y ∈ S : i) x ≤ d ( x ) x , ii) d ( xy ) = d ( xd ( y )) , iii) d ( x ) ≤ , iv) d ( ) = , v) d ( x + y ) = d ( x ) + d ( y ) ,These structures are called domain semirings and not domain dioids because a semiring equippedwith a domain operation is automatically idempotent. Consequences of the axioms of domain semiringsinclude the fact that the image of S under d is precisely the set of fixpoints of d , i.e. S d := { x ∈ S | d ( x ) = x } = d ( S ) , and that S d forms a distributive lattice with + as join and · as meet, bounded by and . It contains thelargest Boolean subalgebra of S bounded by and . We henceforth write p, q, r, . . . for elements of S d and refer to S d as the domain algebra of S . Moreover, S d is a subsemiring of S in the sense that itselement satisfy the semiring axioms, and are in the set, and the set is closed with respect to · and + .Further properties include d ( ) = d ( px ) = p d ( x ) , x ≤ y ⇒ d ( x ) ≤ d ( y ) , for all x, y ∈ S d , and d commutes with all existing sups [6]. . Higher dimensional modal Kleene algebras3.1.4. Boolean domain semirings. A limitation of domain semirings is that complementation in S d cannot be expressed. This requires a notion of antidomain that abstractly describes those elements thatare not in the domain of a particular element. Recall from [6] that a Boolean domain semiring is a dioid ( S, + , · , 0, 1 ) equipped with an antidomain operation ad : S → S that satisfies the following three axioms,for all x, y ∈ S : i) ad ( x ) x = , ii) ad ( xy ) ≤ ad ( x ad ( y )) , iii) ad ( x ) + ad ( x ) = .Setting d = ad , we recover a domain semiring, that is, d satisfies the domain semiring axioms. In thepresence of the operation ad , the subalgebra S d of all fixpoints of d in S is now the greatest Booleanalgebra in S bounded by and ; we have that S d = ad ( S ) and ad acts as Boolean complementation on S d . We therefore denote the restriction of ad to S d by ¬ . We denote the opposite of a semiring S , in which the order of multiplicationhas been reversed, by S op . It is once again a semiring. A codomain (resp. Boolean codomain ) semiring is a semiring equipped with a map r : S → S (resp. ar : S → S ) such that ( S op , r ) (resp. ( S op , ar ) ) is adomain (resp. Boolean domain) semiring.Consider a semiring equipped with a domain operator and a codomain operation. The domain andrange axioms alone do not imply that S d = S r , let alone the compatibility properties d ( r ( x )) = r ( x ) , r ( d ( x )) = d ( x ) , (3.1.6)for every x in S . Indeed, consider the domain and range semiring S = ( { a } , + , · , 0, 1, d , r ) with additiondefined by , multiplication by a = a , domain by d a = and codomain by r a = a . Wehave S d = { 0, 1 } = { 0, a, 1 } = S r and d ( r a ) = = a = r a . The identity r ◦ d = d fails in the oppositesemiring.Recall from [6] that a modal semiring S is a domain semiring that is also a codomain semiring, andsatisfying the compatibility properties (3.1.6). Boolean domain semirings that are also Boolean codomainsemirings are called Boolean modal semirings . In this case, maximality of S d and S r = { x ∈ S | r ( x ) = x } forces the domain and range algebra of S to coincide, so that the extra axioms (3.1.6) are not necessary.We provide a formal proof, as this fact has so far been overlooked. In every Boolean modal semiring the compatible properties (3.1.6) hold.Proof. Let S be a Boolean modal semiring, and x in S . Then d ( r ( x )) = ( ar ( x ) + r ( x )) d ( r ( x ))= ar ( x ) d ( r ( x )) + r ( x ) d ( r ( x ))( ar ( x ) + r ( x ))= + r ( x ) d ( r ( x )) ar ( x ) + r ( x ) d ( r ( x )) r ( x )= + r ( x ) r ( x ) = r ( x ) , proving the first equality in (3.1.6). In the third step, ar ( x ) d ( r ( x )) = because ar ( x ) r ( x ) = and yz = ⇔ yd ( z ) = hold in any Boolean modal semiring. In the fourth step, r ( x ) d ( r ( x )) ar ( x ) = because .1. Modal Kleene algebras d ( r ( x )) ≤ and again ar ( x ) r ( x ) = . Moreover r ( x ) d ( r ( x )) r ( x ) = r ( x ) r ( x ) because d ( y ) y = y holdsin any modal semiring. The proof of the second equality in (3.1.6) follows by opposition.It then follows that S d = S r : if d ( x ) = x , then r ( x ) = r ( d ( x )) = d ( x ) = x ; and r ( x ) = x implies d ( x ) = x by opposition. This forces that S d = S r = { x ∈ S | r ( x ) = x } . A Kleene algebra is a dioid K equipped with an operation (−) ∗ : K → K called Kleene star , satisfying the following axioms. For all x, y, z ∈ K , i) ( unfold axioms ) + xx ∗ ≤ x ∗ and + x ∗ x ≤ x ∗ , ii) ( induction axioms ) z + xy ≤ y ⇒ x ∗ z ≤ y and z + yx ≤ y ⇒ zx ∗ ≤ y .Note that the axioms on the left are the opposites of those on the right. Useful consequences ofAxioms i) and ii) include the following identities for all x, y ∈ K , and i ∈ N , x i ≤ x ∗ x ∗ x ∗ = x ∗ x ∗∗ = x ∗ x ( yx ) ∗ = ( xy ) x ∗ ( x + y ) ∗ = x ∗ ( yx ∗ ) ∗ = ( x ∗ y ∗ ) ∗ , where x i denotes the i -fold multiplication of x with itself, as well as the quasi-identities x ≤ ⇒ x ∗ = ≤ y ⇒ x ∗ ≤ y ∗ xz ≤ zy ⇒ x ∗ z ≤ zy ∗ zx ≤ yz ⇒ zx ∗ ≤ y ∗ z. The Kleene plus is the operation (−) + : K → K defined by x + = xx ∗ .The above notions of domain and codomain extend to Kleene algebras without having to add anyfurther axioms. We thus define a (Boolean) modal Kleene algebra as a Kleene algebra that is also a(Boolean) modal-semiring. Let ( S, + , · , 0, 1, d , r ) be a modal semiring. For x ∈ S and p ∈ S d , we definethe modal diamond operators: | x i p = d ( xp ) , h x | p = r ( px ) . (3.1.10)When S is a Boolean modal semiring, we additionally define modal box operators: | x ] p = ¬| x i ( ¬ p ) , [ x | p = ¬ h x | ( ¬ p ) . (3.1.11)These are modal operators in the sense of Boolean algebras with operators [19] because the followingidentities hold: | x i ( p + q ) = | x i p + | x i q, | x i = h x | ( p + q ) = h x | p + h x | q, h x | = and | x ]( pq ) = | x ] p + | x ] q, | x ] = [ x | ( pq ) = [ x | p + [ x | q, [ x | = It is easy to see that | − i and h − | , as well as | −] and [− | are related by opposition. In a (Boolean) modalKleene algebra, this