Semigroup ForumSpringer-Verlag New York Inc.

Version of July 21, 2000

A Delay Theorem For Pointlikes

Benjamin Steinberg∗

Communicated by J. -E. Pin

Abstract

This paper shows that, for a pseudovariety V of monoids, V∗D has decidablepointlikes if and only if V does. In the process, we develop a theory of pointlikesets for categories and a generalization of the Derived Category Theorem tounderstand how pointlike sets behave with respect to the semidirect product.This paper is intended to be the first of two papers concerning algorithmicproblems for semidirect products of pseudovarieties.

1. Introduction

Since its inception in [11], the notion of a pseudovariety has played a key rolein the classification of finite semigroups; recall that a pseudovariety of semigroups isa class of finite semigroups closed under finite products, taking subsemigroups, andtaking quotients. Thanks to the Krohn-Rhodes Theorem [16], the semidirect productoperator on pseudovarieties of semigroups has received particular attention.

Amongst the most studied semidirect products of pseudovarieties are those ofthe form V ∗D and V ∗Dn where

Dn = [[yx1x2 · · ·xn = x1x2 · · ·xn]]

and D =⋃n Dn [2, 4, 23, 25]. We try to explain, to some degree, their importance. If

V is a pseudovariety of monoids, then LV denotes the pseudovariety of all semigroupswhose submonoids are in V . It is well-known [11] that V ∗ D ⊆ LV . Work byStraubing [23], Therien and Weiss [24], and Tilson [25] led to a solution as to whenequality holds, first in terms of graph congruences [23, 24], and then, more elegantly,in terms of the notion of a pseudovariety of categories [25]. Throughout this paper,we will use Tilson’s [25] as our primary reference as it states this and other results intheir most modern form; it should not be taken that the aforementioned paper is theoriginal source of all these results. The theorem is then that V∗D = LV if and only ifV is local in the sense of Tilson [25]. We now give a list of some local pseudovarieties;we attempt to give credit to the original author, but try to give references to simpleror more modern proofs as well. Local pseudovarieties of monoids include: R , thepseudovariety of R-trivial monoids, due to Eilenberg [11]; Sl , the pseudovariety ofsemilattices, due to Simon [10], see [11] for a more modern proof; any pseudovarietyof bands, due to Jones and Szenderei [14]; and any non-trivial pseudovariety of groupsdue to Straubing, see [25] for a simple proof. On the other hand, a deep result ofKnast [15] shows that J , the pseudovariety of J -trivial monoids, is not local, while aresult of Therien and Weiss [24] shows that Com , the pseudovariety of commutativemonoids, is not local.

Another reason for the importance of semidirect products of the form V ∗Dis that the operator () ∗D takes the monoidal analog of the dot depth hierarchy tothe dot depth hierarchy, see [23]. For instance, there is the result of Knast [15] thatJ ∗D is the pseudovariety of dot depth 1 semigroups and has a decidable membershipproblem.

Finally, there is the Delay Theorem of Tilson [25], based on an earlier resultby Straubing [23], which shows amongst other things that V ∗D is decidable if andonly if gV , the pseudovariety of categories generated by V , is decidable and that asemigroup of delay n is in V ∗D if and only if it is in V ∗Dn .

In the seventies, Henckell and Rhodes introduced the notion of a pointlike set.Roughly speaking, a subset of a semigroup is said to be V -pointlike, for a pseudovariety

∗*The author was supported in part by Praxis XXI scholarship BPD 16306 98 andby FCT through Centro de Matematica da Universidade do Porto.

2 Steinberg

V , if it relates to a point under every relational morphism with a member of V . Onesays that V has decidable pointlikes if, given a finite semigroup and one of its subsetsas input, one can decide if the subset in question is V -pointlike. Recently, the authorand Almeida have shown, independently, that the decidability of pointlikes can leadto the decidability of joins of pseudovarieties (the join V ∨W of V and W is thesmallest pseudovariety containing them both). Recall that a pseudovariety V is calledlocally finite if it has a free object FV(A) on any finite set A . Any finitely generatedpseudovariety is locally finite as are all pseudovarieties of bands.

