The globals of pseudovarieties of ordered semigroups containing <span class="mathjax-formula">$B_2$</span> and an application to a problem proposed by Pin

DOI: 10.1051/ita:2005001

THE GLOBALS OF PSEUDOVARIETIES OF ORDERED SEMIGROUPS CONTAINING B

AND AN APPLICATION

TO A PROBLEM PROPOSED BY PIN

Jorge Almeida

and Ana P. Escada

Abstract. Given a basis of pseudoidentities for a pseudovariety of ordered semigroups containing the 5-element aperiodic Brandt semi- group B2, under the natural order, it is shown that the same basis, over the most general graph over which it can be read, defines the global. This is used to show that the global of the pseudovariety of level 3/2 of Straubing-Th´erien’s concatenation hierarchy has infinite vertex rank.

Mathematics Subject Classification.20M05, 20M07, 20M35.

1. Introduction

Three concatenation hierarchies of rational languages have been considered since the 1970’s: the dot-depth hierarchy introduced by Brzozowski [10], the Straubing-Th´erien hierarchy [32, 33], and the group hierarchy presented in [19].

Pin (see [22]) has observed that all these hierarchies may be regarded as being indexed by half integers, that is numbers of the form n or n+ 1/2 where n is a non-negative integer, and where each level other than level zero is obtained in the following way: the languages of leveln+ 1/2 are finite unions of products of the

Keywords and phrases. Semigroup, pseudovariety, semigroupoid, category, pseudoidentity, dot-depth, concatenation hierarchies.

∗Both authors acknowledge partial support by the project POCTI/32817/MAT/2000, which is co-financed by the European Community Fund FEDER, and by the INTAS grant #99-1224.

Partial support by FCT, through the Centro de Matem´atica da Universidade do Portoand through theCentro de Matem´atica da Universidade de Coimbra, respectively, is also gratefully acknowledged.

1 Centro de Matem´atica, Faculdade de Ciˆencias, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal;[email protected]

2Departamento de Matem´atica, Faculdade de Ciˆencias e Tecnologia, Universidade de Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal.

c EDP Sciences 2005

formL₀a₁L₁. . . a_rL_r,whereL₀, . . . , L_rare languages of levelnanda₀, . . . , a_rare letters¹, and the languages of leveln+ 1 are members of the Boolean closure of the class of languages of leveln+ 1/2. Thus each hierarchy is completely determined by its level zero. The dot-depth hierarchyBn has the finite or cofinite languages ofA⁺ as a basis, the Straubing-Th´erien hierarchyVn is based on the languages∅ and A^∗, and the level zero of the group hierarchy Gn is obtained by taking the group languages. These three hierarchies are infinite and strict (see [11, 22]). The main problem, which is open in almost all cases, is whether each level is decid- able. In fact, for the Straubing-Th´erien hierarchy, this problem has been solved (positively) forn≤3/2 [8, 9, 30] and some partial results are known for the level two [24, 34, 35, 38]. For the other two hierarchies, the membership problem is only known to be decidable for n ≤ 1 [14, 16, 17, 19]. See [31] for related problems concerning the product of rational languages.

Some of the early results concerning the dot-depth and Straubing-Th´erien hi- erarchies follow from deep theorems of Simon [30]. The problem of the effec- tive characterization of thelocally testable languages was solved independently by McNaughton [20] and by Brzozowski and Simon [12]. This characterization im- plies that locally testable languages are of dot-depth one. The graph-theoretic arguments which were implicit in these works have had a strong impact on lan- guage and semigroup theories, namely in Knast’s and Tilson’s results. Knast [17]

obtained a complex theorem on graphs and achieved an effective characterization of the whole class of languages of dot-depth one.

Tilson [37] (inspired by work of Simon [30], Knast [16,17], Th´erien and Weiss [36], and Straubing [33]) developed a theory of (small) categories as partial algebras over graphs and showed its importance to study semigroups. In particular, his Derived Category Theorem turned out to be a powerful tool to deal with semidirect prod- ucts of pseudovarieties of semigroups [7, 27]. In the profinite context, Weil and the first author [7] have used the derived category theorem to describe a basis of pseudoidentities of the semidirect productV∗Wof pseudovarieties of semigroups from a basis of semigroupoid pseudoidentities of the pseudovariety gV of semi- groupoids generated by the semigroups ofV viewed as one-vertex semigroupoids.

The proof of this result, that has come to be known as the basis theorem, has a gap, as has been observed by Rhodes and Steinberg, but the theorem stands in caseW is locally finite orgVhas finite vertex-rank.

Since each level of the considered hierarchies is a positive variety of languages, the variety theorem [13, 21] guarantees a one-to-one correspondence between the levelnof each hierarchy and a pseudovariety of ordered monoids (or semigroups in case of the dot-depth hierarchy). The problem of decidability for a levelnof the hierarchies is now reduced to determine whether the pseudovarietiesBn,VnandGn

are decidable, where Bn, Vn, and Gn denote respectively the pseudovarieties of ordered semigroups (monoids) associated toBn,Vn, andGn. There are interesting

1This is the definition adopted when the empty word is considered, that is for languages of A^∗. Otherwise, for languages of A⁺, one takes instead finite unions of languages of the formu0L1u1. . . Lrur, whereL1, . . . , Lrare languages of levelnandu0, . . . , ur∈A^∗are words withu0 nonempty in caser= 0.

connections betweenVn, Bn andGn: Bn =Vn∗ LI (n >0) [32], where LI is the pseudovariety oflocally trivial semigroups, andGn=Vn∗G(n≥0) [23], whereG is the pseudovariety of all finite groups. It is not immediately clear if these results reduce the study of one hierarchy to another. However, it is known that, for each integern,Bn is decidable if and only ifVn is decidable [33].