can be expressed by the conjugation laws | x i p · q = ⇔ p · h x | q = and | x ] p + q = ⇔ p + [ x | q = In a Boolean modal semiring, boxes and diamonds are related by De Morgan duality by their definition(3.1.11) and additionally by | x i p = ¬| x ]( ¬ p ) , h x | p = ¬ [ x | ( ¬ p ) . (3.1.12) . Higher dimensional modal Kleene algebras Finally, boxes and diamonds are adjoints in Galois connections, expressed by the following relations | x i p ≤ q ⇔ p ≤ [ x | q and h x | p ≤ q ⇔ p ≤ | x ] q. As a consequence, diamonds preserve all existing sups in S , whereas boxes reverse all existing infs tosups, and all modal operators are order preserving. Finally, we mention the properties | xy i = | x i ◦ | y i , h xy | = h y | ◦ h x | , | xy ] = | x ] ◦ | y ] and [ xy | = [ y | ◦ [ x | . For any set X , the structure ( P ( X × X ) , ∪ , ; , ∅ X , Id X , (−) ∗ ) forms a Kleene algebra, called the full relation Kleene algebra over X . The operation ; is the relationalcomposition defined by ( a, b ) ∈ R ; S if and only if ( a, c ) ∈ R and ( c, b ) ∈ S , for some c ∈ X . Therelation Id X = { ( a, a ) | a ∈ X } is the identity relation on X and (−) ∗ the reflexive transitive closureoperation defined, for R = Id X and R i + = R ; R i , by R ∗ = [ i ∈ N R i , The subidentity relations below Id X form its greatest Boolean subalgebra between ∅ X and Id X , which isisomorphic to the power set algebra P ( X ) . Every subalgebra of a full relation Kleene algebra is a relationKleene algebra .The full relation Kleene algebra over X extends to a full relation Boolean modal Kleene algebra over X by defining d ( R ) = { ( a, a ) | ∃ b ∈ X. ( a, b ) ∈ R } and r ( R ) = { ( a, a ) | ∃ b. ( b, a ) ∈ R } . The antidomain and anticodomain maps are then given by complementation ad ( R ) = Id X \ d ( R ) and ar ( R ) = Id X \ r ( R ) , and it is straightforward to check that | R i P = { ( a, a ) | ∃ b ∈ X. ( a, b ) ∈ R ∧ ( b, b ) ∈ P } , | R ] P = { ( a, a ) | ∀ b ∈ X. ( a, b ) ∈ R ⇒ ( b, b ) ∈ P } , which corresponds to the standard relational Kripke semantics of boxes and diamonds. Similar expressionsfor the backward modalities are obtained by swapping ( a, b ) to ( b, a ) in the above expressions. Let P ∗ be the free -category generated by the -polygraph P = ( P , P ) . Then ( P ( P ∗ ) , ∪ , ⊙ , ∅ , , (−) ∗ ) forms a Kleene algebra, the full path (Kleene) algebra K ( P ) over P . Here, composition is defined as φ ⊙ ψ = { u ⋆ v | u ∈ φ ∧ v ∈ ψ ∧ t ( u ) = s ( v ) } for any φ, ψ ∈ P ( P ∗ ) , and is the set of all identity arrows of P . The Kleene star can be defined as φ ∗ = [ i ∈ N φ i .2. n -Dimensional globular Kleene algebras where φ = and φ i + = φ ⊙ φ i . Every subalgebra of the full path Kleene algebra over P is a pathKleene algebra .The full path algebra over P can be extended to a full path Boolean modal Kleene algebra over P bydefining d ( φ ) = { s ( u ) | u ∈ φ } and r ( φ ) = { t ( u ) | u ∈ φ } where x denotes the identity arrow on the object x ∈ P . The antidomain and anticodomain maps arethen given by complementation ad ( φ ) = \ d ( φ ) and ar ( φ ) = \ r ( φ ) . It is then easy to check that | φ i p = { s ( u ) | u ∈ φ ∧ t ( u ) ∈ p } and | φ ] p = { s ( u ) | u ∈ φ ⇒ t ( u ) ∈ p } where p ⊆ is some set of identity arrows. Once again, similar expressions for backward modalities canbe obtained by swapping source and target functions in the right places.The relational and the path model are very similar. In fact the relational model can be obtained fromthe path model by applying a suitable homomorphism of modal Kleene algebras. n -Dimensional globular Kleene algebras We now extend the definitions in the previous sections to a notion of globular n -dimensional modalKleene algebra. First, we define a notion of n -dimensional dioid satisfying lax interchange laws betweenmultiplication operations of different dimensions, similar to those of concurrent Kleene algebra [16]. Wethen extend it with domain operations and add further axioms that capture globularity. Finally we equipthese algebras with star operations for each dimension and impose novel lax interchange laws betweencompositions and stars of different dimensions. n -Dioid. We define a -dioid as a bounded distributive lattice and a -dioid as a dioid. Moregenerally, for n ≥ , an n -dioid is a structure ( S, + , 0, ⊙ i , 1 i ) ≤ i This notation simplifies the reading of proofs when elements of different dimensions areinteracting. For a natural number k ≥ , the k -fold i -multiplication of an element A of S , for ≤ i < n ,is defined by A i = i , A k i = A ⊙ i A ( k − ) i . The axioms ii) and iii) from Section 3.2.1 for n -dioids provide the basic algebraic structure forreasoning about higher-dimensional rewriting systems. Indeed, the dependencies between multiplicationsof different dimensions expressed by the lax interchange laws capture the lifting of the equationalinterchange law for n -categories, while the idempotence of i -multiplication for the j -unit expressescompleteness of the set of j -dimensional cells in an n -category with respect to i -composition. In thisway, these axioms begin to capture the higher dimensional character of polygraphs, as is made clear inSection 3.3.1, in which we provide a model of this structure based on polygraphs. The domain axiom ii) from Section 3.2.4 further captures characteristics of dimension, which are expressed abstractly in thefollowing proposition. For n ≥ , in any domain n -semiring S , for all ≤ i < j < n , the followingconditions hold: i) d j ◦ d i = d i , ii) d j ( i ) = i , iii) i ≤ j , iv) S i ⊆ S j , v) ( S j , + , 0, ⊙ i , 1 i , d i ) is a domain sub semiring of ( S, + , 0, ⊙ i , 1 i , d i ) and d i ( S j ) = S i , vi) ( S j , + , 0, ⊙ k , 1 k , d k ) ≤ k ≤ i is a domain sub ( i + ) -semiring of ( S, + , 0, ⊙ k , 1 i , d k ) ≤ k ≤ i . vii) ( S i , + , 0, ⊙ i , 1 i ) is a -dioid.Proof. The first identity is proved by a simple induction on axiom ii) in (3.2.4). The second one quicklyfollows, since d i ( i ) = i follows from the domain semiring axioms, and thus d j ( i ) = i