Injective envelopes of transition systems and Ferrers languages
IINJECTIVE ENVELOPES OF TRANSITION SYSTEMS AND FERRERSLANGUAGES † MUSTAPHA KABIL AND MAURICE POUZET † Dedicated to the memory of Maurice Nivat
Abstract.
We consider reflexive and involutive transition systems over an ordered al-phabet A equipped with an involution. We give a description of the injective envelopeof any two-element set in terms of Galois lattice, from which we derive a test of itsfiniteness. Our description leads to the notion of Ferrers language. Introduction and presentation of the main results
This paper is about involutive and reflexive transition sytems from a metric point of view.This point of view, inspired from the work of Quilliot (1983), and applied first to posetsand graphs, was initiated by the second author [29] and developped throught the theses ofJawhari (1983), Misane (1984) and several papers [17] (1986), [30] (1994), [33] (1992), [23](1998), [24] (2018). It consists to view arbitrary transition systems as metric spaces. Thedistance between two states is a language instead of a non-negative real. To a transitionsystem M ∶= ( Q, T ) , with set of states Q and set of transitions T over an alphabet A weassociate a map d M from Q × Q into the set ℘( A ∗ ) of languages over A . The value d M ( x, y ) is the language accepted by the automaton A ∶= ( M, { x } , { y }) having x as an initial stateand y as a final state. The set ℘( A ∗ ) is an ordered monoid, the monoid operation being theconcatenation of languages (with neutral element {◻} , the language reduced to the emptyword ◻ ) and the order the reverse of inclusion. The map d M has similar properties of anordinary distance (e.g. it satisfies the triangular inequality). Hence, we may use conceptsand techniques of the theory of metric spaces in the study of transition systems as well asclasses of transition systems. Concepts of balls , hyperconvex metric space and non-expansive maps between metric spaces extend to transition systems and more generally to metricspaces over ℘( A ∗ ) . Due to the fact that joins exist in the set of values, the category ofmetric spaces with the non-expansive maps as morphisms has products. Then, one mayalso define retractions and coretractions , and by considering isometries as approximationsof coretractions, injective metric spaces and absolute retracts . In ordinary metric spaces, thedistance is symmetric. To be closer to this situation, it is convenient to suppose that thevalue of d M ( x, y ) determines the value of d M ( y, x ) ; for that, we suppose that the alphabetis equipped with an involution − and our transition systems M are involutive , in the sensethat ( x, α, y ) ∈ T if and only if ( y, α, x ) ∈ T . Once the involution is extended to A ∗ and thento ℘( A ∗ ) , we have d M ( x, y ) = d M ( y, x ) . Then, one can extend the definion of metric spacesto transition systems in a natural way, see [30]. Date : July 5, 2019.1991
Mathematics Subject Classification.
Primary 06A15, 06D20, 46B85, 68Q70; Secondary 68R15 .
Key words and phrases.
Metric spaces, Injective envelopes, Transition systems, Ferrers languages, Or-dered sets, Interval orders, Well-quasi-order. a r X i v : . [ m a t h . C O ] J u l M.KABIL AND M. POUZET
This work is a continuation of the work published in [22, 23, 24]. We require that transitionsystems M are reflexive , that is every letter occurs to every vertex: ( x, α, x ) ∈ T for every x ∈ Q and α ∈ A . In this case, distances values are final segments of A ∗ equipped withthe subword ordering, that is subsets F of A ∗ such that u ∈ F and u ≤ v for the subwordordering imply v ∈ F . It turns out that several properties of involutive and reflexive systemsand more generally metric spaces over the set F ( A ∗ ) of final segments of A ∗ rely almostuniquely on the structure of F ( A ∗ ) . According to the terminology of Kaarli and Radeleczki[19], this structure is the dual of an integral involutive quantale ( I Q for short); here westick to the name of Heyting algebra that we used in a series of papers. This is a completelattice H with a monoid operation (not necessarily commutative) and an involution − whichis isotone and reverses the operation. In order to be closer to the operation of concatenationof languages we denote by ⋅ the monoid operation. We suppose that the neutral element ofthe monoid, that we denote 1, is the least element of the ordering and we suppose that thedistributivity law below holds(1) ⋀{ p α ⋅ q ∶ α ∈ I } = ⋀{ p α ∶ α ∈ I } ⋅ q . As shown in [17], the notions of injective, absolute retract and hyperconvex spaces overa Heyting algebra coincide, this being essentially due to the fact that the set of the valuesof the distance, being an Heyting algebra, can be equipped with a distance and that everymetric space can be embedded into a power of that metric space. Furthermore, every spacehas an injective envelope.In particular, every metric space over the Heyting algebra F ( A ∗ ) has an injective envelope.The study of such injective envelope was initiated in [23]. It is based on the properties ofthe injective envelope of two-element metric spaces. A large account of its properties wasgiven in [22] and [24]. In this paper, we look at the many facets of this object which havenot been published yet.If F is a final segment of A ∗ , the injective envelope S F of the two-element space { x, y } such that d ( x, y ) = F is associated to an involutive and reflexive transition system. Let M F be this transition system and A F be the automaton ( M F , { x } , { y }) .We characterize first this injective envelope in terms of reflexive and involutive transitionsystems. Theorem 1.
Let
A ∶= ( M, { x } , { y }) be a reflexive and involutive automaton accepting afinal segment F of A ∗ . Then A is isomorphic to A F iff for every reflexive and involutiveautomaton A ′ ∶= ( M ′ , { x ′ } , { y ′ }) which accepts F , the following properties hold:(i) Every automata morphism f ∶ A → A ′ , if any, is an isometric embedding;(ii) The map g ∶ { x ′ , y ′ } → M such that g ( x ′ ) = x and g ( y ′ ) = y extends to a morphism ofautomata from A ′ to A . We will obtain this result as a consequence of a characterization of the injective envelopeamong generalized metric spaces (Theorem 4) given in Section 2.Then, we develop an approach in terms of Galois correspondence.To a final segment F of A ∗ we associate the incidence structure R ∶= ( A ∗ , ρ F , A ∗ ) where ρ is the binary relation on A ∗ defined by uρ F v if the concatenation uv of u and v belongs to F . For each v ∈ A ∗ , let R − ( v ) ∶= { u ∈ A ∗ : uρ F v } and for V ⊆ A ∗ , let R − ∧ ( V ) ∶= ⋂ v ∈ V ρ − F ( v ) .The collection of all R − ∧ ( V ) for V ⊆ A ∗ , once ordered by inclusion, forms a complete latticeGal ( R ) , called the Galois lattice associated to the incidence structure R . As a subset of H ∶= F ( A ∗ ) , it inherits the metric structure d H of H . Containing x ∶= A ∗ and y ∶= F , twoelements which verify d H ( x, y ) = F , this metric space is the injective envelope of { x, y } . NJECTIVE ENVELOPES AND FERRERS LANGUAGES 3
For concrete examples, suppose that F is a finite union of final segments F , . . . F i , . . . F k − and that each F i is generated by X i , a set of words u i of the same length n i , all of the from u i ∶= a i a i ⋯ a in i − with a ij ∈ X i j ⊆ A . Let n ⊗⋯⊗ n k − be the direct product of k chains n , ... n k − where n i ∶= { , ..., n i − } is equipped with the natural ordering, and let F ( n ⊗⋯⊗ n k − ) be the collection of final segments of n ⊗ ⋯ ⊗ n k − ordered by inclusion.We prove (see Section 5): Theorem 2.
As a lattice, the injective envelope S F can be identified with an intersectionclosed subset of the set F ( n ⊗ ⋯ ⊗ n k − ) . Moreover if ↓ u i ∩ ↓ u j = {◻} whenever u i ∈ X i , u j ∈ X j , and i ≠ j , then S F identifies to the full set F ( n ⊗ ⋯ ⊗ n k − ) . We recall that a poset P is well-quasi-ordered , in brief w.q.o., if it well-founded (every nonempty subset contains a minimal element) and contains no infinite antichain. A fundamentalresult of G.Higman [14] asserts that the free ordered monoid A ∗ is w.q.o. whenever thealphabet A is w.q.o.. The set of final segments of a w.q.o. set, once ordered by reverse ofthe inclusion, is well-founded [14]. Hence, if our alphabet A is w.q.o., every final segment F of A ∗ is generated by finitely many words u , ..., u k − , hence has the form mentionedabove. Consequently, the corresponding injective envelope is finite. Concerning its size, letus mention that if k =
2, then F ( n ⊗ n ) has size ( n ⋅ n n ) [ ] ( ) . If n = ... = n k − = 2, then ( F ( n ⊗⋯⊗ n k − ) , ⊇) is isomophic to F D ( k ) , the free distributive lattice with k generators. It is a famous problem, raised by Dedekind, to give an explicit and workableformula for F D ( k ) . The largest exact value known is F D ( ) [38]. An asymptotic formulawas given by Korshunov in 1981[25].For an example, on the two-letter alphabet A = { a, b } , the words u ∶= aa and u ∶= bb give the lattice (ordered by reverse of inclusion) and graph represented on Figure 1. (cid:115)(cid:115)(cid:115) (cid:115)(cid:115)(cid:115) (cid:64)(cid:64)(cid:64)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:64) (cid:0)(cid:0)(cid:0) A ∗ ↑ { b, aa }↑ { ab, ba, aa, bb }↑ { bb, aa }↑ { a, bb } ↑ { a, b } The lattice structure of S F (cid:115)(cid:115)(cid:115) (cid:115)(cid:115)(cid:115) (cid:64)(cid:64)(cid:64)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:64) (cid:0)(cid:0)(cid:0) (cid:85)(cid:30) (cid:11)(cid:77) The graphic structure of S F (cid:18)(cid:73) Figure 1Structural properties of transition systems rely upon algebraic properties of languagesand conversely. In fact, transition systems can be viewed as geometric objects interpretatingthese algebraic properties. An illustration of this claim is given by the following result (seeCorollary 4.9 [23]).
Theorem 3.