For a pseudovarietyV of finite ordered semigroups, V denotes the pseudova- riety of all finite ordered semigroupoids whose finite one-vertex subsemigroupoids with at least one edge lie in V, and gV is the pseudovariety of ordered semi- groupoids generated by the ordered semigroups ofVviewed as ordered one-vertex semigroupoids. A basis of pseudoidentities of ordered semigroupoids which de- fines V can be easily obtained from a given basis for V. However it is not so easy to compute a basis of semigroupoid pseudoidentities forgVgiven such a basis forV. Based on results of Reilly [28], Azevedo, Teixeira and the first author [5]

proved that the problem may be systematically treated when the pseudovarietyV of semigroups contains the 5-element aperiodic Brandt semigroupB₂.

In Section 3, we show that a similar approach works in caseVis a pseudovariety of ordered semigroups containing the ordered inverse semigroupB₂. Indeed, given a basis Σ of pseudoidentities ofV, then the set consisting of the pseudoidentities of Σ over the most general graphs over which they can be read is a basis for the global ofV.

In Section 4, we use our result as well as techniques introduced in [6] to show that the pseudovarietyV³₂ of ordered semigroups, which corresponds to the level 3/2 of the Straubing-Th´erien hierarchy, has infinite vertex rank. Initially, as suggested by J.-E. Pin, our goal was to investigate, using the “basis theorem” for semidirect products, the decidability of the pseudovariety of finite semigroupsG³₂ =V³₂ ∗G, which corresponds to the level 3/2 of the group hierarchy. The infinity of the vertex-rank ofV³₂ renders this problem out of reach for the currently known range of validity of the basis theorem.

2. Preliminaries

We recall in Sections 2.1 and 2.2 some definitions and results concerning ordered semigroups and semigroupoids, free profinite semigroups and semigroupoids, and pseudovarieties of ordered semigroups and semigroupoids. For more details the reader is referred to [3, 4, 15, 22, 27]. Section 2.3 introduces basic facts about inverse semigroups and a version in the “ordered semigroups context” of a result of Reilly [28].

2.1. Ordered semigroups

Anordered semigroupSis a semigroup equipped with a partial order relation≤ such that, for allx, y, z∈S, x≤yimpliesxz ≤yzandzx≤zy. Ahomomorphism of ordered semigroups ϕ : (S,≤) → (T,≤) is a semigroup homomorphism such that, for allx, y ∈S,x≤yimpliesϕ(x)≤ϕ(y). Semigroups are viewed as ordered

semigroups under the equality order relation. An ordered semigroup (S,≤) is an ordered subsemigoupof (T,≤) ifSis a subsemigroup ofT and the order onSis the restriction toSof the order onT. If there is a surjective homomorphism of ordered semigroupsϕ: (S,≤)→(T,≤), then we say that (T,≤) is anordered quotient of (S,≤). Theproduct of a family (S_i,≤i)_i∈I of ordered semigroups is the ordered semigroup (

i∈IS_i,≤) where the multiplication is defined by (s_i)_i∈I(t_i)_i∈I = (s_it_i)_i∈I and the order is given by

(s_i)_i∈I ≤(t_i)_i∈I if and only if, for all i∈I,s_i≤it_i. (1)

Throughout this paper X represents a finite non-empty set. LetX⁺ be the free semigroup onX. Then (X⁺,=) is an ordered semigroup and it is the free ordered semigroup onX.

Atopological semigroupis a semigroup endowed with a topology such that the multiplication is continuous. Finite semigroups are viewed as discrete topological semigroups. For a finite set X, we say that a topological semigroup S is X- generated if a mapping η : X → S is given such that η(X) generates a dense subsemigroup ofS.

For a finite set X, the projective limit of X-generated finite semigroups is denoted by X⁺, and it is called the free profinite semigroup [4]. This semi- group is compact, totally disconnected andX-generatedviathe natural mapping ι:X→X⁺. One may show thatX⁺ has the following universal property: every mapping ϕ:X →S into a finite semigroup S can be extended to a unique con- tinuous homomorphism ˆϕ:X⁺→S such that ˆϕι=ϕ. Moreover, the topological structure ofX⁺ is metrizable, that is X⁺ may be endowed with a metricdsuch thatdinduces the given topology ofX⁺ [4, Prop. 7.4].

Note that the subsemigroup generated byιX is (isomorphic to) the free semi- groupX⁺ so we may assume thatX⁺ is a subsemigroup ofX⁺.

Let u, v ∈ X⁺. We say that a finite ordered semigroup (S,≤) satisfies the (pseudo)identity u≤v and we writeS |=u≤v if, for every continuous homomor- phismϕ:X⁺→S, we haveϕ(u)≤ϕ(v).

A pseudovariety of ordered semigroups is a class of finite ordered semigroups which is closed under taking finite ordered subsemigroups, finitary products and ordered quotients.