using i) . Thethird identity is again a direct consequence, since by ii) we know that i ∈ S j , and that j is the greatestelement of S j . The fourth one follows since x ∈ S i if, and only if, d i ( x ) = x , which is equivalent to d j ( x ) = x by ii) . The fifth identity is verified by noticing that the inclusion S j ֒ → S is a morphism .2. n -Dimensional globular Kleene algebras of domain semirings with the operation ⊙ i . Furthermore, since d i ( S j ) ⊆ S i and S i ⊆ S j , we have d i ( S j ) = S i . Noticing that, in fact, S j ֒ → S is a morphism of domain semirings with the operation ⊙ k forany ≤ k ≤ i gives us vi) . The final result follows from basic properties of domain semirings.Given an n -semiring S , we denote by S op the n -semiring in which the order of each multiplicationoperation has been reversed. An n -semiring S is a codomain n -semiring if S op is a domain n -semiring.The codomain operators are denoted by r i . A modal n -semiring is an n -semiring with domains andcodomains, in which the coherence conditions d i ◦ r i = r i and r i ◦ d i = d i hold for all ≤ i < n . Section 3.1.14 recalls that the path algebra K ( P ) defined as the power set of -cells inthe free category generated by a -polygraph P = ( P , P ) is a model of modal -semiring. The domainalgebra K ( P ) d is isomorphic to the power set of P . As recalled in Section 3.1.3, in the general case ofa domain semiring ( S, + , 0, · , 1, d ) , the domain algebra S d forms a bounded distributive lattice with + as join, · as meet, as bottom and as top. It is for this reason that we consider a -dioid as a boundeddistributive lattice. Indeed, the idempotence and commutativity of the multiplication operation simulatethe properties of a set of identity -cells.Note also that, in Section 3.3, we will construct higher-dimensional path algebras over n -polygraphsand show that these form models of modal n -semirings. In this case it makes sense that ( S i , + , 0, ⊙ i , 1 i ) is a -dioid, since an i -cell f : u → v of an n -category C is a -cell in the hom-category C ( u, v ) . Let S be a modal n -semiring. We introduce forward and backward i -diamond operators defined via (co-)domain operators in each dimension by analogy to (3.1.5). For any ≤ i < n , A ∈ S and φ ∈ S i , we define | A i i ( φ ) = d i ( A ⊙ i φ ) , and h A | i ( φ ) = r i ( φ ⊙ i A ) . (3.2.8)In the absence of antidomains, box operators cannot be expressed in this setting. These diamond operatorshave all of the properties recalled in Section 3.1.9 with respect to i -multiplication and elements of S i . p -Boolean domain semirings. For ≤ p < n , a domain n -semiring ( S, + , 0, ⊙ i , 1 i , d i ) ≤ i 4. Algebraic coherent confluence In this section, we present proofs of the coherent Church-Rosser theorem and coherent Newman’s lemmain the setting of higher-dimensional globular Kleene algebras. These constitute the main results of thisarticle. First, we recall from [5, 28, 29] abstract rewriting properties formulated in modal Kleene algebras.We then formalise notions from higher-dimensional rewriting needed to prove our results, introducing fillers in the setting of globular modal n -Kleene algebras, which correspond to the notion of fillers forpolygraphs defined in Section 2.3.1. We also define the notion of whiskering in modal n -Kleene algebras,analogous to the polygraphic definition in Section 2.1.4 and describe the properties thereof needed for ourproofs. The coherent Church-Rosser theorem is dealt with in Section 4.2, first in Proposition 4.2.7 usingclassical induction and then in Theorem 4.2.8 using only the induction axioms provided by the Kleenestar. In Section 4.3, we define notions of termination and well-foundedness in globular modal p -BooleanKleene algebras and prove Theorem 4.3.2, the coherent Newman’s lemma. Let K be a modal Kleene algebra. An element x ∈ K terminates , or is Noetherian , [5] if p ≤ | x i p ⇒ p = holds for all p ∈ K d . The set of Noetherian elements of K is denoted by N ( K ) . Using the Galoisconnections (3.1.9) yields the following equivalent characterisation: x ∈ K is Noetherian if and only if | x ] p ≤ p ⇒ p = holds for all p ∈ K d . The notions of local confluence, confluence and the Church-Rosser propertyfor rewriting systems can be captured in Kleene algebras as follows. Given elements x, y ∈ K , we saythat the ordered pair ( x, y ) semi-commutes (resp. semi-commutes locally ) if x ∗ · y ∗ ≤ y ∗ · x ∗ ( resp. x · y ≤ y ∗ · x ∗ ) . We say that the ordered pair ( x, y ) semi-commutes modally (resp. semi-commutes locally modally ) if | x ∗ i ◦ | y ∗ i ≤ | y ∗ i ◦ | x ∗ i ( resp. | x i ◦ | y i ≤ | y ∗ i ◦ | x ∗ i ) . It is obvious that (local) commutation implies (local) modal commutation, but not vice versa. Finally wesay that ( x, y ) has the Church-Rosser property if ( x + y ) ∗ ≤ y ∗ x ∗ .26 .2. A coherent Church-Rosser theorem4.1.3. Confluence results in Kleene algebras. The Church-Rosser theorem and Newman’s lemma forabstract rewriting systems are instances of the following formulations in modal Kleene algebras. In thefollowing subsections we prove higher-dimensional generalisations of these results.The Church-Rosser theorem in K [28, Thm. 4] states that, for any x, y ∈ K , the following holds: x ∗ y ∗ ≤ y ∗ x ∗ ⇔ ( x + y ) ∗ ≤ y ∗ x ∗ . Newman’s Lemma in K , with K d a complete Boolean algebra [5], states that for any x, y ∈ K suchthat ( x + y ) ∈ N ( K ) , the following holds: | x i ◦ | y i ≤ | y ∗ i ◦ | x ∗ i ⇔ | x ∗ i ◦ | y ∗ i ≤ | y ∗ i ◦ | x ∗ i . Let K be a globular n -modal Kleene algebra and ≤ i < j < n . Before defining fillers in globularmodal n -Kleene algebras, we first recall the intuition behind the forward diamond operators, defined inSection 3.2.7. Given A ∈ K and φ, φ ′ ∈ K j , recall that by definition | A i j ( φ ) ≥ φ ′ = d j ( A ⊙ j φ ) ≥ φ ′ . In terms of quantification over sets of cells, as for example in the polygraphic model, this signifies that for every element u of φ ′ , there exist elements v of φ and α of A such that the j -source (resp. j -target)of α is u (resp. v ). This observation motivates the definitions in the following paragraph. Given elements φ and ψ of K j , we say that an element A in K is a i) local i -confluence filler for ( φ, ψ ) if | A i j ( ψ ∗ i ⊙ i φ ∗ i ) ≥ φ ⊙ i ψ, ii) left (resp. right ) semi- i -confluence filler for ( φ, ψ ) if | A i j ( ψ ∗ i ⊙ i φ ∗ i ) ≥ φ ⊙ i ψ ∗ i , (resp. | A i j ( ψ ∗ i ⊙ i φ ∗ i ) ≥ φ ∗ i ⊙ i ψ ) , iii) i -confluence filler for ( φ, ψ ) if | A i j ( ψ ∗ i ⊙ i φ ∗ i ) ≥ φ ∗ i ⊙ i ψ ∗ i , iv) i -Church-Rosser filler for ( φ, ψ ) if | A i j ( ψ ∗ i ⊙ i φ ∗ i ) ≥ ( ψ + φ ) ∗ i . In any n -Kleene algebra, the following inequalities hold: ( ψ + φ ) ∗ i ≥ φ ∗ i ⊙ i ψ ∗ i ≥ φ ⊙ i ψ. We may therefore deduce that an i -Church-Rosser filler for ( φ, ψ ) is an i -confluence filler for ( φ, ψ ) andthat an i -confluence filler for ( φ, ψ ) is a local i -confluence filler for ( φ, ψ ) . . Algebraic coherent confluence4.2.2. Remarks. Conditions on the domain and codomain in the above definitions imply an i -dimensionalglobular character of the pair ( φ, ψ ) in the sense that we have the relation | φ ∗ i ⊙ i ψ ∗ i i i ( p ) ≤ | ψ ∗ i ⊙ i φ ∗ i i i ( p ) for all p ∈ K i . Indeed, writing A ′ = A ⊙ j ( ψ ∗ i ⊙ i φ ∗ i ) , we have | φ ∗ i ⊙ i ψ ∗ i i i ( p ) = d i ( φ ∗ i ⊙ i ψ ∗ i ⊙ i p ) ≤ d i ( d j ( A ′ ) ⊙ i p )= d i ( d j ( A ′ ⊙ i p ))= d i ( r j ( A ′ ⊙ i p ))= d i ( r j ( A ′ ) ⊙ i p ) ≤ d i (( ψ ∗ i ⊙ i φ ∗ i ) ⊙ i p ) = | ψ ∗ i ⊙ i φ ∗ i i i ( p ) , where the first step holds by definition of diamonds, the second by the fact that A is an i -confluencefiller and by monotonicity of d i , the third, fourth and fifth by the globularity relations (3.2.14),(3.2.12)and (3.2.15) respectively. The final inequality follows because d ( p · x ) = p · d ( x ) holds in modal Kleenealgebra (see the end of Section 3.1.3). In the case of codomains, its dual implies that r j ( A ′ ) = r j ( A ⊙ j ( ψ ∗ i ⊙ i φ ∗ i )) = r j ( A ) ⊙ j r j ( ψ ∗ i ⊙ i φ ∗ i ) ≤ r j ( ψ ∗ i ⊙ i φ ∗ i ) . The final step is again by definition of the diamond operators. Similar results hold in the case of localand semi-confluence fillers. Thus, φ and ψ commute modally (resp. locally modally) with respect to i -multiplication. For this reason, the confluence filler (local confluence filler) defined in (4.2.1) can berepresented graphically as follows φ ∗ i { { ✇✇✇✇✇✇✇ ψ ∗ i ●●●●●●● A (cid:5) (cid:25) ψ ∗ i (cid:26) (cid:26) ✹✹✹✹✹✹✹✹✹✹ φ ∗ i (cid:4) (cid:4) ✡✡✡✡✡✡✡✡✡✡ { { φ { { ✇✇✇✇✇✇✇ ψ ●●●●●●● A (cid:5) (cid:25) ψ ∗ i (cid:26) (cid:26) ✹✹✹✹✹✹✹✹✹✹ φ ∗ i (cid:4) (cid:4) ✡✡✡✡✡✡✡✡✡✡ { { Let K be a globular modal n -Kleene algebra. Given ≤ i < j < n and φ ∈ K j , the right (resp. left ) i -whiskering of an element A ∈ K by φ is the element A ⊙ i φ ( resp. φ ⊙ i A ) In what follows, we list properties of whiskering and define completions. i) Firstly, it holds that i -whiskering by j -dimensional cells commutes with j -modalities. Indeed, for all A ∈ K and ≤ i < j < n and all φ, ψ, φ ′ , ψ ′ , γ ∈ K j such that φ ′ ≤ φ , ψ ′ ≤ ψ , and d j ( A ) ≤ γ ,we have: φ ′ ⊙ i h A | j ( γ ) ⊙ i ψ ′ = h φ ′ ⊙ i A ⊙ i ψ ′ | j ( φ ⊙ i γ ⊙ i ψ ) (4.2.4)To see this, consider the deductions d j ( φ ′ ⊙ i A ⊙ i ψ ′ ) = φ ′ ⊙ i d j ( A ) ⊙ i ψ ′ ≤ φ ⊙ i γ ⊙ i ψ ⇒ φ ′ ⊙ i A ⊙ i ψ ′ = ( φ ⊙ i γ ⊙ i ψ ) ⊙ j ( φ ′ ⊙ i A ⊙ i ψ ′ ) ⇒ r j ( φ ′ ⊙ i A ⊙ i ψ ′ ) = h φ ′ ⊙ i A ⊙ i ψ ′ | j ( φ ⊙ i γ ⊙ i ψ ) ,28 .2. A coherent Church-Rosser theorem where in the first line we have used the globularity axiom from (3.2.14), as well as the hypothesis d j ( A ) ≤ γ , and in the second one the fact that d ( x ) ≤ p ⇒ px = x , a consequence of axiom i) fromSection 3.1.3. Applying r j to the second line and using the definition of modalities (3.2.8) yields thethird. Again, since d j ( A ) ≤ γ , we have A = γ ⊙ j A , whereby we deduce r j ( φ ′ ⊙ i A ⊙ i ψ ′ ) = φ ′ ⊙ i r j ( γ ⊙ j A ) ⊙ i ψ ′ = φ ′ ⊙ i h A | j ( γ ) ⊙ i ψ ′ , using Axiom (3.2.15). This gives identity (4.2.4), and we have a dual identity for the forwarddiamond. ii) Secondly, we define completions of elements by whiskering. Let A be an i -confluence filler of a pair ( φ, ψ ) of elements in K j . The j -dimensional i -whiskering of A is the following element of K : ( φ + ψ ) ∗ i ⊙ i A ⊙ i ( φ + ψ ) ∗ i . (4.2.5)The j -star of this element is called the i -whiskered j -completion of A . iii) Finally, we have that the i -whiskered j -completion of a confluence filler A , which in the followingparagraph we denote by ^ A , absorbs whiskers, i.e. for any ξ ≤ ( φ + ψ ) ∗ i ξ ⊙ i ^ A ∗ j ≤ ^ A ∗ j and ^ A ∗ j ⊙ i ξ ≤ ^ A ∗ j . (4.2.6)Indeed, by definition of ^ A , we have ξ ⊙ i ^ A ≤ ^ A ≥ ^ A ⊙ i ξ for any ξ ≤ ( φ + ψ ) ∗ i . Using the fact that (−) ∗ j is a lax morphism with respect to i -whiskering by j -dimensional elements, see Section 3.1.8, we deduce ξ ⊙ i ^ A ∗ j ≤ ( ξ ⊙ i ^ A ) ∗ j ≤ ^ A ∗ j , where the last inequality holds by monotonicity of (−) ∗ j . A similar proof shows that ^ A ∗ j ⊙ i ξ ≤ ^ A ∗ j . n -MKA (by induction)). Let K be a globular modal n -Kleene algebra and ≤ i < j < n . Given φ, ψ ∈ K j , an i -confluence filler A of ( φ, ψ ) and any natural number k ≥ , there exists an A k ≤ ^ A ∗ j such that i) r j ( A k ) ≤ ψ ∗ i φ ∗ i , ii) d j ( A k ) ≥ ( φ + ψ ) k i ,where ^ A is the j -dimensional i -whiskering of A .Proof. In this proof, juxtaposition of elements denotes i -multiplication. We reason by induction on k ≥ . For k = , we may take A = i . Indeed, i ≤ j ≤ ^ A ∗ j . 