Let F be a nonempty final segment of A ∗ . If F is the concatenation of finalsegments F , . . . , F n then the automaton A F ∶= ( M F , { x } , { y }) associated to the injective M.KABIL AND M. POUZET envelope S F is the concatenation A F ⋅ ⋅A F n of automata A F i ∶= ( M F i , { x i } , { y i }) associatedto the injective envelope S F i , this concatenation being obtained by identifying each y i with x i + . The fact that an automaton decomposes into such a concatenation can be viewed directlyby looking at states which disconnect the underlying graph. From this follows the uniquenessof such a decomposition. This uniqueness amounts to the fact that the monoid F ( A ∗ ) ∖ {∅} is free. A purely algebraic proof of this result is given in [24].We discuss then the relationship between the minimal deterministic automaton acceptinga final segment F , say M in F , and the automaton A F associated to the injective envelope S F . This minimal automaton is part of A F , but not in an isometric way ( M in F being deter-ministic cannot be reflexive, in general it is not involutive). Among involutive and reflexivetransition systems accepting a given final segment F , we consider those with a minimumnumber of states and among those, the ones with a maximal number of transitions, thatwe call Minmax automata. Exemples given by Mike Main and communicated by MauriceNivat [27] show that contrarily to the case of deterministic automata, these automata arenot unique.We introduce Ferrers languages. A language L over A ∗ is Ferrers if(2) xx ′ ∈ L and yy ′ ∈ L imply xy ′ ∈ L or yx ′ ∈ L for all x, x ′ , y, y ′ ∈ A ∗ . The class of Ferrers languages is closed under complement but not under concatenation.Still, if F , . . . F n are Ferrers and each F i is a final segment of A ∗ then the concatenation F ⋯ F n is Ferrers (Corollary 19).We prove that a final segment F of A ∗ is Ferrers if and only if the injective envelope S F is totally orderable, that is there is a linear order ⪯ on S F such that d ( x, z ) ⪯ d ( x, y ) and d ( z, y ) ⪯ d ( x, y ) for all x ⪯ z ⪯ y . (Theorem 23).Over a finite alphabet A ∗ , Boolean combinations of final segments of A ∗ are called piece-wise testable languages . They have been characterized by Simon [35] by the fact that theirsyntactical monoid is J -trivial. The Boolean algebra of piecewise testable languages is in-cluded into the Boolean algebra generated by rational Ferrers languages. Indeed, over afinite alphabet, every final segment is a finite union of rational Ferrers languages. But, onan alphabet with at least two letters, there are rational Ferrers languages which are notpiecewise testable (e.g., L ∶= A ∗ b on A ∶= { a, b } ). We do not know if they are dot-depth one.This paper is organized as follows. Properties of metric spaces over a Heyting algebra andtheir injective envelopes are summarized in section 2. In section 3, we introduce the Heytingalgebra F ( A ∗ ) . In section 4 we consider transition systems as metric spaces. In section5 we describe the injective envelope of a two-element metric spaces over F ( A ∗ ) ; we proveTheorem 1 and 2 and conclude the section by a counterexample about Minmax automatadue to M.Main. Ferrers languages are introduced in section 6.The results developped here have been presented at the International Conference onDiscrete Mathematics and Computer Science (DIMACOS’11) organized by A. Boussa¨ıri,M. Kabil, and A. Taik in Mohammedia (Morocco) May, 5-8, 2011. They were never pub-lished; a part of it was included into the Th`ese d’´Etat defended by the first author [22].2. Metric spaces over a Heyting algebra
Basic facts.
The following is extracted from [24] (for more details, see [23]). Let H bea Heyting algebra and let E be a set. A H -distance on E is a map d ∶ E —→ H satisfyingthe following properties for all x, y, z ∈ E :(1) d ( x, y ) = ⇐⇒ x = y , NJECTIVE ENVELOPES AND FERRERS LANGUAGES 5 (2) d ( x, y ) ≤ d ( x, z ) ⋅ d ( z, y ) ,(3) d ( x, y ) = d ( y, x ) . The pair ( E, d ) is called a H - metric space . If there is no danger of confusion we will denoteit E. A H -distance can be defined on H . This fact relies on the classical notion of residuation .Let v ∈ H . Given β ∈ H , each of the sets { r ∈ H ∶ v ≤ r ⋅ β } and { r ∈ H ∶ v ≤ β ⋅ r } has a leastelement, that we denote respectively ⌈ v ⋅ β − ⌉ and ⌈ β − ⋅ v ⌉ (note that ⌈ β − ⋅ v ⌉ = ⌈ ¯ v ⋅ ( ¯ β ) − ⌉ ).It follows that for all p, q ∈ H , the set D ( p, q ) ∶= { r ∈ H ∶ p ≤ q ⋅ ¯ r and q ≤ p ⋅ r } has a least element, namely ⌈ ¯ p ⋅ ( ¯ q ) − ⌉ ∨ ⌈ p − ⋅ q ⌉ , that we denote d H ( p, q ) . As shown in [17],the map ( p, q ) —→ d H ( p, q ) is a H− distance.Let ( E, d ) and ( E ′ , d ′ ) be two H− metric spaces. Recall that a map f ∶ E —→ E ′ is a contraction (or a non-expansive map) from ( E , d ) to ( E ′ , d ′ ) provided that d ′ ( f ( x ) , ( f ( y )) ≤ d ( x, y ) holds for all x, y, ∈ E . The map f is an isometry if d ′ ( f ( x ) , ( f ( y )) = d ( x, y ) for all x, y, ∈ E . We say that E and E ′ are isomorphic , a fact that we denote E ≅ E ′ , if there is asurjective isometry from E onto E ′ .Let (( E i , d i )) i ∈ I be a family of H -metric spaces. The direct product ∏ i ∈ I ( E i , d i ) , is themetric space ( E, d ) where E is the cartesian product ∏ i ∈ I E i and d is the ”sup” (or (cid:96) ∞ )distance defined by d (( x i ) i ∈ I , ( y i ) i ∈ I ) = ⋁ i ∈ I d i ( x i , y i ) .For a H -metric space E , x ∈ E and r ∈ H , we define the ball B E ( x, r ) as the set { y ∈ E ∶ d ( x, y ) ≤ r } . We say that E is convex if the intersection of two balls B E ( x , r ) and B E ( x , r ) is non-empty provided that d ( x , x ) ≤ r ⋅ r . We say that E is hyperconvex ifthe intersection of every family of balls ( B E ( x i , r i ) ) i ∈ I is non-empty whenever d ( x i , x j ) ≤ r i ⋅ r j for all i, j ∈ I . For an example, (H , d H ) is a hyperconvex H -metric space and every H -metric space embeds isometrically into a power of (H , d H ) [17]. This is due to the factthat for every H -metric space ( E, d ) and for all x, y ∈ E the following equality holds: d ( x, y ) = ⋁ z ∈ E d H ( d ( z, x ) , d ( z, y )) . The space E is a retract of E ′ , in symbols E ⊲ E ′ , if there are two contractions f ∶ E —→ E ′ and g ∶ E ′ —→ E such that g ○ f = id E (where id E is the identity map on E ). In this case, f isa coretraction and g a retraction . If E is a subspace of E ′ , then clearly E is a retract of E ′ ifthere is a contraction from E ′ to E such g ( x ) = x for all x ∈ E. We can easily see that everycoretraction is an isometry. A metric space is an absolute retract if it is a retract of everyisometric extension. The space E is said to be injective if for all H -metric space E ′ and E ′′ , each contraction f ∶ E ′ —→ E and every isometry g ∶ E ′ —→ E ′′ there is a contraction h ∶ E ′′ —→ E such that h ○ g = f . We recall that for a metric space over a Heyting algebra H , the notions of absolute retract, injective, hyperconvex and retract of a power of (H , d H ) coincide [17].2.2. Injective envelope.
A contraction f ∶ E —→ E ′ is essential it for every contraction g ∶ E ′ —→ E ′′ , the map g ○ f is an isometry if and only if g is isometry (note that, inparticular, f is an isometry). An essential contraction f from E into an injective H -metricspace E ′ is called an injective envelope of E . We will rather say that E ′ is an injectiveenvelope of E . We can view an injective envelope of a metric space E as a minimal injective H -metric space containing (isometrically) E . Two injective envelopes of E are isomorphicvia an isomorphism which is the identity over E . This allows to talk about ”the” injectiveenvelope of E ; we will denote it by N ( E ) . A particular injective envelope of E will be M.KABIL AND M. POUZET called a representation of N ( E ) . The construction of injective envelope is based upon thenotion of minimal metric form . A weak metric form is every map f ∶ E —→ H satisfying d ( x, y ) ≤ f ( x ) ⋅ f ( y ) for all x, y ∈ E . This is a metric form if in addition f ( x ) ≤ d ( x, y ) ⋅ f ( y ) for all x, y ∈ E. Equivalently, f is a metric form if and only if d H ( d ( x, y ) , f ( x )) ≤ f ( y ) forall x, y ∈ E . A (weak) metric form is minimal if there is no other (weak) metric form g satisfying g ≤ f (that is g ( x ) ≤ f ( x ) for all x ∈ E ). Since every weak metric form majorizes ametric form, the two notions of minimality coincide. As shown in [17] every H -metric spacehas an injective envelope; the space of minimal metric forms is a representation of it, (cf.Theorem 2.2 of [23] ) .We give below a new characterization of the injective envelope. Theorem 4.