Pin and Weil have extended Reiterman’s theorem [29] to pseudovarieties of ordered semigroups [25]. For a set Σ of pseudoidentities, we denote by [[Σ]] the class of all finite ordered semigroups which satisfy all pseudoidentities of Σ.

Theorem 2.1. Let V be a class of finite ordered semigroups. ThenV is a pseu- dovariety of ordered semigroups if and only if there exists a setΣ of pseudoiden- tities over finite sets such thatV= [[Σ]].

2.2. Ordered semigroupoids

A graph is a set G =V(G) ∪^◦ E(G) consisting of two sorts of elements, ver- tices and edges, endowed with two operations α, ω : E(G) → V(G) which give respectively the beginning and end vertices of each edge. For a, b ∈ E(G), we say that a and b are coterminal if α(a) = α(b) and ω(a) = ω(b), and they are consecutive if ω(a) = α(b). A path in a graph Γ is a finite sequencea₁. . . a_n of edges of Γ such thata_ianda_i+1are consecutive, fori= 1, . . . , n−1. Graphs with one vertex may be identified with their sets of edges. Agraph homomorphism is a mapping ϕ : G → H between two graphs respecting sorts and operations. A subgraph H of G is a graph contained inG such that the inclusion H → Gis a graph homomorphism.

By asemigroupoid we mean a graphS endowed with a partial associative mul- tiplication onE(S) given by: for s, t∈E,st is defined if and only ifω(s) =α(t) and, then,α(st) =α(s) andω(st) =ω(t). IfSadmits a local identity at each ver- tex, then we say thatSis acategory. For verticesc, d∈V(S), thehom-set S(c, d) is the set of edgess∈E(S) such thatα(s) =c andω(s) =d; in casec=dwe put S(c) =S(c, d).

For a semigroupoidS, a binary relationτ on E(S) is said to becompatible if, for allx, y ∈E(S),

(i) if (x, y)∈τ thenxandy are coterminal;

(ii) if (x, y)∈τ and x, zare consecutive edges, then (xz, yz)∈τ;

(iii) if (x, y)∈τ and z, xare consecutive edges, then (zx, zy)∈τ.

Acongruence τ on a semigroupoidS is an equivalence relation onE(S) which is compatible. Thequotient semigroupoid S/τ has V(S/τ) = V(S) and E(S/τ) = E(S)/τ, with the induced mapsαandω and composition of edges.

We say that the semigroupoid S is an ordered semigroupoid if it is equipped with a compatible partial order relation≤onE(S). It is important to notice that two edges must be coterminal to be comparable under≤.

For a finite graph Γ, thefree semigroupoid Γ⁺on Γ has as vertex-setV(Γ) and as edges the non-empty paths of Γ.

Ahomomorphism of ordered semigroupoidsis a mapϕbetween (S,≤) and (T,≤) which respects sorts, operations and the partial order, that is,ϕ(x)≤ϕ(y) when- everx≤y. A homomorphism of ordered semigroupoidsϕ: S →T is full if the restriction of ϕ to each hom-set S(v₁, v₂) is surjective, ϕ is order-faithful if for every two coterminal edgesx, y, ϕ(x)≤ϕ(y) implies x≤y, and it is said to be a quotient if ϕ is full and ϕ|_V(S) is bijective. Note that, if S is a semigroupoid andτ is a congruence onS, then thecanonical homomorphism η :S→S/τ is a quotient homomorphism.

An ordered semigroupoid S is said to divide an ordered semigroupoid T if there are an ordered semigroupoid U, a quotient homomorphism U → S and an order-faithful homomorphism U → T. The product of a family (S_i)_i∈I of semigroupoids has set of vertices

i∈IV(S_i), set of edges

i∈IE(S_i), and the partial operationsα,ω, and edge composition are defined component-wise. If the factorsS_iare ordered semigroupoids, then so is the product under the order given

by (1). Thecoproduct of (S_i)_i∈I is obtained by taking the disjoint union of theS_i, both for the set of vertices, the set of edges, and the partial operationsα,ω, and edge composition. If the cofactors S_i are ordered semigroupoids then so is the coproduct under the (disjoint) union of the order relations of the cofactors.

A pseudovariety of ordered semigroupoids is a class of finite ordered semi- groupoids containing the one-element semigroup, which is closed under taking divisors of ordered semigroupoids, and finitary direct products and coproducts.

The pseudovariety of all finite ordered semigroupoids is denoted byOSd.

An ordered semigroup S is viewed as an ordered semigroupoid by taking the set of edges S with both ends at an added vertex. Conversely, for an ordered semigroupoidSand a vertexvofS, the setS(v) of allloopsat vertexvconstitutes an ordered semigroup called thelocal semigroup ofS atv.

The pseudovariety of ordered semigroupoids generated by a given pseudovari- ety V of ordered semigroups is called the global of V and is denoted gV. Note that gV is the smallest pseudovariety of ordered semigroupoids whose ordered semigroups are precisely those ofV. The largest such pseudovariety is called the localofVand is denotedV; it consists of all finite ordered semigroupoids such that all local semigroups are members ofV. A pseudovarietyVof ordered semigroups is said to belocal ifgV=V.

Atopological semigroupoid S is a semigroupoid endowed with a topology with respect to which the partial operationsα,ω, and edge multiplication are continu- ous. Finite semigroupoids equipped with the discrete topology become topological semigroupoids.