29 . Algebraic coherent confluence Furthermore, we have d j ( A ) = i = ( φ + ψ ) i and r j ( A ) = i ≤ ψ ∗ i φ ∗ i . Supposing that A k − isconstructed, we set A k = (( φ + ψ ) A k − ) ⊙ j ( A ′ φ ∗ i ) , where A ′ = A ⊙ j ( ψ ∗ i φ ∗ i ) . We first show that d j ( A k ) ≥ ( φ + ψ ) k i as follows d j ( A k ) = d j ((( φ + ψ ) A k − ) ⊙ j ( A ′ φ ∗ i )) , = d j ((( φ + ψ ) A k − ) ⊙ j d j ( A ′ φ ∗ i )) , = d j ((( φ + ψ ) A k − ) ⊙ j d j ( A ′ ) φ ∗ i ) , ≥ d j ((( φ + ψ ) A k − ) ⊙ j φ ∗ i ψ ∗ i φ ∗ i ) , = d j (( φ + ψ ) A k − ) , = ( φ + ψ ) d j ( A k − ) , = ( φ + ψ )( φ + ψ ) ( k − ) i , = ( φ + ψ ) k i , · ( φ + ψ ) / / ψ ∗ i (cid:27) (cid:27) ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ · o o ( φ + ψ ) k − / / ψ ∗ i ✾✾✾✾✾✾✾✾✾ (cid:28) (cid:28) ✾✾✾✾✾✾✾✾✾ A ′ φ ∗ i ( φ + ψ ) A k − · C C φ ∗ i ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ · C C φ ∗ i ✞✞✞✞✞✞✞✞✞ · where the first step is given by definition of A k , the second by axiom ii) from (3.1.3), the third by globu-larity (3.2.14). The inequality in the fourth step is by hypothesis that A is an i -confluence filler, and thefifth is a consequence of the fact that (( φ + ψ ) A k − ) ⊙ ( φ ∗ i ψ ∗ i φ ∗ i ) = ( φ + ψ ) A k − , which is in turn a consequence of the following: r j (( φ + ψ ) A k − ) = ( φ + ψ ) r j ( A k − ) ≤ φ ∗ i ψ ∗ i φ ∗ i . The sixth step is again a consequence of globularity (3.2.14), the seventh follows from the inductionhypothesis, and the last equality is by definition of the k -fold i -multiplication.Now we show r j ( A k ) ≤ ψ ∗ i φ ∗ i : r j ( A k ) = r j ((( φ + ψ ) A k − ) ⊙ j ( A ′ φ ∗ i ))= r j ( r j (( φ + ψ ) A k − ) ⊙ j ( A ′ φ ∗ i )) ≤ r j (( φ + ψ ) ψ ∗ i φ ∗ i ⊙ j ( A ′ φ ∗ i )) ≤ r j (( φ ∗ i ψ ∗ i φ ∗ i ) ⊙ j ( A ′ φ ∗ i ))= r j ( d j ( A ′ φ ∗ i ) ⊙ j ( A ′ φ ∗ i ))= r j ( A ′ ) φ ∗ i ≤ ψ ∗ i φ ∗ i φ ∗ i = ψ ∗ i φ ∗ i . The first equality holds by definition of A k , the second by axiom ii) from Section 3.1.3 (for codomains),the third by the induction hypothesis, the fourth by φ ≤ φ ∗ i and ψψ ∗ i = ψ ∗ i . The fifth step holdssince A is an i -confluence filler, the sixth by the fact that d ( x ) · x = x , a consequence of axiom i) fromSection 3.1.3. Finally, as recalled in Section 4.2.2, r j ( A ′ ) = r j ( A ⊙ j ( ψ ∗ i ⊙ i φ ∗ i )) = r j ( A ) ⊙ j r j ( ψ ∗ i ⊙ i φ ∗ i ) ≤ r j ( ψ ∗ i ⊙ i φ ∗ i ) ,30 .2. A coherent Church-Rosser theorem which gives step seven since ψ ∗ i ⊙ i φ ∗ i ∈ K j . The final step is due to φ ∗ i ⊙ i φ ∗ i = φ ∗ i , a consequenceof the Kleene star axioms.To conclude, we must also show that A k ≤ ^ A ∗ j . By whisker absorption, described in (4.2.3), and thefact that A ′ ≤ A ≤ ^ A , we have A ′ φ ∗ i ≤ ^ Aφ ∗ i = ^ A, and ( φ + φ ) A k − ≤ ( φ + ψ ) ^ A ∗ j ≤ ^ A ∗ j . Thus, A k = (( φ + ψ ) A k − ) ⊙ j ( Aφ ∗ i ) ≤ ^ A ∗ j ⊙ j ^ A ∗ j = ^ A ∗ j , which completes the proof.We now reprove this theorem using the implicit induction of Kleene algebra. n -MKA). Let K be a globular n -modal Kleenealgebra and ≤ i < j < n . Given φ, ψ ∈ K j and an i -confluence filler A ∈ K of ( φ, ψ ) , we have | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≥ ( φ + ψ ) ∗ i , where ^ A is the j -dimensional i -whiskering of A . Thus ^ A ∗ j is an i -Church-Rosser filler for ( φ, ψ ) .Proof. As in the previous proof, i -multiplication will be denoted by juxtaposition. Let φ, ψ be in K j , for , and A in K be an i -confluence filler of ( φ, ψ ) , with ≤ i < j . By the left i -star inductionaxiom, see Section 3.1.8, we have i + ( φ + ψ ) | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≤ | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ⇒ ( φ + ψ ) ∗ i ≤ | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) The inequality i ≤ ψ ∗ i φ ∗ i ≤ | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) holds. Indeed, by the unfold axiom from Section 3.1.8,we have i ≤ ψ ∗ i , i ≤ φ ∗ i , giving the first inequality, and j ≤ ^ A ∗ j . The latter implies that id S d j = | j i j ≤ | ^ A ∗ j i j , which gives ψ ∗ i φ ∗ i ≤ | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) . It then remains to show that ( φ + ψ ) | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≤ | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) . By distributivity, we may prove this for each of the summands: − In the case of whiskering by φ on the left: φ | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≤ | φ ^ A ∗ j i j ( φψ ∗ i φ ∗ i ) ≤ | φ ^ A ∗ j i j ( | A i j ( ψ ∗ i φ ∗ i ) φ ∗ i ) ≤ | φ ^ A ∗ j i j ( | Aφ ∗ i i j ( ψ ∗ i φ ∗ i φ ∗ i )) ≤ | φ ^ A ∗ j ⊙ j Aφ ∗ i i j ( ψ ∗ i φ ∗ i ) ≤ | ^ A ∗ j ⊙ j ^ A i j ( ψ ∗ i φ ∗ i ) ≤ | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) · φ / / ψ ∗ i (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ · o o ( φ + ψ ) ∗ i / / ψ ∗ i ❇❇❇❇❇❇❇ ❇❇❇❇❇❇❇ Aφ ∗ i φ ^ A ∗ j · > > φ ∗ i ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ · ? ? φ ∗ i ⑧⑧⑧⑧⑧⑧⑧ · The first step is given by whiskering properties from (4.2.3), the second by the hypothesis that A is an i -confluence filler and that φψ ∗ i ≤ φ ∗ i ψ ∗ i . The third step is again by whiskering, and thefourth follows by definition of diamonds and axiom ii) from (3.1.3). The fifth follows by whiskerabsorption, (4.2.3), and the last step follows from the unfold axiom from (3.1.8), since it impliesthat x · x ∗ ≤ x ∗ . . Algebraic coherent confluence − In the case of whiskering by ψ on the right: ψ | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≤ | ψ ^ A ∗ j i j ( ψψ ∗ i φ ∗ i ) ≤ | ψ ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≤ | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) . · ψ / / ψ ∗ i * * · o o ( φ + ψ ) ∗ i / / ψ ∗ i ❇❇❇❇❇❇❇ ❇❇❇❇❇❇❇ j ψ ^ A ∗ j · > > φ ∗ i ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ ·· The first step is again by whiskering properties from Section 4.2.3, the second by the fact that ψψ ∗ i ≤ ψ ∗ i which as explained above is a consequence of the unfold axiom recalled in Section 3.1.8. Finally, whiskerabsorption justifies the last inequality. Note that in Theorem 4.2.7, the elements A k verify | A k i j ( ψ ∗ i φ ∗ i ) ≥ ( φ + ψ ) k i ,meaning that scanning backward along A k from ψ ∗ i φ ∗ i , we see at least all of the "zig-zags" in φ and ψ of length k , whereas in Theorem 4.2.8, the inequality | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≥ ( φ + ψ ) ∗ i means that scanningback from ψ ∗ i φ ∗ i , we see at least all of the zig-zags in φ and ψ of any length. However, the elements A k from Theorem 4.2.7 satisfy in addition h A k | j (( φ + ψ ) k i ) ≤ ψ ∗ i φ ∗ i . This formulation is of interest, since it coincides with the intuition of paving from zigzags ( φ + ψ ) k i to the confluences ψ ∗ i φ ∗ i . However, this sort of