Let ( E, d ) be a metric space over H and X ⊆ E . Then ( E, d ) is isomorphic tothe injective envelope of ( X, d ↾ X ) iff for every metric space ( E ′ , d ′ ) , X ′ ⊆ E ′ , every isometry f ∶ ( X, d ↾ X ) onto ( X ′ , d ′↾ X ′ ) :(i) Every non-expansive map f ∶ ( E, d ) → ( E ′ , d ′ ) , if any, which extends f is an isometricembedding of ( E, d ) into ( E ′ , d ′ ) ;(ii) The isometry f − extends to a non-expansive map from ( E ′ , d ′ ) into ( E, d ) .Proof. Suppose that ( E, d ) is isomorphic to the injective envelope of ( X, d ↾ X ) . Since ( E, d ) is injective, the isometry f − extends to a non-expansive map from ( E ′ , d ′ ) into ( E, d ) . Thisproves that ( ii ) holds. The proof that ( i ) holds relies on the properties of metric forms. Toprove that f is an isometry embedding amounts to prove the equality:(3) d ′ ( f ( x ) , f ( y )) = d ( x, y ) for all x, y ∈ E . Set X ′′ ∶= f ( X ) . For every x ∈ E , let h x ∶ X → H and g x ∶ X → H be themaps defined by setting h x ( z ) = d ( z, x ) and g x ( z ) = d ′ ( f ( z ) , f ( x )) for z ∈ X . Since E is theinjective envelope of X , h x is a minimal metric form; since f is non-expansive and inducesan isometry from X onto X ′′ , the map g x is a metric form below h x , hence h x = g x . Itfollows that(4) d ( z, x ) = d ( f ( z ) , f ( x )) for every z ∈ X .Let x, y ∈ E . By construction of the injective envelope, its elements identify to minimalmetric forms over X , hence x and y identify respectively to h x and h y and(5) d ( x, y ) = d H ( h x , h y ) ∶= Sup { d H ( d ( z, x ) , d ( z, y )) ∶ z ∈ X } . Set α ∶= d ′ ( f ( x ) , f ( y )) . Let z ∈ E . By definition, d H ( d ′ ( f ( z ) , f ( x )) , d ′ ( f ( z ) , f ( y ))) = Min D ( d ′ ( f ( z ) , f ( x )) , d ′ ( f ( z ) , f ( y ))) where D ( d ′ ( f ( z ) , f ( x )) , d ′ ( f ( z ) , f ( y ))) = { r ∈ H ∶ d ′ ( f ( z ) , f ( x )) ≤ d ′ ( f ( z ) , f ( y )) ⋅ r and d ′ ( f ( z ) , f ( y )) ≤ d ′ ( f ( z ) , f ( x )) ⋅ r } . From the triangu-lar inequality, we have d ′ ( f ( z ) , f ( y )) ≤ d ′ ( f ( z ) , f ( x ))⋅ α and d ′ ( f ( z ) , f ( x )) ≤ d ′ ( f ( z ) , f ( y ))⋅ α . Hence, α ∈ D ( d ′ ( f ( z ) , f ( x )) , d ′ ( f ( z ) , f ( y ))) . The inequality d H ( d ′ ( f ( z ) , f ( x )) , d ′ ( f ( z ) , f ( y ))) ≤ α follows. Consequently,Sup { d H ( d ′ ( f ( z ) , f ( x )) , d ′ ( f ( z ) , f ( y ))) ∶ z ∈ E } ≤ α. With Equalities (4) and (5)we get: d ( x, y ) = Sup { d H ( d ( z, x ) , d ( z, y )) ∶ z ∈ E } ≤ α . Since f is non expansive, we have α ≤ d ( x, y ) . Thus d ( x, y ) = α that is Equality (3) holds.Conversely, suppose that ( i ) and ( ii ) hold. Let ( E ′ , d ′ ) be the injective envelope of ( X, d ↾ X ) and f be the identity map from ( X, d ↾ X ) onto itself. Applying ( ii ) , the map f − extends to a non-expansive map g from ( E ′ , d ′ ) into ( E, d ) . Since ( E ′ , d ′ ) is injective, the NJECTIVE ENVELOPES AND FERRERS LANGUAGES 7 map f extends to a non-expansive map f from ( E, d ) into ( E ′ , d ′ ) . The map f ○ g is non-expansive and is the identity on X . Since ( E ′ , d ′ ) is the injective envelope of ( X, d ↾ X ) , f ○ g is the identity on E ′ (note that elements of E ′ identify to minimal metric forms over X ),hence g is injective and f is surjective. Now by ( i ) , f is an isometry on its image. Hence f is an isometry of ( E, d ) onto ( E ′ , d ′ ) . Thus ( E, d ) is the injectyive envelope of ( X, d ↾ X ) asclaimed. (cid:3) Up to Theorem 6, we include the few facts we need about injective envelopes of two-element metric spaces ( see [23] for proofs).Let H be a Heyting algebra and v ∈ H . Let E ∶= { x, y } be a two-element H -metricspace such that d ( x, y ) = v . We denote by N v the injective envelope of E . We give threerepresentations of it. For the fist one, we consider the set of minimal metric forms over E .That is, in this case, the set of minimal pairs h ∶= ( h x , h y ) ∈ H such that h x ⋅ h y ≥ v , the set H being equipped with the product ordering. Each element z ∈ N v identifies to the pair ( d N v ( x, z ) , d N v ( y, z )) ; in particular, x and y identify to ( , v ) and to ( v, ) respectively. Weequip H with the supremum distance: d H (( u , u ) , ( u ′ , u ′ )) ∶= d H ( u , u ′ ) ∨ d H ( u , u ′ ) . With the induced distance, N v becomes a metric space. If ( h x , h y ) and ( h ′ x , h ′ y ) are twoelements of N v , their distance is d H ( h x , h ′ x ) ∨ d H ( h y , h ′ y ) . In fact,(6) d H ( h x , h ′ x ) = d H ( h y , h ′ y ) . The proof is easy: We prove that if h x ≤ h ′ x ⋅ r for some r ∈ H , then h ′ y ≤ h y ⋅ r . This and thecorresponding inequality with y replacing x will leads to (6). Suppose that h x ≤ h ′ x ⋅ r . Wehave v ≤ h x ⋅ h y ≤ h ′ x ⋅ r ⋅ h y and thus r ≤ h ′ x ⋅ r ⋅ h y . Since v ≤ h ′ x ⋅ h ′ y , we have v ≤ h ′ x ⋅( h ′ y ∧( r ⋅ h y )) by distributivity. Since ( h ′ x , h ′ y ) is a minimal metric form above v , we have h ′ y = h ′ y ∧ ( r ⋅ h y ) ,that is h ′ y ≤ h y ⋅ r .Due to the fact that in a minimal metric form ( h x , h y ) each component determines theother, we may prefer an other presentation of N v as a subset of H . Set S v ∶= {⌈ v ⋅ β − ⌉ ∶ β ∈ H} ;equipped with the ordering induced by the ordering over H this is a complete lattice. Apair ( h x , h y ) belongs to N v if and only if h x = ⌈ v ⋅ h y − ⌉ and h y = ⌈ h − x ⋅ v ⌉ . This yields acorrespondence between N v and S v .Now, in several instances, e.g. in the case of the sum of two metric spaces (see subsection2.3), it is preferable to consider the set C v of of all pairs ( u , u ) ∈ H such that v ≤ u ⋅ u .Once equipped with the ordering induced by the product ordering on H , we can considerthe set N ′ v of its minimal elements. Each minimal element ( u , u ) yields the minimalmetric form ( u , u ) (and conversely). The distance over H is different from the previouscase. We have to equip H with the product of the distance d H with the distance d ′H definedon H by d ′H ( u , u ′ ) ∶= d H ( u , u ′ ) . Lemma 1. (Lemma 2.3, Proposition 2.7 of [23] ) The space N v equipped with the supremumdistance and the set S v equipped with the distance induced by the distance over H are injectiveenvelopes of the two-element metric spaces {( , v ) , ( v, )} and { , v } respectively. Thesespaces are isometric to the injective envelope of E ∶= { x, y } . Theorem 5. (Theorem 2.9 [23] ) Let E be a H -metric space. If E is the injective envelopeof a two-element set then E contains no proper isometric subspace. Proposition 1. (Corollary 3.3 [23] ) The following properties are equivalent:
M.KABIL AND M. POUZET (i) The injective envelope of any finite metric space is finite;(ii) The injective envelope of any two-element metric space is finite.
A metric space is linearly orderable if there is a linear ordering ⪯ on E such that d ( x, z ) ⪯ d ( x, y ) and d ( z, y ) ⪯ d ( x, y ) for all x, y, z ∈ E with x ⪯ z ⪯ y. Proposition 2. (Fact 5 [23] ) Let u ∈ H . The space S u is linearly orderable if and only ifthe ordering is induced by the order on H or by its reverse. We say that a metric space is finitely indecomposable if for every finite family ( E i ) i ∈ I ofmetric spaces, E ⊲ ∏ i ∈ I E i implies E ⊲ E i for some i ∈ I. This notion was used by E. Corominasfor posets [7].
Theorem 6. (Theorem 3.8 [23] ) Let E be a finite absolute retract. The following propertiesare equivalent:(i) E is finitely indecomposable;(ii) E is the injective envelope of a two-element metric space { x, y } such that the distance d ( x, y ) is join-irreducible in H . Sum of metric spaces.
Let ( E , d ) and ( E , d ) be two disjoint H -metric spacesand let x ∈ E , x ∈ E . If we endow the set { x , x } with a H -distance d ′ , then we candefine a H -distance d on E ∶= E ∪ E as follows: ● If x, y ∈ E i with i ∈ { , } then d ( x, y ) = d i ( x, y ); ● If x ∈ E i , y ∈ E j with i, j ∈ { , } and i ≠ j , then d ( x, y ) = d i ( x, x i ) ⋅ d ′ ( x i , x j ) ⋅ d j ( x j , y ) . In particular, we can identify x and x which amounts to set d ′ ( x , x ) = E and E are not disjoint, we replace it by two disjoint copies E ′ , E ′ ( eg E ′ i ∶= E i × { i }) . Identifying the corresponding elements x ′ , x ′ , we obtain a H -metric space thatwe denote ( E , d ) ⋅ ( E , d ) . Alternatively, we may suppose that E and E have onlyone element in common, say z , , and we define the distance d on E ∪ E by setting d ( x, y ) ∶= d i ( x, z , ) ⋅ d j ( z , , y ) if x ∈ E i , y ∈ E j , i /= j , and d ( x, y ) ∶= d i ( x, y ) if x, y ∈ E i .We consider now objects consisting of a H -metric space and two distinguished elements.Given two such objects, say E ∶= (( E , d ) , x , y ) and E ∶= (( E , d ) , x , y ) , set E ⋅ E ∶= (( E, d ) , x, y ) where ( E, d ) is the space obtained by taking disjoint copies ( E ′ , d ′ ) and ( E ′ , d ′ ) of ( E , d ) and ( E , d ) , respectively, and by identifying the corresponding elements y ′ and x ′ and setting x ∶= x ′ and y ∶= y ′ . Definition 7.
Let H be Heyting. A pair ( v , v ) ∈ H is summable if S v ⋅ v is isomorphicto S v ⋅ S v . Definition 8.