The following theorem is proved by the author in [19, 21] and independentlyby Almeida in [3].

Theorem 1.1. Let W be a locally finite pseudovariety of semigroups with a com-putable bound on the order of FW(A) for any finite set A . Let V be a pseudovariety ofsemigroups with decidable pointlikes (or even decidable pointlikes pairs). Then V∨Whas decidable membership.

This then motivates our main theorem:

Theorem 1.2. Let V be a pseudovariety of monoids. Then V has decidablepointlikes if and only if V ∗D has decidable pointlikes.

In fact, it will follow from our results that V has decidable pointlike pairs ifand only if V∗D does. Non-locally finite pseudovarieties of monoids which are knownto have decidable pointlikes include: J , shown independently by the author [19] andby Almeida and Zeitoun [8], and many of its subpseudovarieties, see the author’s [21];R by Almeida and Silva [6]; G (the pseudovariety of groups) by Ash [9]; and Gp (thepseudovariety of p-groups for p a prime) by the author [22]. All pseudovarieties ofbands, being locally finite with a computable bound on the order of the free objecton any set, have decidable pointlikes. The author shows in [20] that any decidablepseudovariety of abelian groups has decidable pointlike pairs. The results of [21]then imply that any decidable pseudovariety of commutative monoids has decidablepointlike pairs (since any such pseudovariety is a join of a pseudovariety of commutativeJ -trivial monoids with a decidable pseudovariety of abelian groups). Thus we candraw the following corollaries:

Corollary 1.3. The pseudovarieties LG , LGp , LSl , LR , and J ∗D have decid-able pointlikes.

Corollary 1.4. Let H be a decidable pseudovariety of abelian groups and C adecidable pseudovariety of commutative monoids, then LH and C ∗D have decidablepointlike pairs.

Of course, these corollaries can then be combined with Theorem 1.1 to deducevarious decidability results for joins.

To prove Theorem 1.2, we use the following proof scheme: first we generalizethe notion of pointlike sets to categories, study how pointlikes for V and gV relate,and show the existence of a Bonded Normal Form for Pointlikes (BNFP); then wegeneralize the Derived Category Theorem [25] to a result about pointlike sets; finallywe generalize the Delay Theorem itself to a result about pointlike sets. The BNFPrequires some technical results about profinite categories which will, along with theBNFP, be used in a sequel paper showing that various other pseudovarieties, besidesD , preserve decidability of pointlike sets (and even stronger algorithmic properties)under semidirect product. We relegate these results to the end of the paper.

2. Preliminaries and Notation

A semigroup is a set S with an associative binary operation. A monoid is asemigroup with an identity 1. If A is a finite set, A+ will denote the free semigroupon A and consists of all non-empty words in A under concatenation, while A∗ willdenote the free monoid and is obtained from A+ by adjoining the empty word.

A relational morphism ϕ : S → T of semigroups (monoids) is a relationϕ : S → 2T whose graph #ϕ , the set of elements (s, t) such that t ∈ sϕ , is asubsemigroup (submonoid) of the product S × T , which projects onto the first factorS . If the projection to the second factor T is injective, then the relational morphismis called a division. Let us define a relation ϕ : S → T of semigroups (monoids)to be multiplicative if sϕs′ϕ ⊆ (ss′)ϕ (1 ∈ 1ϕ), to be fully defined if sϕ 6= Ø for

Steinberg 3

all s ∈ S , and to be injective if sϕ ∩ s′ϕ 6= Ø implies s = s′ . Then a relationalmorphism is a fully defined, multiplicative relation and a division is an injectiverelational morphism. It is easy to see that the composition of relational morphisms,respectively, divisions is another relational morphism, respectively, division. It is alsoeasy to see that ϕ : S → T is a relational morphism if and only if there is a semigroupR , a surmorphism α : R � S and a morphism β : R → T such that ϕ = α−1β . Ifthere is a division from S to T , we say that S is a divisor of T .