inequality cannot be expected of the j -dimensional i -completion of A , since in general, using the path algebra intuition, ^ A ∗ j contains cells which go fromzigzags to zigzags. In conclusion, the fact that the diamonds scan all possible future or past states meansthat we must formulate as in Theorem 4.2.8 when considering completions, or construct the elementspaving precisely what we would like as in Theorem 4.2.7. Let K be a globular modal n -Kleene algebra. Given φ, ψ ∈ K j , for i < j < n , forany semi- i -confluence filler A ∈ K we have | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≥ ( φ + ψ ) ∗ i , where ^ A is the j -dimensional i -whiskering of A .Proof. In the case of a left semi-confluence filler, the proof is identical. If A is a right semi-confluencefiller, we use the right i -star axiom and the proof is given by symmetry. n -Kleene algebra n -semirings. We define the notion of termination, or Noethericity, in a modal n -semiring K as an extension of the notion of termination in modal Kleene algebras, recalled in (4.1.1).Given ≤ i < j < n , an element φ ∈ K j is said to be i -Noetherian or i -terminating if p ≤ | φ i i p ⇒ p ≤ .3. Newman’s lemma in globular modal n -Kleene algebra holds for all p ∈ K i . The set of i -Noetherian elements of K is denoted by N i ( K ) . When K is a modal p -Boolean semiring, we recall that as a consequence of the adjunction between diamonds and boxes, seeSection 3.1.9, we obtain an equivalent formulation of Noethericity in terms of the forward box operator: φ ∈ N i ( K ) ⇐⇒ ∀ p ∈ K i , | φ ] i p ≤ p ⇒ i ≤ p. We also define a notion of well-foundedness ; φ is said to be i -well-founded if it is i -Noetherian in theopposite n -semiring of K . p -Boolean MKA). Let K be a globular p -Boolean modal Kleene algebra, and ≤ i ≤ p < j < n , such that i) ( K i , + , 0, ⊙ i , 1 i , ¬ i ) is a complete Boolean algebra, ii) K j is continuous with respect to i -restriction, i.e. for all ψ, ψ ′ ∈ K j and every family ( p α ) α ∈ I ofelements of K i such that sup I ( p α ) exists, we have ψ ⊙ i sup I ( p α ) ⊙ i ψ ′ = sup I ( ψ ⊙ i p α ⊙ i ψ ′ ) . Let ψ ∈ K j be i -Noetherian and φ ∈ K j i -well-founded. If A is a local i -confluence filler for ( φ, ψ ) , then | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≥ φ ∗ i ψ ∗ i , i.e. ^ A ∗ j is a confluence filler for ( φ, ψ ) .Proof. We denote i -multiplication by juxtaposition. First, we define a predicate expressing restricted j -paving. Given p ∈ K i , let RP ( p ) ⇔ | ^ A ∗ i j ( ψ ∗ i φ ∗ i ) ≥ φ ∗ i pψ ∗ i . By completeness of K i , we may set r := sup { p | RP ( p ) } . By continuity of i -restriction, we may infer RP ( r ) . Furthermore, by downward closure of RP , we have the following equivalence: RP ( p ) ⇐⇒ p ≤ r. This in turn allows us to make the following deductions: ∀ p. ( RP ( | φ i i p ) ∧ RP ( h ψ | i p ) ⇒ RP ( p )) ⇔ ∀ p. ( | φ i i p ≤ r ∧ h ψ | i p ≤ r ⇒ p ≤ r ) ⇔ ∀ p. ( p ≤ [ φ | i r ∧ p ≤ | ψ ] i r ⇒ p ≤ r ) ⇔ [ φ | i r ≤ r ∧ | ψ ] i r ≤ r Thus, it suffices to show ∀ p. ( RP ( | φ i i p ) ∧ RP ( h ψ | i p ) ⇒ RP ( p )) in order to conclude that r = i , byNoethericity (resp. well-foundedness) of ψ (resp. φ ).Let p ∈ K i , set | φ i i ( p ) = p φ and h ψ | i ( p ) = p ψ and suppose that RP ( p φ ) and RP ( p ψ ) hold. Notethat we have φp = d i ( φp ) φp = | φ i i ( p ) φp ≤ p φ φ, 33 . Algebraic coherent confluence since d ( x ) x = x by axiom i) from Section 3.1.3 and p ≤ i . We have a similar inequality for ψ , that is pψ ≤ ψp ψ . These inequalities, along with the unfold axioms from Section 3.1.8, give φ ∗ i pψ ∗ i ≤ φ ∗ i p + φ ∗ i φpψψ ∗ i + pψ ∗ i ≤ φ ∗ i p + φ ∗ i p φ φψp ψ ψ ∗ i + pψ ∗ i . p · φ ∗ i H H ✒✒✒✒✒✒✒✒✒✒✒✒ p ψ (cid:22) (cid:22) ✲✲✲✲✲✲ φ H H ✑✑✑✑✑✑ ψ ∗ i (cid:21) (cid:21) ✱✱✱✱✱ · φ ∗ i I I ✒✒✒✒✒ · p ψ ∗ i (cid:22) (cid:22) ✱✱✱✱✱✱✱✱✱✱✱✱ · The outermost summands are below | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) . Indeed, id S j = | j i j ≤ | ^ A ∗ j i j since j ≤ ^ A ∗ j , p ≤ i and φ ∗ i , ψ ∗ i ≤ ψ ∗ i φ ∗ i .For the middle summand, we calculate φ ∗ i p φ φψp ψ ψ ∗ i ≤ φ ∗ i p φ | A i j ( ψ ∗ i φ ∗ i ) p ψ ψ ∗ i ≤ | φ ∗ i p φ Ap ψ ψ ∗ i i j ( φ ∗ i p φ ψ ∗ i φ ∗ i p ψ ψ ∗ i ) ≤ | φ ∗ i p φ ^ Ap ψ ψ ∗ i i ( | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) φ ∗ i p ψ ψ ∗ i ) ≤ | ^ A i ( | ^ A ∗ j φ ∗ i p ψ ψ ∗ i i j ( ψ ∗ i φ ∗ i p ψ ψ ∗ i )) ≤ | ^ A ⊙ j ^ A ∗ j φ ∗ i p ψ ψ ∗ i i j ( ψ ∗ i φ ∗ i p ψ ψ ∗ i ) ≤ | ^ A ⊙ j ^ A ∗ j i j ( ψ ∗ i φ ∗ i p ψ ψ ∗ i ) , p ψ (cid:25) (cid:25) ✸✸✸✸✸✸ · φ B B ✆✆✆✆✆✆✆ ψ ∗ i ✽✽✽ (cid:27) (cid:27) ✽✽✽ ^ A · ψ ∗ i (cid:23) (cid:23) ✵✵✵✵✵✵✵ · φ ∗ i D D ✟✟✟✟✟✟✟ ψ ∗ i (cid:27) (cid:27) ✼✼✼✼✼✼✼ ^ A ∗ j · φ ∗ i ☞☞☞ E E ☞☞☞ ·· φ ∗ i ☎☎☎ B B ☎☎☎ The first step is by the local i -confluence filler hypothesis, the second by whiskering properties fromSection 4.2.3 and the third by RP ( p φ ) . The fourth step is again by whiskering properties, and the fifthfollows from axiom ii) in Section 3.1.3 and the definition of diamond operators. The final step is bywhisker absoprtion, see Section 4.2.3. By similar arguments, we have | ^ A ⊙ j ^ A ∗ j i j ( ψ ∗ i φ ∗ i p ψ ψ ∗ i ) ≤ | ^ A ⊙ j ^ A ∗ j i j ( ψ ∗ i | ^ A ∗ j i j ( ψ ∗ i φ ∗ i )) ≤ | ^ A ⊙ j ^ A ∗ j i j ( | ψ ∗ i ^ A ∗ j i j ( ψ ∗ i φ ∗ i )) ≤ | ^ A ⊙ j ^ A ∗ j ⊙ j ψ ∗ i ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) ≤ | ^ A ⊙ j ^ A ∗ j ⊙ j ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) p ψ (cid:28) (cid:28) ✽✽✽✽✽✽ · φ ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ψ ∗ i ❁❁❁ (cid:30) (cid:30) ❁❁❁ ^ A · ψ ∗ i (cid:26) (cid:26) ✺✺✺✺✺✺✺ · φ ∗ i A A ☎☎☎☎☎☎☎☎ ψ ∗ i (cid:30) (cid:30) ❁❁❁❁❁❁❁ ^ A ∗ j · φ ∗ i ✞✞✞ C C ✞✞✞ ^ A ∗ j ·· φ ∗ i ⑧⑧⑧ ? ? ⑧⑧⑧ ψ ∗ i ❆❆❆❆❆❆❆❆ · φ ∗ i C C ✟✟✟✟✟✟✟ · φ ∗ i A A ☎☎☎☎☎☎ Indeed, the first step follows from RP ( p ψ ) , and the second by whiskering properties. The third step followsfrom axiom ii) in Section 3.1.3 and the definition of diamond operators as in the preceding calculation.The final step follows from whisker absorption. Finally, we observe that ^ A ⊙ j ^ A ∗ j ⊙ j ^ A ∗ j ≤ ^ A ∗ j ,34 .4. Application