Let H ′ be an initial segment of H which is also a submonoid. We say that H ′ has the decomposition property if every pair ( v , v ) ∈ H ′ × H ′ is summable. Several examples of metric spaces over a Heyting algebra are given in [17]. We brieflyexamine some of these examples w.r.t. to their injective envelopes and the sum operation.Ordinary metric spaces enter in his frame. Add a largest element +∞ to the set R + ofnon-negative reals, extend the + operation in the natural way, take the identity for theinvolution. Then R + ∪ {+∞} becomes a Heyting algebra and the metric spaces over it arejust direct sums of ordinary metric spaces. The injective envelope of such a space is thedirect sum of the injective envelope of its factors. The injective envelope of a two-elementmetric spaces E ∶= { x, y } with r ∶= d ( x, y ) is isometric to the segment [ , r ] si r < +∞ and to E if r = +∞ . Trivially, R + has the decomposition property. The sum of two convex (resp. NJECTIVE ENVELOPES AND FERRERS LANGUAGES 9 injective) metric spaces with a common vertex is convex (resp. injective). The reader willfind in [9] a description of injective envelopes of finite ordinary metric spaces and interestingcombinatorial properties as well (see also [10]).Now, let
H ∶= { a, b, , } ordered by 0 < a, b < a incomparable to b . The operation isthe join, the involution exchange a and b . Metric spaces over H correspond to ordered sets.As shown by Banaschewski and Bruns, every poset P has an injective envelope, namely itsMacNeille completion [1]. Hence, if P has two elements, its injective envelope is P wheneverthese two elements are comparable, otherwise this is P augmented of a smallest and a largestelement. Convexity property does not hold for H , hence the sum of two injective with acommon element does not need to be injective.Next, suppose that H is a complete meet-distributive lattice, the operation is the joinand the involution is the identity. For example, if H ∶= R ∪ {+∞} , metric spaces over H aredirect sums of ultrametric spaces. Metric spaces over Boolean algebras have been introducedby Blumenthal [3]. Let B be a Boolean algebra, let the operation be the supremum andthe involution be the identity. Although B is not necessary complete, residuation allows todefine a distance on B setting d B ( p, q ) ∶= p ∆ q where ∆ denotes the symmetric difference.From this follows that the interval [ , u ] ∶= { v ∈ B ∶ ≤ v ≤ u } with the distance inducedby d B is the injective envelope of every pair { p, q } of vertices of B such that p ∧ q = p ∨ q = u . Hence if B is finite, the number of elements of the injective envelope of a 2-elementmetric space is a power of 2 hence the decomposition property does not hold. In Section3, we give an example for which this decomposition property holds, namely the algebra F ○ ( A ∗ ) (for more examples of generalisations of metric spaces, see [4], [5]).3. The Heyting algebra F ( A ∗ ) Let A be a set. Considering A as an alphabet whose members are letters , we write aword α with a mere juxtaposition of its letters as α = a a ...a n − where a i are letters from A for 0 ≤ i ≤ i − . The integer n is the length of the word α and we denote it ∣ α ∣ . Hencewe identify letters with words of length 1. We denote by ◻ the empty word, which is theunique word of length zero. The concatenation of two word α ∶= a ⋯ a n − and β ∶= b ⋯ b m − is the word αβ ∶= a ⋯ a n − b ⋯ b m − . We denote by A ∗ the set of all words on the alphabet A . Once equipped with the concatenation of words, A ∗ is a monoid, whose neutral elementis the empty word, in fact A ∗ is the free monoid on A . A language is any subset X of A ∗ . We denote by ℘( A ∗ ) the set of languages. We will use capital letters for languages. If X, Y ∈ ℘( A ∗ ) we may set XY ∶= { αβ ∶ α ∈ X, β ∈ Y } (and use Xy and xY instead of X { y } and { x } Y ). With this operation, which extends the concatenation operation on A ∗ , the set ℘( A ∗ ) is a monoid (the set {◻} is the neutral element). Ordered by inclusion, this is a (join)lattice ordered monoid. Indeed, concatenation distributes over arbitrary union, namely: ( ⋃ i ∈ I X i ) Y = ⋃ i ∈ I X i Y. This monoid is residuated. Let
X, Y, F ∈ ℘( A ∗ ) . As it is customary, we set X − F ∶= { y ∈ A ∗ ∶ Xy ⊆ F } and F Y − ∶= { x ∈ A ∗ ∶ xY ⊆ F } . We recall that(7) X ( X − F ) ⊆ F and ( F Y − ) Y ⊆ F. Set ρ F ∶= {( x, y ) ∈ A ∗ × A ∗ } and R ∶= ( A ∗ , ρ F , A ∗ ) , then R ( x ) = x − F and R − ( y ) = F y − .Thus, Gal ( R ) , the Galois lattice of R , is the set { F Y − ∶ Y ⊆ A ∗ } ordered by inclusion. Thisis a complete lattice. The meet is the intersection, the largest element is A ∗ .In the sequel, we study the metric structure of Gal ( R ) when F is a final segment of themonoid A ∗ , this monoid being equipped with the Higman ordering. We suppose from now that the alphabet A is ordered and equipped with an involution − preserving the order. The involution extends to A ∗ : we set for every α ∶= a ⋯ a n − , α ∶= a n − ⋯ a . Note that αβ = βα for all α, β ∈ A ∗ . We order A ∗ with the Higman ordering: if α and β are two elements in A ∗ such α ∶= a ⋯ a n − and β ∶= b ⋯ b m − then α ≤ β if thereis an injective and increasing map h from { , ..., n − } to { , ..., m − } such that for each i , 0 ≤ i ≤ n −
1, we have a i ≤ b h ( i ) . Then A ∗ becomes an ordered monoid with respect tothe concatenation of words. Let F ( A ∗ ) be the collection of final segments of A ∗ (that is X ∈ F ( A ∗ ) if X ⊆ A ∗ and α ≤ β, α ∈ X implies β ∈ X ) . The set F ( A ∗ ) is stable w.r.t.the concatenation of languages: if X, Y ∈ F ( A ∗ ) , then XY ∈ F ( A ∗ ) . Clearly, the neutralelement is A ∗ . The set F ( A ∗ ) ordered by inclusion is a complete lattice (the join is theunion, the meet is the intersection). Concatenation distributes over union. Order F ( A ∗ ) byreverse of the inclusion, denote X ≤ Y instead of X ⊇ Y , extend the involution − to F ( A ∗ ) ,set X = { α ∶ α ∈ X } , denote by X ⋅ Y the concatenation XY and set 1 ∶= A ∗ then: Theorem 9.
The set F ( A ∗ ) ∶= ( F ( A ∗ ) , ≤ , ⋅ , , − ) is a Heyting algebra. We may then define metric spaces over F ( A ∗ ) and study injective objects and particularlyinjective envelopes.According to Corollary 4.9 p.177 of [23] we have: Theorem 10.
Let F , F , F be final segment of A ∗ such that F = F ⋅ F . If F ≠ ∅ then F = F ⋅ F ⇐⇒ S F ≅ S F ⋅ S F According to Definition 8, this means that F ○ ( A ∗ ) ∶= F ( A ∗ )∖{∅} has the decompositionproperty.Among metric spaces over F ( A ∗ ) are those coming from reflexive and involutive transitionsytems. They are introduced in the next section.4. Transition systems as metric spaces
Let A be a set. A transition system on the alphabet A is a pair M ∶= ( Q , T ) where T ⊆ Q × A × Q. The elements of Q are called states and those of T transitions . Let M ∶= ( Q, T ) and M ′ ∶= ( Q ′ , T ′ ) be two transition systems on the alphabet A . A map f ∶ Q —→ Q ′ is a mor-phism of transition systems if for every transition ( p, a, q ) ∈ T , we have ( f ( p ) , a, f ( q )) ∈ T ′ .When f is bijective and f − is a morphism from M ′ to M , we say that f is an isomorphism .The collection of transition systems over A , equipped with these morphisms, form a category.This category has products. If ( M i ) i ∈ I is a family of transition systems, M i ∶= ( Q i , T i ) , thentheir product M is the transition system ( Q, T ) where Q is the direct product ∏ i ∈ I Q i and T is defined as follows: if x ∶= ( x i ) i ∈ I and y ∶= ( y i ) i ∈ I are two elements of Q and a is a letter,then ( x, a, y ) ∈ T if and only if ( x i , a, y i ) ∈ T for every i ∈ I .An automaton A on the alphabet A is given by a transition system M ∶= ( Q, T ) andtwo subsets I, F of Q called the set of initial and final states . We denote the automatonas a triple ( M, I, F ) . A path in the automaton A ∶= (
M, I, F ) is a sequence c ∶= ( e i ) i < n ofconsecutive transitions, that is of transitions e i ∶= ( q i , a i , q i ⋅ ) . The word α ∶= a ⋯ a n − is the label of the path, the state q is its origin and the state q n its end . One agrees to define foreach state q in Q a unique null path of length 0 with origin and end q . Its label is the emptyword ◻ . A path is successful if its origin is in I and its end is in F . Finally, a word α on thealphabet A is accepted by the automaton A if it is the label of some successful path. The language accepted by the automaton A , denoted by L A , is the set of all words accepted by NJECTIVE ENVELOPES AND FERRERS LANGUAGES 11 A . Let A ∶= (
M, I, F ) and A ′ ∶= ( M ′ , I ′ , F ′ ) be two automata. A morphism from A to A ′ isa map f ∶ Q —→ Q ′ satisfying the two conditions:(1) f is morphism from M to M ′ ;(2) f ( I ) ⊆ I ′ and f ( F ) ⊆ F ′ .If, moreover, f is bijective, f ( I ) = I ′ , f ( F ) = F ′ and f − is also a morphism from A ′ to A ,we say that f is an isomorphism and that the two automata A and A ′ are isomorphic .To a metric space ( E, d ) over F ( A ∗ ) , we may associate the transition system M ∶= ( E, T ) having E as set of states and T ∶= {( x, a, y ) ∶ a ∈ d ( x, y ) ∩ A } as set of transitions. Noticethat such a transition system has the following properties: for all x, y ∈ E and every a, b ∈ A with b ≥ a :1) ( x, a, x ) ∈ T ;2) ( x, a, y ) ∈ T implies ( y, a, x ) ∈ T ;3) ( x, a, y ) ∈ T implies ( x, b, y ) ∈ T. We say that a transition system satisfying these properties is reflexive and involutive (cf.[33, 23]). Clearly if M ∶= ( Q, T ) is such a transition system, the map d M ∶ Q × Q —→ F ( A ∗ ) where d M ( x, y ) is the language accepted by the automaton ( M, { x } , { y }) is a distance. Wehave the following: Lemma 2.