A topological semigroupoid isprofinite if it is a projective limit of finite semi- groupoids; its topology is said to be theprofinite topology. For a finite graph Γ, a topological semigroupoidSis said to be Γ-generated if there exists a graph homo- morphismϕ: Γ→Ssuch thatϕ_|V(Γ)is injective and the subgraph ofSgenerated byϕ(Γ) is dense.

We denote by Γ⁺ the projective limit of all Γ-generated finite semigroupoids.

Note thatΓ⁺is a Γ-generated semigroupoidviathe natural mappingφ: Γ→Γ⁺. The semigroupoidΓ⁺ has the usual universal property: for every homomorphism ϕ: Γ→S into a profinite semigroupoid there exists a unique continuous homo- morphism ˆϕ : Γ⁺ → S such that ˆϕφ=ϕ. Moreover, we may endow Γ⁺ with a metricdsuch that the topology induced bydis the profinite topology [15].

By a V-pseudoidentity over a graph Γ we mean an ordered pair of the form (x ≤ y,Γ) where x and y are coterminal edges of Γ⁺. A finite ordered semi- groupoidS satisfies the pseudoidentity (x≤ y,Γ) and we write S |= (x≤ y,Γ) if, for each continuous homomorphism of semigroupoids ϕ : Γ⁺ → S, we have ϕ(x)≤ϕ(y).

Given a set Σ of pseudoidentities over finite graphs it is readily checked that the classV = [[Σ]], consisting of all finite ordered semigroupoids S which satisfy all pseudoidentities from Σ, is a pseudovariety of ordered semigroupoids. Then we also say that the pseudovarietyV is defined by Σ or that Σ is abasis of pseu- doidentities forV. The analog of Reiterman’s Theorem in this context states that

every pseudovariety of ordered semigroupoids has a basis of pseudoidentities. The proof of the ordered case can be readily obtained from the unordered case [7, 15]

using the necessary adaptations to accommodate the order as in [25].

Following [1], we say that a pseudovarietyV of semigroupoids hasvertex-rank orv-rank nifnis the smallest non-negative integer such thatVadmits a basis of pseudoidentities over graphs with at mostnvertices. If no such integer nexists, then we say thatVhasinfinite v-rank.

2.3. Inverse semigroups

In this subsection, we recall and extend some results on inverse semigroups.

The reader unfamiliar with this topic may wish to consult [18] for a much more thorough introduction.

A semigroupS is inverse if, for everys∈S there is a unique inverse s⁻¹ ∈S such thatss⁻¹s=sands⁻¹ss⁻¹=s⁻¹. Thenatural orderon a inverse monoidS is defined as follows:

x≤y if and only ifx=ey for somee=e²∈S.

An inverse semigroup endowed with the natural order is an ordered semigroup and it is said to be anordered inverse semigroup.

Inverse semigroups considered as algebras with the binary operation of multi- plication and the unary operation of inversion form the varietyI defined by the equations

xx⁻¹x=x, (x⁻¹)⁻¹=x, xx⁻¹yy⁻¹=yy⁻¹xx⁻¹.

Note that homomorphisms of (inverse) semigroups respect the natural order on inverse semigroups.

Denote byX⁻¹a set disjoint fromXand in one-to-one correspondencex→x⁻¹ withX. We callX⁻¹the set of formal inversesof elements ofX. Forx∈X, we let (x⁻¹)⁻¹=x. There is a free inverse semigroup overX denoted byF I_X. It is well known that

F I_X (X∪X⁻¹)⁺/τ

whereτ is the congruence on (X∪X⁻¹)⁺ generated by the set uu⁻¹u, u

,

uu⁻¹zz⁻¹, zz⁻¹uu⁻¹

: u, z ∈

X∪X⁻¹₊ .

Letw∈ (X ∪X⁻¹)^∗. Thecontent of w, denotedc(w), is the set ofx∈X such that either xor x⁻¹ occur in w. A word w is said to be reduced if w does not contain factorsxx⁻¹ or x⁻¹xfor any x∈X. If w=y₁. . . y_n (y_i ∈X ∪X⁻¹) is reduced then 1 and every wordy_i. . . y_j (1≤i≤j≤n) aresegments ofw, and 1 and every segment ofwwhich begins at y₁ areinitial segments ofw. Let K_X be the set of all words of (X∪X⁻¹)⁺ of the form

w=a₁a⁻¹₁ . . . a_na⁻¹_n g

wherea₁, . . . , a_n, gare reduced words, noa_iis an initial segment of anya_j (j=i) andgis an initial segment ofa₁.

Theorem 2.2.

1. For everyw∈F I_X, there isu∈K_X such thatw=uτ.

2. Letu=a₁a⁻¹₁ . . . a_ma⁻¹_mg andv=b₁b⁻¹₁ . . . b_nb⁻¹_n hbe words ofK_X. Then uτ =vτ if and only if{a₁, . . . , a_m}={b₁, . . . , b_n}andg=h.

We say thatu∈K_X is acanonical form ofuτ ∈F I_X. By abuse of notation, we may also useuforuτ.

LetI be a finite set. The setI×I∪ {0}with multiplication given by (i, j)(i, j) = (i, j) ifi=j

0 otherwise

and 0a=a0 = 0, for anya∈I×I∪ {0}is theI×I aperiodic Brandt semigroup and is denoted byB_I orB_|I|. All aperiodic Brandt semigroups are inverse. LetB be the subvariety of the varietyI of all inverse semigroups generated byB₂.