in rewriting and thus by monotonicity of the diamond operator we may conclude that φ ∗ i p φ φψp ψ ψ ∗ i ≤ | ^ A ∗ j i j ( ψ ∗ i φ ∗ i ) . We have thereby shown that ∀ p ( RP ( p φ ) ∧ RP ( p ψ ) ⇒ RP ( p )) and thus that r = i , concluding theproof. Similarly to the discussion from Remark 2.3.10 in the context of polygraphs, we remarkhere that the proofs of Theorems 4.2.8 and 4.3.2 are similar to those of the analogous -dimensionalresults for modal Kleene algebra found in [5, 28]. Indeed, if we look exclusively at the induction axiomsand deductions applied to j -dimensional cells, we obtain the same proof structures as in the case ofmodal Kleene algebras. This indicates that the structure of globular modal n -Kleene algebra is a naturalhigher dimensional generalisation of modal Kleene algebras in which proofs of coherent confluence maybe calculated. The consistency of the abstract, algebraic results from the previous sections with thepoint-wise, polygraphic results from Section 2.3 are made explicit in the next section. In this section we interpret the theorems from the preceding section in terms of polygraphs. We fix an n -polygraph P and a cellular extension Γ of P ⊤ n . Recall from [1] that a Kleene algebra with converse is a Kleene algebra K equippedwith an involution (−) ∨ : K → K that distributes through addition, acts contravariantly on multiplication,commutes with the Kleene star, i.e. ( a + b ) ∨ = a ∨ + b ∨ , ( a · b ) ∨ = b ∨ · a ∨ , ( a ∗ ) ∨ = ( a ∨ ) ∗ , ( a ∨ ) ∨ = a, and satisfies the inequality a ≤ aa ∨ a . When the underlying Kleene algebra is a modal Kleene algebra,we say that it is a modal Kleene algebra with converse, see also [4]. ( n, p ) -Kleene algebra. A modal ( n, p ) -Kleene algebra K is a modal n -Kleene algebra equippedwith operations (−) ∨ j : K j + → K j + for p ≤ j < n − and an operation (−) ∨ n − : K → K , satisfyingthe axioms listed above relative to the appropriate multiplication operation, i.e. for all φ, ψ ∈ K j + , ( φ + ψ ) ∨ j = φ ∨ j + ψ ∨ j , ( φ ⊙ j ψ ) ∨ j = ψ ∨ j ⊙ j φ ∨ j , ( φ ∗ j ) ∨ j = ( φ ∨ j ) ∗ j , ( φ ∨ j ) ∨ j = φ, φ ≤ φ ⊙ j φ ∨ j ⊙ j φ, and (−) ∨ n − satisfies the above axioms with j = n − and for any elements of K . Note that for φ ∈ K i with i < j , we have φ ∨ j = φ . This is a consequence of the fact that ⊙ j is idempotent for elements of K i . . Algebraic coherent confluence4.4.3. Conversion in K ( P, Γ ) . The modal ( n + ) -Kleene algebra K ( P, Γ ) , as defined in Section 3.3.1,is a modal ( n + 1, n − ) -Kleene algebra. Indeed, for any φ ∈ K ( P, Γ ) n and A ∈ K , let φ ∨ n − := { u − | u ∈ φ } and A ∨ n = { α − | α ∈ A } . This operation is well defined in the following sense: If φ ∈ K ( P, Γ ) n , then φ is a set of cells of dimensionless than or equal to n . Given a cell v of dimension i < n , its n -inverse is itself, since we consider itas an identity. Given a cell u of dimension n , we know that u − is well defined since if u ∈ P ⊤ n then u − ∈ P ⊤ n . Similarly for the case of (−) ∨ n . Γ -coherence properties as fillers. Recall that Γ and P ∗ n are themselves elements of K ( P, Γ ) , andthat in Proposition 3.3.2 we observed that Γ c = ( n ⊙ n − ( · · · ⊙ ( ⊙ ( ⊙ Γ ⊙ ) ⊙ ) ⊙ · · · ) ⊙ n − n ) , where Γ c is the set of cells of Γ in context. In the following, we will denote by P cn the set of rewritingsteps generated by P n , which can be expressed in K ( P, Γ ) as P cn = ( n − ⊙ n − ( · · · ⊙ ( ⊙ ( ⊙ P n ⊙ ) ⊙ ) ⊙ · · · ) ⊙ n − n − ) . The construction of K ( P, Γ ) is compatible with Γ -coherence properties in the following sense: With Γ ′ := ( Γ c ) ∗ n , the following equivalences hold: i) Γ is a (local) confluence filler for P ⇐⇒ Γ ′ is a (local) ( n − ) -confluence filler for (( P cn ) ∨ n − , P cn ) , ii) Γ is a Church-Rosser filler for P ⇐⇒ Γ ′ is an ( n − ) -Church-Rosser filler for (( P cn ) ∨ n − , P cn ) .Proof. Let us prove the equivalence in the case of (global) confluence.Suppose that Γ is a confluence filler for P . An element f − ⋆ n − g ∈ ( P cn ) ∨ n − ⊙ n − P cn correspondsto a branching ( f, g ) . By hypothesis, there exists an α ∈ P ⊤ n [ Γ ] such that s n ( α ) = f − ⋆ n − g and α is an n -composition of rewriting steps so α ∈ Γ ′ . Furthermore, the n -target of α is a confluence, so α ∈ Γ ′ ⊙ n ( P cn ⊙ n − ( P cn ) ∨ n − ) . In equations, this means that ( P cn ) ∨ n − ⊙ n − P cn ⊆ d n (cid:16) Γ ′ ⊙ n ( P cn ⊙ n − ( P cn ) ∨ n − ) (cid:17) = | Γ ′ i n (cid:16) P cn ⊙ n − ( P cn ) ∨ n − (cid:17) , i.e. Γ ′ is an ( n − ) -confluence filler for (cid:16) ( P cn ) ∨ n − , P cn (cid:17) .Conversely, if Γ ′ is an ( n − ) -confluence filler for (cid:16) ( P cn ) ∨ n − , P cn (cid:17) , then given some branching ( f, g ) ,we know that f − ⋆ n − g ∈ d i Γ ′ ⊙ n ( P cn ⊙ n − ( P cn ) ∨ n − ) . This means there exists some cell α ∈ Γ ′ with n -source f − ⋆ n − g and whose n -target is a confluence. Since α ∈ Γ ′ , we know that it is a compositionof rewriting steps of Γ . With this we conclude that P is Γ -confluent.The other cases are similarly deduced.Due to this compatibility, we may deduce the following theorems, that is Theorems 2.3.4 and 2.3.7,as corollaries of our main results: EFERENCES n -polygraphs). Let P be an n -polygraph and Γ a cellular extensionof P ⊤ n . Then Γ is a confluence filler for P if, and only if, Γ is a Church-Rosser filler for P .Proof. Suppose first that Γ is a confluence filler for P . Using the result and notations from Proposi-tion 4.4.5, we know that Γ ′ is an ( n − ) -confluence filler for (( P cn ) ∨ n − , P cn ) . We apply Theorem 4.2.8 to K ( P, Γ ) for i = n − and j = n , obtaining that b Γ ′∗ n is an ( n − ) -Church-Rosser filler for (( P cn ) ∨ n − , P cn ) .Observing that ( P cn + ( P cn ) ∨ n − ) ∗ n − = P ⊤ n , we have b Γ ′∗ n = (cid:16) P ⊤ n ⊙ n − ( Γ c ) ∗ n Âă ⊙ n − P ⊤ n (cid:17) ∗ n ⊆ (cid:16) ( P ⊤ n ⊙ n − Γ c Âă ⊙ n − P ⊤ n ) ∗ n (cid:17) ∗ n = Γ ′ , where the first step is by definition, the second uses the fact that the n -star is a lax morphism for ( n − ) -multiplication, see Section 3.2.16, and the third uses the fact that Γ c absorbs whiskers and that ( A ∗ n ) ∗ n = A ∗ n . Since additionally, Γ ′ ⊆ b Γ ′∗ n , Γ ′ is an ( n − ) -Church-Rosser filler for (( P cn ) ∨ n − , P cn ) .By Proposition 4.4.5, this allows us to conclude that Γ is a Church-Rosser filler for P .For the trivial direction, suppose that Γ is a Church-Rosser filler for P . We deduce by Proposition 4.4.5that Γ ′ is an ( n − ) -Church-Rosser filler for (( P cn ) ∨ n − , P cn ) . As pointed out at the end of Section 4.2.1,this means that Γ ′ is an i -confluence filler for (( P cn ) ∨ n − , P cn ) , by which we conclude that Γ is a confluencefiller for P . n -polygraphs). Let P be a terminating n -polygraph and Γ a cellularextension of P ⊤ n . Then Γ is a local confluence filler for P if, and only if, Γ is a confluence filler for P .Proof. Suppose that Γ is a local confluence filler for P . Using the result and notations from Proposi-tion 4.4.5, we know that Γ ′ is an ( n − ) -local confluence filler for (( P cn ) ∨ n − , P cn ) . We apply Theorem 4.3.2to K ( P, Γ ) for i = n − and j = n , obtaining that b Γ ′∗ n is an ( n − ) -confluence filler for (( P cn ) ∨ n − , P cn ) .As in the proof of the previous theorem, we have that b Γ ′∗ n = Γ ′ , allowing us to conclude that Γ is aconfluence filler for P , again by Proposition 4.4.5.For the trivial direction, suppose that Γ is a confluence filler for P . As above, we deduce that Γ ′ is an ( n − ) -Church-Rosser filler for (( P cn ) ∨ n − , P cn ) . Again, as pointed out at in Section 4.2.1, this meansthat Γ ′ is a local i -confluence filler for (( P cn ) ∨ n − , P cn ) , by which we conclude that Γ is a local confluencefiller for P via Proposition 4.4.5. References [1] S. L. Bloom, Z. Ésik, and G. Stefanescu. Notes on equational theories of relations. Algebra Universalis ,33(1):98–126, 1995.[2] R. Book and F. Otto. String-rewriting systems . Springer-Verlag, 1993.[3] A. Burroni. Higher-dimensional word problems with applications to equational logic. Theor. Comput. Sci. ,115(1):43–62, 1993.[4] J. Desharnais, B. Möller, and G. Struth. Kleene algebra with domain. ACM Trans. Comput. Log. , 7(4):798–833,2006.[5] J. Desharnais, B. Möller, and G. Struth. Algebraic notions of termination. Log. Methods Comput. Sci. , 7(1),2011. EFERENCES [6] J. Desharnais and G. Struth. Internal axioms for domain semirings. Sci. Comput. Program. , 76(3):181–203,2011.[7] H. Doornbos, R. C. Backhouse, and J. van der Woude. A calculational approach to mathematical induction. Theor. Comput. Sci. , 179(1-2):103–135, 1997.[8] S. Gaussent, Y. Guiraud, and P. Malbos. Coherent presentations of Artin monoids. Compos. Math. ,151(5):957–998, 2015.[9] Y. Guiraud. Termination orders for three-dimensional rewriting. J. Pure Appl. Algebra , 207(2):341–371,2006.[10] Y. Guiraud, E. Hoffbeck, and P. Malbos. Convergent presentations and polygraphic resolutions of associativealgebras. Math. Z. , 293(1-2):113–179, 2019.[11] Y. Guiraud and P. Malbos. Higher-dimensional categories with finite derivation type. Theory Appl. Categ. ,22(18):420–478, 2009.[12] Y. Guiraud and P. Malbos. Coherence in monoidal track categories. Math. Structures Comput. Sci. , 22(6):931–969, 2012.[13] Y. Guiraud and P. Malbos. Higher-dimensional normalisation strategies for acyclicity. Adv. Math. , 231(3-4):2294–2351, 2012.[14] Y. Guiraud and P. Malbos. Polygraphs of finite derivation type. Math. Structures Comput. Sci. , 28(2):155–201,2018.[15] N. Hage and P. Malbos. Knuth’s coherent presentations of plactic monoids of type A. Algebr. Represent.Theory , 20(5):1259–1288, 2017.[16] T. Hoare, B. Möller, G. Struth, and I. Wehrman. Concurrent Kleene algebra and its foundations. J. Log.Algebr. Program. , 80(6):266–296, 2011.[17] P. Höfner and G. Struth. Algebraic notions of nontermination: omega and divergence in idempotent semirings. J. Log. Algebr. Program. , 79(8):794–811, 2010.[18] G. Huet. Confluent reductions: abstract properties and applications to term rewriting systems. J. Assoc.Comput. Mach. , 27(4):797–821, 1980.[19] B. Jónsson and A. Tarski. Boolean algebras with operators. I. Amer. J. Math. , 73:891–939, 1951.[20] D. Kozen. Kleene algebra with tests. ACM Trans. Program. Lang. Syst. , 19(3):427–443, 1997.[21] S. Mac Lane. Natural associativity and commutativity. Rice Univ. Studies , 49(4):28–46, 1963.[22] F. Métayer. Resolutions by polygraphs. Theory Appl. Categ. , 11(7):148–184, 2003.[23] S. Mimram. Towards 3-dimensional rewriting theory. Log. Methods Comput. Sci. , 10(2):2:1, 47, 2014.[24] M. Newman. On theories with a combinatorial definition of “equivalence”. Ann. of Math. (2) , 43(2):223–243,1942.[25] C. C. Squier, F. Otto, and Y. Kobayashi. A finiteness condition for rewriting systems. Theoret. Comput. Sci. ,131(2):271–294, 1994. EFERENCES [26] J. D. Stasheff. Homotopy associativity of H -spaces. I, II. Trans. Amer. Math. Soc. 108: 275-292; ibid. ,108:293–312, 1963.[27] R. Street. Limits indexed by category-valued -functors. J. Pure Appl. Algebra , 8(2):149–181, 1976.[28] G. Struth. Calculating Church-Rosser proofs in Kleene algebra. In Relational methods in computer science ,volume 2561 of Lecture Notes in Comput. Sci. , pages 276–290. Springer, 2002.[29] G. Struth. Abstract abstract reduction. J. Log. Algebr. Program. , 66(2):239–270, 2006.[30] G. Struth. Modal tools for separation and refinement. Electron. Notes Theor. Comput. Sci. , 214:81–101, 2008.[31] Terese. Term rewriting systems , volume 55 of Cambridge Tracts in Theoretical Computer Science . CambridgeUniversity Press, 2003.[32] J. von Wright. Towards a refinement algebra. Sci. Comput. Program. , 51(1-2):23–45, 2004. C ameron Calk [email protected] École PolytechniqueBâtiment Alan Turing1 rue Honoré d’Estienne d’Orves91128 Palaiseau, France Eric Goubault [email protected] École PolytechniqueBâtiment Alan Turing1 rue Honoré d’Estienne d’Orves91128 Palaiseau, France Philippe Malbos [email protected] Univ Lyon, Université Claude Bernard Lyon 1CNRS UMR 5208, Institut Camille Jordan43 Blvd. du 11 novembre 1918F-69622 Villeurbanne cedex, France Georg Struth g.struth@sheffield.ac.uk