Let ( E, d ) be a metric space over F ( A ∗ ) . The following properties are equiva-lent:(i) The map d is of the form d M for some reflexive and involutive transition system M ∶=( E, T ) ;(ii) For all α, β ∈ A ∗ and x , y ∈ E , if α ⋅ β ∈ d ( x, y ) , then there is some z ∈ E such that α ∈ d ( x, z ) and β ∈ d ( z, y ) . The category of reflexive and involutive transition systems with the morphisms definedabove identify to a subcategory of the category having as objects the metric spaces andmorphisms the contractions. Indeed:
Fact 1.
Let M i ∶= ( Q i , T i ) ( i = , ) be two reflexive and involutive transition systems. Amap f ∶ Q —→ Q is a morphism from M to M if only if f is a contraction from ( Q , d M ) to ( Q , d M ) . From this fact, we can observe that if ( M i ) i ∈ I is a family of transition systems M i ∶=( Q i , T i ) then the metric space ( Q, d ) associated to the transition system ∏ i ∈ I M i , productof the M i ’s, is the product of metric spaces ( Q i , d i ) associated to the transition systems ( Q i , T i ) .Injective objects satisfy the convexity property stated in ( ii ) of Lemma 2. Hence, if F isa final segment of A ∗ , the distance d on the injective envelope S F coincide with the distance d F associated with the transition system M F ∶= ( S F , T F ) where T F ∶= {( p, a, q ) ∶ a ∈ d ( p, q ) ∩ A } . We denote by A F the automaton ( M F , { x } , { y }) , where x ∶= A ∗ and y ∶= F .From the existence of the injective envelope, we get: Theorem 11.
For every F ∈ F ( A ∗ ) there is an involutive and reflexive transition system M ∶= ( Q, T ) , an initial state x and a final state y such that the language accepted by theautomaton A = ( M, { x } , { y }) is F. Moreover, if A is well-quasi-ordered then we may choose Q to be finite. Proof.
Take M ∶= M F , x ∶= A ∗ and y ∶= F . If A is well-quasi-ordered then A ∗ is also well-quasi-ordered (Higman[14]), hence the final segment F has a finite basis, that is, there arefinitely many words α , ..., α n − such that F = { α ∶ α i ≤ α for some i < n } . (cid:3) Since injective objects in the category of metric spaces satisfy the convexity propertystated in ( ii ) of Lemma 2, their distance is the distance associated with a transition system.Thus we may reproduce Theorem 4 almost verbatim. We get: Theorem 12.
Let M ∶= ( Q, T ) be a transition system, X ⊆ Q . Then ( M, d M ) is isomorphicto the injective envelope of ( X, d M ↾ X ) iff for every reflexive and involutive transition system M ′ ∶= ( Q ′ , T ′ ) , X ′ ⊆ Q ′ , every isometry f ∶ ( X, d M ↾ X ) onto ( X ′ , d M ′ ↾ X ′ ) :(i) Every non-expansive map f ∶ ( M, d M ) → ( M ′ , d M ′ ) , if any, which extends f is anisometric embedding of ( M, d M ) into ( M ′ , d M ′ ) ;(ii) The isometry f − extends to a non-expansive map from ( M ′ , d M ′ ) into ( M, d M ) . Taking for X a 2-element subset { x, y } of Q , Theorem 4 translates to Theorem 1 statedin the introduction.4.1. Minimal automaton and Minmax automata.
We suppose now that the alphabet A is finite. We refer to [34] Subsection 3.3 p. 111-118 for the construction of the minimalstate deterministic automaton. If F ∈ F ( A ∗ ) and u ∈ A ∗ , the left residual is u − F ∶= { v ∈ A ∗ ∶ uv ∈ F } . The minimal automaton M in F recognizing F has Q F ∶= { u − F ∶ u ∈ A ∗ } asset of states. Its initial state is x ∶= F , its final state is y ∶= A ∗ and the transition function δ F associate to the pair ( u − F, a ) ∈ Q F × A the state ( ua ) − F .One can check that: Lemma 3.
The map i ∶ M in F → A F defined by i ( Y ) ∶= ⋂ y ∈ Y F y − is a morphism of automata.Proof. Suppose that ( u − F, a, ( ua ) − F ) is a transtion in the automaton M in F . We provethat ( i ( u − F ) , a, i (( ua ) − F )) is a transition in the automaton A F . This is equivalent to a ∈ d ( i ( u − F ) , i (( ua ) − F ) . The last condition amounts to 1) i ( u − F ) a ⊆ i (( ua ) − F ) and 2) i (( ua ) − F a ⊆ i ( u − F ) . For 1), let v ∈ i ( u − F ) , we claim that va ∈ i (( ua ) − F ) , that is foreach word w , if uaw ∈ F , then vaw ∈ F . Since v ∈ i ( u − F ) , from uaw ∈ F , we have vaw ∈ F ,as required. The inclusion 2) follows directly from the fact that F is a final segment.The equalities i ( F ) = A ∗ and i ( A ∗ ) = F are obvious. (cid:3) In general, the transition system associated to
M in F is neither reflexive nor involutive.Among all involutive and reflexive transition systems M ∶= ( Q, T ) with initial state x andfinal state y such that the language accepted by the automaton ( M, { x } , { y }) is F , we selectthose with the least number of states and among those transition systems, we select thosewith a maximum number of transitions; we call minmax transition systems these transitionsystems. The automaton corresponding to a minmax transition system is a minmax au-tomaton . In other terms, an automaton A ∶= ( M, { x } , { y }) is minmax if ( a ) it is reflexiveand involutive, ( b ) for every reflexive and involutive automaton A ′ = ( M ′ , { x ′ } , { y ′ }) suchthat d M ( x, y ) = d M ′ ( x ′ , y ′ ) , one has ∣ Q ∣ ≤ ∣ Q ′ ∣ , moreover ( c ) ∣ Q ∣ = ∣ Q ′ ∣ implies ∣ T ∣ ≥ ∣ T ′ ∣ . Proposition 3.
Let M ∶= ( Q, T ) be transition system and F be a final segment of A ∗ . If A ∶= ( M, { x } , { y }) is a minmax automaton which accepts F , then A is isomorphic to aninduced automaton A ′ of A F , that is there is some Q ′ ⊆ S F containing A ∗ and F such thatthe automaton A ′ ∶= ( M F ↾ Q ′ , { A ∗ } , { F }) is isomorphic to A . NJECTIVE ENVELOPES AND FERRERS LANGUAGES 13
Proof.
To the automaton
A ∶= ( M, { x } , { y }) , we associate the metric space ( Q, d M ) inducedby M . Identifying x with A ∗ and y with F = d M ( x, y ) , the space S F identifies with theinjective envelope of the 2-element space { x, y } . Let i be the identity mapping on { x, y } .This is a partial non-expansive mapping from ( Q, d M ) into S F . Since S F is injective, thispartial map extends to a non-expansive mapping f defined on Q . Since f is non-expansive,this map is an automata morphism. Let Q ′ ∶= f ( Q ) and consider the transition system M ′ ∶= ( Q ′ , S F ↾ Q ′ ) . We have F = d ( x, y ) ≤ d M ′ ( x, y ) ≤ d M ( x, y ) = F. Thus d M ′ ( x, y ) = F. Since ∣ Q ∣ is minimal, it follows that ∣ Q ∣ = ∣ Q ′ ∣ proving that f is injective. Now since ∣ T ∣ ismaximum, we have ∣ T F ↾ Q ′ ∣ = ∣ T ∣ proving that f is an automata isomorphism from A to A ′ ∶= ( M ′ , { x } , { y }) . (cid:3) Let M ∶= ( Q , T ) and M ∶= ( Q , T ) be two transition systems such that Q and Q have only the element y in common. Let A ∶= ( M , { x } , { y }) and A ∶= ( M , { x } , { z }) be two automata. We denote A ⋅ A the automaton ( M, { x } , { y }) where M = ( Q, T ) Q = Q ∪ Q and T = T ∪ T . Theorem 13.
Let F be a non-empty final segment of A ∗ . If F = F F ,with F and F final segments of A ∗ , then an automaton A with initial state x and final state y accepting F is minmax if and only if it decomposes into A ⋅ A where A and A are two minmaxautomata accepting respectively F between x and z and F between z and y .Proof. From Theorem 10, we have S F ⋅ S F ≅ S F F . According to Proposition 3, a minmaxautomaton A accepting F is a subautomaton of A F = A F ⋅ A F . Thus A decomposes into A ⋅ A where A is a subautomaton of A F and A is a subautomaton of A F . Since A isminmax, both A and A are minmax. Conversely, assume that A and A are minmax.Let A be a minmax automaton accepting F ⋅ F . Then A decomposes into A ′ ⋅ A ′ where A ′ and A ′ are minmax. The automaton A and A ′ (resp. A and A ′ ) have the same numberof states and transitions. That is A ⋅ A is minmax. (cid:3) Example 14. (Mike Main,1989, communicated by Maurice Nivat [27] ). We give an exampleof two non-isomorphic minmax automata accepting the same language L ∈ F ( A ∗ ) .Let A = { a, b, c, a ′ , b ′ , c ′ } with a = a ′ , b = b ′ and c = c ′ . Consider the automata representedon Figure 2. To each of these automata, we associate the involutive and reflexive automataobtained by replacing each transition ( p , α, q ) by ( p , α, q ) and ( q , α, p ) and adding a loopat every vertex.The language accepted by each of these automata between x and y is L =↑{ ab, ac, ba, bc, ca, cb } . As it is easy to check, these automata are minmax but not isomorphic. (cid:115) (cid:115) (cid:115)(cid:115)(cid:115) (cid:45) b (cid:18) a (cid:82) c (cid:82) b (cid:18) a (cid:45) a (cid:45) c (cid:82) c (cid:18) b x y (cid:115) (cid:115) (cid:115)(cid:115)(cid:115) (cid:45) b (cid:18) b (cid:82) a (cid:82) a (cid:18) c (cid:45) a (cid:45) c (cid:18) c (cid:82) b x y Figure 25.
A description of the injective envelope
We describe the injective envelope S F in Galois lattice terms. We derive a test of itsfiniteness which leads to the notion of Ferrers language.5.1. Incidence structures.
We recall first some basic facts about incidence structures,Galois lattices and Ferrers relations. We follow the exposition given in [31].An incidence structure R is a triple ( V, ρ, W ) where ρ is a subset of the product V × W. We set ρ − ∶= {( x, y ) ∶ ( y, x ) ∈ ρ } and R − ∶= ( W, ρ − , V ) , that we call the dual of R . Wedenote by ¬ ρ the relation V × W ∖ ρ and ¬ R the resulting incidence structure.A subset of V × W of the form X × Y is a rectangle . Let ( X, Y ) ∈ ℘( V ) × ℘( W ) . We set R ∧ ( X ) ∶= { y ∈ W ∶ xρy for all x ∈ X } , R − ∧ ( Y ) ∶= { x ∈ V ∶ xρy for all y ∈ Y } . We recallthat the set X × Y is a maximal rectangle included into ρ if and only if X = R − ∧ ( Y ) and Y = R ∧ ( X ) . The Galois lattice
Gal ( R ) of R is the collection, ordered by inclusion, of subsetsof V of the form R − ∧ ( Y ) for Y ∈ ℘( W ) . This is a complete lattice; the largest element is V ( = R − ∧ (∅) ). Then, Gal ( R − ) is the collection, ordered by inclusion, of subsets of W ofthe form R ∧ ( X ) for X ∈ ℘( V ) . We recall the important fact that Gal ( R ) and Gal ( R − ) are dually isomorphic. Since Gal ( R ) consists of intersections of sets of the form R − ( y ) for y ∈ W , Gal ( R ) is finite if and only if the set of R − ( y ) for y ∈ W is finite; since Gal ( R ) isdually isomorphic to Gal ( R − ) , it is finite if and only if the set of R ( x ) for x ∈ V is finite.We give two examples from the theory of ordered sets. Fact 2. If R ∶= ( P, ≤ , P ) is an ordered set, then Gal ( R ) , the MacNeille completion of P ,is a complete lattice in which every member is a join and a meet of elements of P . And Gal (¬ R ) is the set of final segments of P ordered by inclusion. Now, we mention the facts we need.Let R := ( V, ρ, W ) and R ′ ∶= ( V ′ , ρ ′ , W ′ ) be two incidence structures. According toBouchet [2], a coding from R into R ′ is a pair of maps f ∶ V —→ V ′ , g ∶ W —→ W ′ such that xρy ⇐⇒ f ( x ) ρ ′ g ( y ) . NJECTIVE ENVELOPES AND FERRERS LANGUAGES 15
Bouchet’s Coding Theorem [2] below is a striking illustration of the links between codingand embedding.
Theorem 15.
Let ( T, ≤) be a complete lattice and R be an incidence structure. Then R has a coding into ( T, ≤ , T ) if and only if Gal ( R ) is embeddable in T . The basic facts about coding we need are the following:
Fact 3.
Let ( f, g ) be a coding from R into R ′ .(a) ( f, g ) is a coding from ¬ R to ¬ R ′ ; (b) Gal ( R ) is embeddable into Gal ( R ′ ) ;(c) If f is surjective, then Gal ( R ) identifies with an intersection closed subset of Gal ( R ′ ) .Proof. ( a ) Immediate consequence of the definition. ( b ) Follows from Bouchet’s Theorem. ( c ) The map X —→ X ′ = R ′− ∧ ( R ′∧ ( X )) is an embedding from Gal ( R ) into Gal ( R ′ ) whichpreserves non-empty intersections. If f is surjective, then the least element of the Galoislattice is preserved, hence Gal ( R ) identifies with an intersection closed subset of Gal ( R ′ ) . (cid:3) Fact 4.
Let ( f i , g i ) i be a family of codings from ( V, ρ i , W ) into ( V i , θ i , W i ) . Then, the pair ( Π i f i , Π i g i ) is a coding from ( V, ⋃ i ρ i , W ) into ( Π i V i , ¬ Π i ¬ θ i , Π i W i ) . Proof.
Clearly ( Π i f i , Π i g i ) is a coding from ( V, ⋂ i ρ i , W ) into ( Π i V i , Π i θ i , Π i W i ) . Indeed, if x ∶= ( x i ) i and y ∶= ( y i ) i , then x Π i θ i y means x i θ i y i for all i . From Fact 3 (a) we get that ( Π i f i , Π i g i ) is a coding from ( V, ⋃ i ρ i , W ) into ( Π i V i , ¬ Π i ¬ θ i , Π i W i ) . (cid:3) Corollary 16.
If for every i , V i = W i and θ i is of the form ¬ ≤ i for some ordering ≤ i on V i , then there is a coding from ( V, ⋃ i ρ i , W ) into ( Π i V i , ¬ ≤ , Π i V i ) where ≤ is the productordering on Π i V i . Fact 5.
Let R ∶= ( V, ρ, W ) be an incidence structure. Then Gal ( R ) is finite if and only if Gal (¬ R ) is finite.Proof. For each y ∈ V , (¬ R ) − ( y ) = V ∖ R − ( y ) . Since Gal ( R ) is made of intersections ofsets of the form R − ( y ) , if Gal ( R ) is finite, the collection of such sets is finite, hence thecollection of sets of the form (¬ R ) − ( y ) is finite too. Since Gal (¬ R ) is made of intersectionsof these sets, it is finite. (cid:3) Fact 6.
Let R i := ( V i , ρ i , W i ) be a family of incidence structures. Then Gal ( Π i R i ) embedsinto Π i Gal ( R i ) . Proof.
Let y ∶= ( y i ) i ∈ Π i W i . We have ( Π i R i ) − ( y ) = Π i R − i ( y i ) . Hence, if p j ∶ Π i V i —→ V j denotes the j th projection, then for each X ∈ Gal ( Π i R i ) , we have ( p i ( X )) i ∈ Π i Gal ( R i ) . This defines an embedding, proving our claim. If each member X of Gal ( Π i R i ) is non-empty,then this embedding is an isomorphism. (cid:3) Fact 7.
Suppose V i = V, W i = W for each i ∈ I and I finite. If Gal ( R i ) is finite for each i ∈ I, then Gal (( V, ⋃ i ρ i , W )) is finite.Proof. According to Fact 5, this amounts to prove that Gal (( V, ⋂ i ¬ ρ i , W )) is finite. Fact4 yields that ( V, ⋃ ρ i , W ) has a coding into ( V V , ¬ Π i ¬ ρ i , W W ) . According to ( a ) of Fact3, ( V, ∩¬ ρ i , W ) has a coding into ( V V , Π i ¬ ρ i , W W ) ≡ Π i ( V, ¬ ρ i , W ) . According to ( b ) ofFact 3, Gal (( V, ⋂ i ¬ ρ i , W )) embeds into Gal ( Π i ( V, ¬ ρ i , W )) , which in turns embeds intoΠ i Gal ( V, ¬ ρ i , W ) from Fact 6. From Fact 5, each Gal ( V, ¬ ρ i , W ) is finite. The resultfollows. (cid:3) Languages and their Galois lattices.
Galois lattices arose from group theory andgeometry. We show here how they interact with language theory.We represent subsets of the free monoid by incidence structures as follows.Let A be an alphabet and L be a subset of A ∗ . Denoting by xy the concatenation of thewords x, y ∈ A ∗ , define a binary relation ρ L on A ∗ by: xρ L y ⇐⇒ xy ∈ L and set R L ∶= ( A ∗ , ρ L , A ∗ ) .We mention without proof some properties of this association.It preserves Boolean operations, that is: Fact 8. (1) ρ A ∗ ∖ L = A ∗ × A ∗ ∖ ρ L . Hence, R A ∗ ∖ L = ¬ R L .(2) ρ ⋃ i L i = ⋃ i ρ L i and ρ ⋂ i L i = ⋂ i ρ L i Not every binary relation ρ on A ∗ can be of the form ρ L . For an example, if ρ containsa pair ( u, v ) with u, v ∈ A ∗ , then ρ contains ρ { w } where w ∶= uv . In fact, if ρ is a binaryrelation on A ∗ , set π ρ ∶= { uv ∶ ( u, v ) ∈ ρ } ; then ρ π ρ is the least binary relation of the form ρ L containing ρ . In particular, ρ { w } is the least relation of the form ρ L containing a pair ( u, v ) such that w = uv .In the Boolean lattice made of relations of the form ρ L , the atoms are of the form ρ { w } where w is any word in A ∗ . They form a partition of A ∗ × A ∗ . Every binary relation of theform ρ L is an union of some blocks of this partition.We will say more about this association in Section 6.We conclude this subsection by the following property.If B is a subset of A ∗ , we set R L ↾ B ∶= ( B, ρ L ∩ B × B, B ) . Fact 9.
The Galois lattices of R L and and R L ↾ A ∗ ∖ L are isomorphic provided that L is afinal segment of A ∗ . Proof.
Observe that for every subset X of A ∗ , we have: R − ∧ ( R ∧ ( X )) = R − ∧ ( R ∧ ( X ∖ L )) . (cid:3) Proof of the first part of Theorem 2.
From Corollary 16, and c ) of Fact 3, in orderto prove the first part of Theorem 2, namely that S F can be identified to an intersectionclosed subset of the set F ( n ⊗ ⋯ ⊗ n k − ) , all that we need is to prove that for each i , there isa coding ( f i , g i ) , with f i surjective, from ( A ∗ , ρ F i , A ∗ ) into ( n i , ¬ ≤ , n i ) . But, this is false.From Fact 9, a coding from ( A ∗ ∖ F, ρ
F i , A ∗ ∖ F ) into ( n i , ¬ ≤ , n i ) suffices. Let X , ..., X n − be non-empty subsets of A and let X ∶= X ⋅ ⋅ ⋅ X n − (that is the set of words x ⋯ x n − with x i ∈ X i ) and let F ∶=↑ X . Let n ∶= { , ..., n − } be equipped with the natural ordering. Let f ∶ A ∗ ∖ F —→ n and g ∶ A ∗ ∖ F —→ n defined as follows: For v ∈ A ∗ ∖ F, if there is m ∈ N suchthat v ∈↑ ( X ⋅ ⋅ ⋅ X m − ) then f ( v ) is the largest m having this property, otherwise f ( v ) = w ∈ A ∗ ∖ F , if there is p ∈ N such that v ∈↑ { X p − ⋅ ⋅ ⋅ X n − } then g ( v ) is the least p having this property, otherwise g ( v ) = n − . By a straightforward verification, we have thefollowing:
Fact 10.