The following is well known.

Lemma 2.3. Let I be an arbitrary set. ThenB_I belongs toB.

Leta∈F I_X have canonical form

a=a₁a⁻¹₁ . . . a_na⁻¹_n g and define

S[a] = n i=1

{b= 1 :b is an initial segment ofa_i}.

Note that, ifS[a] ={b1, . . . , b_k}thena= (b₁b⁻¹₁ . . . b_kb⁻¹_k g)τ.

Reilly [28] defines two relationsγ_a andδ_a onY =X∪X⁻¹as follows:

• (x, y)∈γ_a if and only if x, y ∈ S[a] or there existsu∈ F I_X∪ {1} such thatux⁻¹y∈S[a] or uy⁻¹x∈S[a].

• δ_a is the reflexive and transitive closure ofγ_a.

Sinceγ_a is a symmetric relation it follows thatδ_a is an equivalence relation.

Ifa_i =a_i1. . . a_ini for i = 1, . . . , n and g =g₁. . . g_r with a_ij, g_k ∈Y, then we puts(a) =a₁₁δ_a and

e(a) = s(a) ifg= 1, g⁻¹_r δ_a otherwise.

Note that, for alli, j= 1, . . . , n, a_i1, a_j1∈S[a] so (a_i1, a_j1)∈δ_a.

We say that a varietyV of inverse semigroupssatisfies u≤v, withu, v∈F I_X if, for every homomorphismϕ:F I_X→S withS∈ V, we haveϕu≤ϕv.

The following result generalizes the necessary condition of Theorem 3.3 of [28].

Proposition 2.4. Let a, b∈F I_X. If B satisfies b≤a then the following condi- tions hold:

(i) c(a)⊆c(b);

(ii) δ_a ⊆δ_b; (iii) s(a)⊆s(b);

(iv) e(a)⊆e(b).

Proof. Let us show thatc(a)⊆c(b). If there isx∈c(a) which is not inc(b) then, taking the homomorphism ϕ: F I_X →B₂ given by ϕ(x) = 0, andϕ(y) = (1,1) for everyy∈X\ {x}, we haveϕ(a) = 0 and ϕ(b) = (1,1), which contradicts our hypothesis.

LetI=Y /δ_b and let us defineθ:X →B_I such that θx= (xδ_b, x⁻¹δ_b). Then there exists a homomorphism ˆθ:F I_X →B_I which extendsθ.

Since B satisfies b ≤ a, by hypothesis, and B_I ∈ B by Lemma 2.3, we have θbˆ ≤θa.ˆ

First we will show that ˆθb= 0. Suppose thatb=b₁b⁻¹₁ . . . b_mb⁻¹_mgis a canonical form ofband, for eachi,b_i =b_i1. . . b_ipi is the reduced form ofb_i. Let 1≤j < p_i. Sinceb_i1. . . b_i,j+1 ∈S[b], it follows that (b⁻¹_ij , b_i,j+1)∈γ_b ⊆δ_b. This allows us to obtain

θ(bˆ _ijb_i,j+1) = (θb_ij)(θb_i,j+1) = (b_ijδ_b, b⁻¹_ij δ_b)(b_i,j+1δ_b, b⁻¹_i,j+1δ_b)

= (b_ijδ_b, b⁻¹_i,j+1δ_b)= 0.

By [28, Lem. 2.2], this implies that ˆθb_i= 0 for everyi, thus ˆθ(b_ib⁻¹_i )= 0, and so θ(bˆ _ib⁻¹_i ) = (θb_i1). . .(θb_ipi)(θb_ipi)⁻¹. . .(θb_i1)⁻¹

= (b_i1δ_b, b_i1δ_b)

= (s(b), s(b)).

Ifg= 1 thengis an initial segment ofb₁so ˆθb= ˆθg= 0. Ifg= 1 then θ(b) = ˆˆ θ(b₁b⁻¹₁ )ˆθ(b₂b⁻¹₂ ). . .θ(bˆ _mb⁻¹_m) = (s(b), s(b))= 0.

Since ˆθb≤θaˆ and ˆθb= 0, we have ˆθa= ˆθb.

Let us show thatδ_a ⊆δ_b. Letx, y ∈Y such that (x, y)∈γ_a. Suppose thata= a₁a⁻¹₁ . . . a_na⁻¹_n his a canonical form ofaand leta₁=a₁₁. . . a_1n1 (a_1i ∈X∪X⁻¹) be the reduced form ofa₁. Ifx∈S[a] then ˆθ(xx⁻¹aa⁻¹) = ˆθ(aa⁻¹) which implies thatxδ_b=a₁₁δ_b. So, if x, y∈S[a], then we have

xδ_b=a₁₁δ_b=yδ_b.

Suppose now that there existsu∈F I_X∪ {1} such thatux⁻¹y ∈S[a]. We have θ(uxˆ ⁻¹y(ux⁻¹y)⁻¹a) = ˆθa and ˆθa= 0 so

0= ˆθ(x⁻¹y) = (x⁻¹δ_b, xδ_b)(yδ_b, y⁻¹δ_b)

which implies thatxδ_b=yδ_b. This shows that γ_a ⊆δ_b, thusδ_a⊆δ_b.

Let us show thats(a)⊆s(b) ande(a)⊆e(b). First, since ˆθa= ˆθb= 0, we must havea₁₁δ_b =b₁₁δ_b, which implies thats(a)⊆s(b) by (ii).

Suppose next thatg=g₁. . . g_randh=h₁. . . h_sare reduced forms ofg andh, respectively. Ifg= 1 andh= 1, then

0= ˆθb= ˆθg= (g₁δ_b, g⁻¹₁ δ_b). . .(g_rδ_b, g_r⁻¹δ_b) = (g₁δ_b, g_r⁻¹δ_b).

Similarly, ˆθa= (h₁δ_b, h⁻¹_s δ_b). Since ˆθa= ˆθb, we haveg⁻¹_r δ_b=h⁻¹_s δ_b, soe(a)⊆e(b) again by (ii). Ifg= 1 and h= 1 then ˆθa= (h₁δ_b, h⁻¹_s δ_b) and ˆθb= (b₁₁δ_b, b₁₁δ_b).

Since ˆθa= ˆθb, we haveh⁻¹_s δ_b=b₁₁δ_b, thuse(a)⊆e(b). The caseg= 1 andh= 1 is similar. Finally, ifg= 1 =hthene(a) =s(a)⊆s(b) =e(b).

3. Pseudoidentities satisfied by the globals of pseudovarieties containing B

In this section we extend some results of Section 5 of [5] from pseudovarieties of semigroups to pseudovarieties of ordered semigroups.

Let X be a finite set and u ∈ X⁺. Let ρ_u be the equivalence relation over Y =X∪X⁻¹generated by the relation

{(x⁻¹, y) : xyis a factor ofu}. (2) Since factors of length 2 can be recognized by finite semigroups, the correspondence u∈ X⁺ → ρ_u ∈ P(Y²) defines a continuous map, where P(Y²) is viewed as a discrete space.

Thecontent ofu, c(u), is the set ofx∈X such that xis a factor of u, that is u=u₁xu₂for some u₁, u₂∈X^∗.

Definition 3.1. [5] Foru∈X⁺andY =X∪X⁻¹, letA_ube the graph defined by V(A_u) =Y /ρ_u

A_u(xρ_u, yρ_u) ={z:z∈X,(x, z)∈ρ_u,(z⁻¹, y)∈ρ_u}.

Note that eachz∈X gives one edge andE(A_u) =X.

This definition ofA_uintroduces some edgesx /∈c(u) which do not play any role when we test pseudoidenties overA_u, but it simplifies some technical arguments.

Lemma 3.2. Let X be a finite set. Ifu∈X⁺ thenρ_u=δ_u.

Proof. Suppose thatu=x₁. . . x_n withx_i∈X, i= 1, . . . , n. Thenu=uu⁻¹uis a canonical form ofu∈F I_X. It follows that (x, y)∈γ_u if and only ifx=y=x₁ or (exactly) one the two-letter wordsx⁻¹y andy⁻¹xis a factor ofu. This shows thatγ_ucontains the relation (2) and is contained is the reflexive symmetric closure of (2). Henceγ_u and (2) generate the same equivalence relation.

For a finite setX andu∈X⁺, we denote byt₁(u) the unique letter which is a suffix of u, that is,u=u₁t₁(u) for some u₁∈X^∗. Dually,i₁(u) is the unique letter which is aprefix ofu, which means thatu=i₁(u)u₂for some u₂∈X^∗. Theorem 3.3. Let X be a finite set and u, v∈X⁺. The following conditions are equivalent:

(a) B₂ satisfiesu≤v.

(b) We haveρ_v⊆ρ_u,(i₁(u), i₁(v))∈ρ_u and(t₁(u)⁻¹, t₁(v)⁻¹)∈ρ_u. (c) There is a graph homomorphism θ:A_v→A_u such that

(i) θ(i₁(v)ρ_v) =i₁(u)ρ_u andθ(t₁(v)⁻¹ρ_v) =t₁(u)⁻¹ρ_u, and (ii) for every z∈A_v(xρ_v, yρ_v),θ(z) =z.

Proof. First, we prove that (a) is equivalent to (b). Recall that uandv are limits of sequences of words, say (u_n) and (v_n), respectively. By continuity of ρ, i₁, and t₁, we may assume that, for every n, ρ_un = ρ_u, ρ_vn = ρ_v, i₁(u_n) = i₁(u), i₁(v_n) =i₁(v),t₁(u_n) =t₁(u), andt₁(v_n) =t₁(v).

AsB₂ is finite, there ispsuch that, for eachn≥p,B₂satisfies the pseudoiden- tities u=u_n andv =v_n. By Lemma 3.2 and Proposition 2.4, if B₂ satisfies the inequality u_n ≤v_n then Condition (b) holds foru_n and v_n. Thus ifB₂ satisfies u≤v then Condition (b) holds.

To prove the converse, letη :X⁺→B₂ be an arbitrary continuous homomor- phism of ordered semigroups. If η(u) = 0, the minimum of B₂, then obviously η(u)≤η(v). So, assume thatη(u)= 0. By continuity ofη, we may assume that η(u_n) =η(u) for everyn. It is clear that Condition (b) holds foru_n andv_n. Our goal is to show thatη(u_n) =η(v_n) and thusη(u) =η(v), by continuity ofη. Hence we may assume thatu, v∈X⁺.