The map f is surjective and the pair ( f, g ) form a coding from ( A ∗ ∖ F, ρ F , A ∗ ∖ F ) into ( n , ¬ ≤ , n ) . This proves the first part of Theorem 2.
NJECTIVE ENVELOPES AND FERRERS LANGUAGES 17
Proof of the second part of Theorem 2.
Let F ∶= F ∪ ... ∪ F k − , let X i ⊆ F such that F i =↑ X i and let X i , ..., X i ni − ⊆ A such that X i ⋅ ⋯ ⋅ X i ni − = X i . From Fact10 above, for each i, we have a coding ( f i , g i ) from ( A ∗ ∖ F i , ρ F i , A ∗ ∖ F i ) into ( n i , ¬ ≤ , n i ) .The restrictions of f i and g i to A ∗ ∖ F give a coding that we still denote ( f i , g i ) from ( A ∗ ∖ F i , ρ F i , A ∗ ∖ F i ) into ( n i , ¬ ≤ , n i ) and f i is still surjective. From Fact 4, ( Π f i , Π g i ) is a coding from R F ∶= ( A ∗ ∖ F, ∪ ρ F i , A ∗ ∖ F ) into R ′ ∶= ( n ⊗ ... ⊗ n k − , ¬ ≤ , n ⊗ ... ⊗ n k − ) where ≤ is the natural ordering on the direct product n ⊗ ... ⊗ n k − . Since for each i, f i issurjective, Π f i is surjective and from ( c ) of Fact 3, Gal ( R F ) is identified with an intersectionclosed subsets of Gal ( R ′ ) which is F ( n ⊗ ... ⊗ n k − ) by Fact 2. ◻ An explicit isomorphism.
For reader’s convenience, let us describe explicitly theisomorphism between S F and an intersection closed subset of F ( n ⊗ ... ⊗ n k − ) . For sakeof simplicity, let us suppose that F is finitely generated (which is the case if A is w.q.o.).Let u , . . . , u k − ∈ A ∗ , n ∶= ∣ u ∣ , . . . , n k − ∶= ∣ u k − ∣ , and F ∶=↑ { u , ..., u k − } . Let v ∈ A ∗ ; foreach i ∈ { , ..., k − } , let v ′′ i be the largest suffix of u i such that v ′′ i ≤ v and let v ′ i be theunique prefix of u i such that u i = v ′ i v ′′ i . Set s ( v ) ∶= { x ∶= ( x , ..., x k − ) ∈ n ⊗ ... ⊗ n k − suchthat x i < ∣ v i ∣ for all i } and τ ( v ) ∶= n ⊗ ... ⊗ n k − ∖ s ( v ) . Define ϕ ∶ S F —→ F ( n ⊗ ... ⊗ n k − ) by setting ϕ ( X ) ∶= ⋂ τ ( v ) v ∈ A ∗ ,X ⊆ ρ − ( v ) . For w ∶= w ...w m ∈ A ∗ and (cid:96) < ∣ w ∣ , let w ∣ (cid:96) = w ...w (cid:96) − be the restriction of w to the first (cid:96) letters. Let x ∶= ( x , ..., x k − ) ∈ n ⊗ ... ⊗ n k − , let µ ( x ) =↑ { u i ∣ x i − ∶ ≤ i ≤ k − } be the finalsegment of A ∗ generated by words of the form u i ∣ x i − . Define Ψ ∶ F ( n ⊗ ... ⊗ n k − ) —→ S F by settingΨ ( Y ) ∶= ⋂ µ ( v ) x ∈ n ⊗ ... ⊗ n k − ∖ Y The maps ϕ and Ψ preserve intersections and Ψ ○ ϕ = id S F . Finiteness of the injective envelope.Proposition 4.
Let F be a final segment of A ∗ . The following conditions are equivalent:(i) The injective envelope S F is finite;(ii) F is finite union F = F ∪ ... ∪ F i ∪ ... ∪ F k − of final segments, each F i generated bya set X i of words u i of the same length n i all of the form u i = u i ...u i j ...u i ni − with u i j ∈ X i j ⊆ A. Proof. ( i ) ⇒ ( ii ) . Let us suppose that S F is finite. Let A F be the corresponding reflexiveand involutive automaton with an initial state x ∶= A ∗ and a final state y ∶= F . Let Q ∗ F bethe set of finite sequences s ∶= ( s , ..., s n ) such that all s i are distinct states, s ∶= x, s n ∶= y, ( s i , a i , s i + ) ∈ T F for some letter a i . Since S F is finite this set is finite too. For each s = ( s , ..., s n ) let X s j ∶= { a ∈ A ∶ ( s j , a, s j + ) ∈ T F } and let X s ∶= { a ...a n − ∶ a j ∈ X s j } andlet F s ∶=↑ X s . It is easy to check that F = ⋃ s ∈ Q F F s , hence has the form mentioned above. ( ii ) ⇒ ( i ) . Apply Theorem 2. (cid:3) Theorem 17.
Let
H ∶= F ( A ∗ ) be the Heyting algebra made of final segments of A ∗ . Thefollowing conditions are equivalent:(i) A is well-quasi-ordered; (ii) The injective envelope N ( E ) of every finite metric space E is finite.Proof. ¬( i ) (cid:212)⇒ ¬( ii ) . Let F be a final segment of A ∗ . According to Proposition 4, theinjective envelope S F is infinite whenever for each decomposition F = ∪ F i , some F i cannot begenerated by a set of words having a bounded length. This is the case if F is generated by aninfinite antichain X made of words of unbounded length. If A is not w.q.o., then there is aninfinite bad sequence of letters, say a , ..., a n , .... The set X = { a , a a , a a a , a a a a , ... } is such an example. ( i ) (cid:212)⇒ ( ii ) . According to Proposition 1, it suffices to show that the injective envelope of atwo-element metric space is finite. Let F be a final segment of A ∗ . If A is w.q.o., F satisfiescondition ( ii ) of Proposition 4, hence S F is finite. (cid:3) Comments.
Proposition 4 and Theorem 17 are special instances of two basic facts oflanguage theory concerning rational languages, namely Myhill-Nerode Theorem (see [34]Theorem 2.3, p.247) and a result of Ehrenfeucht-Haussler-Rozenberg (Theorem 3.3 of [12],see [34] Theorem 5.3, p.296). For brevity, let us put these results together in the context ofmonoids. Let us recall that if L is a subset of a monoid M , the residuals of L are sets of theform Lv − ∶= { u ∈ M ∶ uv ∈ L } for v ∈ M and of the form u − L ∶= { v ∈ M ∶ uv ∈ L } for u ∈ M .The syntactic congruence of L is the largest congruence ≡ L on M for which L is an unionof classes. The set L is recognisable if it is the union of classes of a congruence on M whichhas finitely many classes(alias finite index). The aforementioned results read as follows: Theorem 18.
For a subset L of a monoid M , the following properties are equivalent:(i) L is recognisable;(ii) The set of residuals { Lv − ∶ v ∈ M } is finite;(iii) The set of residuals { u − L ∶ u ∈ M } is finite;(iv) The syntactic congruence ≡ L has finite index;(v) L is a final segment of a well-quasi-ordered set on M which is compatible with themonoid operation. The equivalences from ( i ) to ( iv ) is Myhill-Nerode Theorem. The equivalence with ( v ) is Ehrenfeucht-Haussler-Rozenberg Theorem.Let R ∶= ( M, ρ L , M ) be the incidence structure where ρ L ∶= {( u, v ) ∈ M × M ∶ uv ∈ L } .Then uρ L v ⇐⇒ uv ∈ L and R ( u ) = u − L , R − ( v ) = Lv − for all u, v ∈ M . Conditions ( ii ) and ( iii ) in the theorem above express both that the Galois lattice Gal ( R ) is finite.According to Higman’s Theorem, if the alphabet A is w.q.o., A ∗ equipped with theHigman ordering is w.q.o. Since this ordering is compatible with concatenation, we mayapply ( v ) to every final segment F of A ∗ , obtaining that the Galois lattice Gal ( R ) is finite.Since the domain of this lattice is S F , implication ( i ) (cid:212)⇒ ( ii ) of Theorem 17 follows.6. Interval orders, Ferrers relations and injective envelope
We record the characterization of interval orders and Ferrers relations. These two notionsare intertwined. Interval orders are those orderings for which the irreflexive part is a Ferrersrelation. Ferrers relations have been introduced by J.Riguet [32]. Interval orders are studiedby Fishburn in [13]. Part of the characterization of these relations given below is due toWiener [39]. Let C be a chain; an interval of C is any subset of I such that x ∈ I, y ∈ I and x ≤ z ≤ y imply z ∈ I. ( ) The collection
Int C of non-empty intervals of C is ordered as follows: X < Y if x < y for all x ∈ X, y ∈ Y. ( ) NJECTIVE ENVELOPES AND FERRERS LANGUAGES 19
Let P be a poset. The ordering on P, or P itself, is an interval order if P is orderisomorphic to a collection of non-empty intervals of some chain C , ordered by condition ( ) . To each incidence structure R ∶= ( V, ρ, W ) we associate a poset B ( R ) ∶= ( P, ≤) defined asfollows:The domain of P is V × { } ∪ W × { } , for u = ( x, i ) , v = ( y, j ) ∈ P, the order relation isdefined by: u < v if i < j and xρy. ( ) Let R ∶= ( V, ρ, W ) be an incidence structure. We say that R is Ferrers or ρ is a Ferrersrelation if R satisfies one of the following conditions (see [13]): Proposition 5.