Suppose thatxyis a factor of v. Sinceρ_v⊆ρ_u, then there are an indexnand z₁, z₂, . . . , z_2n∈X such that the words

z₀z₁, z₂z₁, z₂z₃, . . . , z_2nz_2n−1, z_2nz_2n+1,

are factors ofu, wherez₀=xandz_2n+1=y. Ifη(z_i) = (r_i, s_i) withr_i, s_i∈ {1,2}, thenη(u)= 0 implies thats₀ =r₁ =s₂ =r₃ =· · · =s_2n =r_2n+1. Sincev is a word, it follows thatη(v)= 0, and soη(u) andη(v) lie in the sameJ-class. Fur- thermore, in this case a similar process may be used to show that (i₁(u), i₁(v))∈ρ_u implies thatη(u) andη(v) areR-related, and (t1(u)⁻¹, t₁(v)⁻¹)∈ρ_uimplies that η(u) andη(v) areL-related. Hence we haveη(u) =η(v).

Let us show that (b) is equivalent to (c). Suppose that Condition (b) holds.

Sinceρ_v⊆ρ_u, we may define a mapϕ:A_v→A_u by ϕ(xρ_v) =xρ_u, forxρ_v∈V(A_v),

ϕ(z) =z, forz∈A_v(xρ_v, yρ_v).

It is clear that ϕ is a graph homomorphism. By (b), it is immediate that ϕ(i₁(v)ρ_v) =i₁(v)ρ_u=i₁(u)ρ_uand ϕ(t₁(v)⁻¹ρ_v) =t₁(v)⁻¹ρ_u=t₁(u)⁻¹ρ_u.

Conversely, suppose thatθ:A_v →A_uis a graph homomorphism which verifies the conditions of (c). Forx∈X, we have

x∈A_u(xρ_u, x⁻¹ρ_u), x∈A_u(θ(xρ_v), θ(x⁻¹ρ_v))

andxgives one edge inA_u, thenxρ_u=θ(xρ_v). It follows that, if (x, y)∈ρ_v then θ(xρ_v) =θ(yρ_v), that isxρ_u =yρ_u. Thusρ_v⊆ρ_u.

Ifx∈i₁(v)ρ_v thenxρ_u=θ(xρ_v) =θ(i₁(v)ρ_v) =i₁(u)ρ_u soi₁(v)ρ_v ⊆i₁(u)ρ_u. Similarly, we havet₁(v)⁻¹ρ_v⊆t₁(u)⁻¹ρ_u. Foru∈X⁺, letϕ_u:A_u→X be the natural graph homomorphism, where the finite setX is viewed as a graph with one vertex. Then there is a unique continuous homomorphism of semigroupoidsϕ_u :A⁺_u →X⁺such thatϕ_u extendsϕ_u, that is the following diagram commutes:

A_u ^φu //

ϕu

A⁺_u

ϕu

X //X⁺

By Proposition 2.3 of [2],ϕ_u is faithful. We define thecontent ofw∈A⁺_u as being the content ofϕ_u(w).

Lemma 3.4. Let u ∈ X⁺ and let ϕ_u be the faithful homomorphism of semi- groupoids described above. Thenϕ_u restricted toE(A⁺u)is injective andu=ϕ_u(˜u) for someu˜∈A⁺_u(i₁(u)ρ_u, t₁(u)⁻¹ρ_u)).

Proof. Letϕ =ϕ_u. Note that, if w= a₁. . . a_n (a_i ∈ X) is a finite path onA_u then ˆϕ(w) =w∈X⁺.

Let us show that ˆϕis injective. Suppose that ˆϕ(w₁) = ˆϕ(w₂) for somew₁, w₂∈ E(A⁺_u). Let (s_n) and (r_n) be sequences of finite paths onA_uwhich converge tow₁ andw₂respectively. Since the functionsα,ωand content are continuous, we may assume that, for all n,α(w₁) =α(s_n), α(w₂) = α(r_n), ω(w₁) =ω(s_n),ω(w₂) = ω(r_n), c(w₁) = c(s_n), andc(w₂) = c(r_n). As ˆϕis continuous, the sequences of words ( ˆϕ(s_n)) and ( ˆϕ(r_n)) converge to ˆϕ(w₁) and ˆϕ(w₂), respectively. Since i₁ and t₁ are continuous, we may assume that, for all n, i₁( ˆϕ(s_n)) = i₁( ˆϕ(w₁)),

i₁( ˆϕ(r_n)) =i₁( ˆϕ(w₂)), t₁( ˆϕ(s_n)) =t₁( ˆϕ(w₁)) and t₁( ˆϕ(r_n)) =t₁( ˆϕ(w₂)). There- fore, for alln, we have

i₁( ˆϕ(s_n)) =i₁( ˆϕ(w₁)) =i₁( ˆϕ(w₂)) =i₁( ˆϕ(r_n)), and t₁( ˆϕ(s_n)) =t₁( ˆϕ(w₁)) =t₁( ˆϕ(w₂)) =t₁( ˆϕ(r_n)).

It follows that α(w₁) = α(s_n) = i₁( ˆϕ(s_n))ρ_u = i₁( ˆϕ(r_n))ρ_u = α(r_n) = α(w₂) and, similarly,ω(w₁) =ω(w₂). This shows that w₁ andw₂ are coterminal hence w₁=w₂, since ˆϕis faithful.

Finally, we show that u ∈ ϕ(ˆ A⁺_u). Suppose that u = a₁. . . a_n with a_i ∈ X.

For eachi < n, a_ia_i+1 is a factor of u so (a⁻¹_i , a_i+1) ∈ ρ_u, so that a_i and a_i+1 are consecutive edges, that is ω(a_i) = α(a_i+1). Hence u may be viewed as a finite path and ˆϕ(u) = u, as we have observed above. If u is the limit of a sequence (u_n) of words then, by continuity of the functions involved, we may assume that c(u_n) = c(u), i₁(u_n) = i₁(u), t₁(u_n) = t₁(u), and ρ_un = ρ_u (so A_un=A_u) for alln. By compactness ofA⁺_u, there is a subsequence of (u_n) which converges to some ˜u∈A⁺_u(i₁(u)ρ_u, t₁(u)⁻¹ρ_u). As ˆϕis continuous and ˆϕ(u_n) =u_n

for alln, we conclude that ˆϕ(˜u) =u.

By Lemma 3.4, for a givenu ∈ X⁺, there is a unique ˜u∈ E(A⁺_u) such that

ϕ_u(˜u) =uso we will abuse notation and denote ˜ubyu.

Let u, v ∈ X⁺ and let θ : A_v → A_u be a homomorphism of graphs. By the universal property ofA⁺_v there exists a unique continuous homomorphism ˆθsuch that the following diagram commutes.

A_v ^φv //

θ

A⁺_v

θˆ

A_u

φu

// A⁺_u

Corollary 3.5. Let u, v∈X⁺ be such that B₂ satisfiesu≤v, and let ϕ_u be the homomorphism of graphs described in Lemma 3.4. Then there exists a unique edge v ∈A_u(i₁(u)ρ_u, t₁(u)⁻¹ρ_u)such that ϕ_u(v) =v.

Proof. By Lemma 3.4, there is (a unique) ˜v ∈ A_v(i₁(v)ρ_v, t₁(v)⁻¹ρ_v) such that

ϕ_v(˜v) =v, whereϕ_v:A⁺_v →X⁺is a homomorphism of semigroupoids described in Lemma 3.4. By Theorem 3.3, there exists a homomorphism of graphsθ:A_v→A_u which satisfies

θ(i₁(v)ρ_v) =i₁(u)ρ_u, θ(t₁(v)⁻¹ρ_v) =t₁(u)⁻¹ρ_u (3)

θ(z) =z, for everyz∈E(A_v). (4)

Sinceϕ_v(z) =z=ϕ_u(θ(z)) for everyz∈E(A_v), the following diagram commutes:

A⁺_v ^θˆ //

ϕAvAAAAAA A A⁺_u

ϕu

X⁺

Letv = ˆθ(˜v). Thenv∈A_u(i₁(u)ρ_u, t₁(u)⁻¹ρ_u), by (3), and

ϕ_u(v) =ϕ_u(ˆθ(˜v)) =ϕ_v(˜v) =v.

To complete the proof it is suffices to recall thatϕ_u is faithful.

Taking into account Lemma 3.4 and Corollary 3.5, we will abuse notation and denote ˆθ(˜v) byv.

LetSbe a semigroupoid. If|V(S)|>1 then the consolidated semigroup S_cdis the setS_cd=E(S)∪ {0},with the multiplication defined by

ss= s·s if ωs=αs, 0 otherwise,

and 0·a = a·0 = 0, for everya ∈ S_cd. From this point, we will omit the · to represent the operation in the consolidated semigroup. If |V(S)| = 1 then the consolidated semigroup ofS isS itself, viewed as a semigroup.

If S is an ordered semigroupoid, then S_cd is an ordered semigroup under the order given bys≤s ifs≤s inS, and 0≤afor everya∈S_cdin case 0∈S_cd.

The following lemma is adapted from [5].

Lemma 3.6. Let u, v∈X⁺ and suppose that the ordered semigroup B₂ satisfies the pseudoidentity u ≤ v. Then a finite semigroupoid S satisfies (u ≤v, A_u) if and only ifS_cdsatisfies u≤v.

Proof. By Theorem 3.3 there exists a graph homomorphism θ : A_v → A_u that satisfies Condition (3) of Corollary 3.5. Thus, by Lemma 3.4 and Corollary 3.5, uand v represent edges of A⁺_u from i₁(u)ρ_u to t₁(u)⁻¹ρ_u. Hence (u≤v, A_u) is indeed a semigroupoid pseudoidentity.

Let S be a finite semigroupoid and let (u_n) and (v_n) be sequences of words ofX⁺ converging respectively to uandvin X⁺. We may assume that, for alln, i₁(u_n) =i₁(u), t₁(u_n) = t₁(u), i₁(v_n) =i₁(v), t₁(v_n) =t₁(v), A_un =A_u, A_vn = A_v, S_cd satisfies u=u_n, v =v_n, and S satisfies (u=u_n, A_u) and (v =v_n, A_u).

This reduces the proof to the caseu, v∈X⁺. SinceB₂ satisfies u≤v, it is clear thatc(v)⊆c(u). Without loss of generality, we may assume thatX =c(u).

Suppose first that S_cd |= u ≤ v and consider an arbitrary semigroupoid ho- momorphism ϕ : A⁺_u → S. Define a homomorphism η : X⁺ → S_cd by taking η(z) =ϕ(z) for eachz∈X.