Let R ∶= ( V, ρ, W ) be an incidence structure. The following conditions areequivalent:(i) The set { R ( x ) ∶ x ∈ V } is totally ordered by inclusion;(ii) The set { R − ( y ) ∶ y ∈ W } is totally ordered by inclusion;(iii) The Galois lattice Gal ( R ) is totally ordered by inclusion;(iv) R has a coding into a chain;(v) The poset B ( R ) does not embed the direct sum ⊕ of two copies of the -elementchain ;(vi) The ordering on B ( R ) is an interval order;(vii) xρy and x ′ ρy ′ imply xρy ′ or x ′ ρy for all x, x ′ ∈ V , y, y ′ ∈ W. We say that a language L is Ferrers if the relation ρ L is Ferrers.According to condition ( vii ) of Proposition 5, this amounts to the following condition: xx ′ ∈ L and yy ′ ∈ L imply xy ′ ∈ L or yx ′ ∈ L for all x, x ′ , y, y ′ ∈ A ∗ . ( ) The study of Ferrers relations leads to the notion of Ferrers dimension of a binary relation:the least number of Ferrers relations whose intersection is this relation. The study of thisnotion, initiated by Bouchet in his thesis [2], yields numerous interesting results (e.g., see[6, 8]). This suggest to look at the same direction in the theory of languages. But, wemay notice that contrarily to the case of relations, not every language is Ferrers, or is anintersection of Ferrers languages. In fact, if w is any word, ρ { w } is a Ferrers relation only if w = ◻ ( indeed, let R ∶= ( A ∗ , ρ { w } , A ∗ ) , then Gal ( R ) = {{ u } ∶ u prefix of w } ∪ {∅ , A ∗ } , thus if w /= ◻ , the Galois lattice has at least two incomparable elements, namely { w } and {◻} plus atop and a bottom, thus it is not a chain). Thus, if w /= ◻ , { w } is not a Ferrers language andsince it is a singleton, it is not a union of Ferrers languages, hence its complement A ∗ ∖ { w } is not an intersection of Ferrers languages.Since the complement of a Ferrers relation is Ferrers, the complement of a Ferrers languageis Ferrers (apply ( ) of Fact 8). The concatenation of two Ferrers languages is not Ferrers ingeneral. For a simple minded example, let A ∶= { a, b } , let U ∶= { a n ∶ n ≥ } , U ′ ∶= { b n ∶ n ≥ } .Each of these languages is Ferrers, but the concatenation U U ′ is not: let x ∶= a b , y ∶= b and x ′ ∶= a , y ′ ∶= ab . Then xy = x ′ y ′ = a b ∈ U U ′ but neither xy ′ = a bab nor x ′ y = ab belong to U U ′ .Recall that for two words u and v , u is a prefix of v if v = uw for some word w ; similarly, u is a suffix of v if v = wu for some word w . Fact 11.
Let U and U ′ be two subsets of A ∗ . If U, U ′ are Ferrers and U is a final segmentfor the prefix ordering or U ′ is a final segment for the suffix ordering then the concatenation U U ′ is Ferrers.Proof. Let L ∶= U U ′ , let xx ′ ∈ L and yy ′ ∈ L . We prove that either xy ′ or x ′ y belong to L .There are four cases to consider; we only consider two, the others being similar. Case 1. x = x x , y = y y with x , y ∈ U and x x ′ ∈ U ′ and y y ′ ∈ U ′ . Since U ′ is Ferrers,either x y ′ ∈ U ′ or y x ′ ∈ U ′ . In the former case xy ′ ∈ U U ′ whereas in the latter yx ′ ∈ U U ′ . Case 2. x = x x , y ′ = y ′ y ′ with x ∈ U, yy ′ ∈ U, x x ′ ∈ U ′ and y ′ ∈ U ′ . If U is a finalsegment for the prefix ordering, then x x y ′ ∈ U since x belongs to U and is a prefix of x x y ′ . Thus xy ′ = x x y ′ y ′ ∈ U U ′ . If U ′ is a final segment for the suffix ordering, then x y ′ y ′ ∈ U ′ since y ′ belongs to U ′ and is a suffix of y ′ . Thus xy ′ = x x y ′ y ′ ∈ U U ′ . (cid:3) Corollary 19.
The concatenation of finitely many Ferrers final segments of A ∗ is a Ferrersfinal segment. We recall that an ideal of an ordered set P is any non-empty initial segment I which isup-directed (that is any two elements x and y of I have an upper bound z in I ). Filtersare defined dually. Ideals of P are the join-irreducible elements of the lattice I ( P ) ofinitial segments of P . Ideals of the poset A ∗ equipped with the Higman ordering have beendescribed when A is finite by Jullien [18] and by us [20] when A is an ordered alphabetpossibly infinite. According to Jullien, an elementary ideal of A ∗ is any set of the form J ∪ {◻} for some non empty ideal J of A , a star-ideal is any set of the form I ∗ for someinitial segment I of A . Products of ideals are ideals. It is proved in [20] that every ideal is afinite product of elementary and star-ideals if and only if the alphabet is well-quasi-ordered. Fact 12.
Ideals and filters of A ∗ are Ferrers.Proof. Let I be an ideal of A ∗ . Suppose xx ′ , yy ′ ∈ I . We prove that xy ′ or yx ′ ∈ I . Let z ∈ I such that xx ′ , yy ′ ≤ z. Let z be the least prefix of z such that x, y ≤ z and z thecorresponding suffix, i.e., z = z z . If x ′ ≤ z or y ′ ≤ z , then since x, y ≤ z then xy ′ or yx ′ ∈ I . If not, then since xx ′ ≤ z and yy ′ ≤ z , we have x, y ≤ z − , where z − is obtained from z by deleting its last letter. This contradicts the choice of z . Hence, I is Ferrers. Let F be a filter of A ∗ . Let xx ′ , yy ′ ∈ F . Let z ∈ F such that z ≤ xx ′ and z ≤ yy ′ . Write z = z x z x ′ with z x ≤ x , z x ′ ≤ x ′ and z = z y z y ′ with z y ≤ y , z y ′ ≤ y ′ . Either z y ′ ≤ z x ′ or z x ′ ≤ z y ′ . Inthe first case, since z y ≤ y and z y ′ ≤ z x ′ ≤ x ′ we have z = z y z y ′ ≤ yx ′ , hence yx ′ ∈ F . In thesecond case we obtain similarly xy ′ ∈ F . Hence F is Ferrers. (cid:3) For every u ∈ A ∗ , the initial segment ↓ u of A ∗ is an ideal and the final segment ↑ u of A ∗ is a filter, hence: Corollary 20.
For every u ∈ A ∗ , the initial segment ↓ u of A ∗ and the final segment ↑ u of A ∗ are Ferrers. Since { u } =↓ u ⋂ ↑ u , we have: Corollary 21.
For every u ∈ A ∗ , { u } is an intersection of two Ferrers languages and A ∗ ∖ { u } is an union of two Ferrers languages. If A is w.q.o., every final segment is finitely generated and every ideal is a finite unionof ideals, hence from the second part of Corollary 20 and from the first part of Fact 12, weobtain. Corollary 22. If A is w.q.o., every final segment if a finite union and a finite intersectionof Ferrers languages. NJECTIVE ENVELOPES AND FERRERS LANGUAGES 21
From Corollary 19 and Fact 12 follows:
Proposition 6.
Final segments of A ∗ which are finite product of complement of ideals of A ∗ are Ferrers. According to Corollary 21, for every u ∈ A ∗ , A ∗ ∖{ u } is an union of two Ferrers languages,hence: Proposition 7.
Every language is a union, possibly infinite, of a family of intersections oftwo Ferrers languages.
According to Corollary 22, if the alphabet is w.q.o. (and particularly, if it is finite),Boolean combinations of final segments, alias piecewise testable languages, are Booleancombination of rational Ferrers languages. ● If the alphabet A consists of one letter, say a , these two Boolean algebras coincide .Indeed, if L is Ferrers then with respect to the natural order on A ∗ ∶= { a n ∶ n ∈ N } , it isconvex. Otherwise, there are n < p < m ∈ N such that a n , a m ∈ L , a p /∈ L . Choosing n, m with the difference m − n minimum, we have a q /∈ L for n < p < m . Let X ∶= ( a n ) − L and Y ∶= ( a m − ) − L . Then ◻ and a m − n ∈ X but ◻ /∈ X , whereas a ∈ Y but ◻ /∈ Y . Hence, X and Y are incomparable with respect to inclusion, contradicting the fact that L is Ferrers.Being convex, L is the intersection of an initial segment segment with a final segment, thusit is piecewise testable. ● If A ∶= { a, b } , with a /= b , then L ∶= A ∗ b is rational and Ferrers and not piecewise testable .Indeed, let Q L ∶= { u − L ∶ u ∈ A ∗ } , then Q L has two elements, namely L and L ′ ∶= {◻} ∪ L (in fact a − L = L , b − L = L ′ , a − L ′ = L , b − L ′ = L ′ . The fact that L is not piecewisetestable follows from Stern’criterium ([36] Theorem 1.2): the sequence ( u n ) n ∈ N defined by u n ∶= ( ba ) n b and u n + ∶= ( ba ) n + is increasing for the subword ordering while a n ∈ L and a n + /∈ L for all n ∈ N .The language L above has dot-depth one. Is this general? That is: Question.
Do rational Ferrers languages have dot-depth one?
We relate Ferrers piecewise testable languages and structural properties of transitionsystems.
Theorem 23.
Let F be a final segment of A ∗ . The following conditions are equivalent:(1) F is Ferrers;(2) The space S F is linearly orderable.Proof. Proposition 5 and Proposition 2. (cid:3)
Corollary 24.
The finitely indecomposable absolute retracts are linearly orderable.Proof.
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Laboratoire Math´ematiques et Applications, D´epartement de Math´ematiques, Facult´e des Sci-ences et Techniques, Universit´e Hassan II -Casablanca, BP 146 Mohammedia, Morocco.
E-mail address : [email protected] Univ. Lyon, Universit´e Claude-Bernard Lyon1, CNRS UMR 5208, Institut Camille Jordan, 43bd. 11 Novembre 1918, 69622 Villeurbanne Cedex, France and Mathematics & Statistics Depart-ment, University of Calgary, Calgary, Alberta, Canada T2N 1N4
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