A pseudovariety of semigroups (monoids) is then a class of finite semigroups(monoids) closed under finite direct products and divisors. A pseudovariety V is calledlocally finite if, for every finite set A , there is a semigroup (monoid) FV(A) in V withthe usual universal property with respect to members of V .

We will need to talk about categories from the viewpoint of [25]. A (directed)graph A consists of a vertex set V (A), an edge set E(A), and two functions α, ω :E(A)→ V (A) which choose the initial and terminal vertex of each edge respectively.If A is a graph and a1, a2 ∈ V (A), we denote by A(a1, a2) the set of all edges froma1 to a2 ; such a set is called a hom set. Edges e, f are composable if eω = fα . A(directed) path from a1 to a2 is a, possibly empty, sequence of composable edges suchthat the first edge begins at a1 and the last edge ends at a2 . The path is calleda loop if a1 = a2 ; so there is an empty loop at each vertex. A graph A is said tobe strongly connected if every pair of vertices can be joined by a directed path. Themaximal strongly connected subgraphs of a graph are called the strongly connected orbonded components. An edge e of a graph is called a transition edge if eα and eω liein different strongly connected components. A graph is called connected if every pairof vertices can be joined by an undirected path. The maximal connected subgraphsare called connected components.

A graph morphism ϕ : A → B , consists of two functions, written (by abuseof notation) ϕ : V (A) → V (B) and ϕ : E(A) → E(B) such that, for e ∈ E(A),eϕα = eαϕ and eϕω = eωϕ . A graph morphism ϕ : A → B is called faithfulif its restriction to each hom set is injective. It is called full if the induced mapϕ : A(a1, a2) → B(a1ϕ, a2ϕ) is surjective for all a1, a2 ∈ V (A). A faithful morphismis called an embedding if it is one-to-one on vertices. A full morphism is called aquotient morphism if it is bijective on vertices. A graph isomorphism is a faithful,quotient morphism. A subgraph is called full if the inclusion map is full.

A category consists the following data:

• A graph A ;

• A morphism µ : E(A) ×ω,α E(A) → E(A) (where E(A) ×ω,α E(A) = {(e, f) ∈E(A)|eω = fα} is the set of composable arrows), called composition, written asjuxtaposition, satisfying:

(ef)α = eα;

(ef)ω = fω;

(ef)g = e(fg) when both sides are defined;

• A map 1() : V (A)→ E(A) such that 1a ∈ A(a, a) is the unique local identity ata .

A morphism ϕ : C → D of categories is then a graph morphism whichpreserves multiplication: (ef)ϕ = eϕfϕ , 1cϕ = 1cϕ . A morphism is called faithful,etc., if the underlying graph morphism bears that name.

We note that a one vertex category is a monoid, and that any monoid can beviewed as a one vertex category in the obvious way. In general, the hom set C(c, c) ofa category C is a monoid called the local monoid at c .

Let A be a graph. Then the free category on A , denoted A∗ , is defined by:

V (A∗) = V (A);

E(a1, a2) = {p|p is a path from a1 to a2};

with the usual path multiplication. The empty paths are the local identities. It iseasy to see that the inclusion of A is universal amongst graph morphisms of A into

4 Steinberg

a category. A category C is said to be A-generated if there is a quotient morphismϕ : A∗ → C . If A is a subgraph of a category C , then the smallest subcategory of Ccontaining A is A-generated and denoted 〈A〉 .

A congruence ≡ on a category C is an equivalence relation on E(C) suchthat if e ≡ f , then e and f are coterminal, and if aeb is defined, then aeb ≡ afb .The quotient category C/≡ has the same vertices as C , but the edge set is the setof equivalence classes. The natural map from C to C/≡ is a quotient map and itis easy to see that every quotient map arises in this manner. Furthermore, everymorphism can be factored as a quotient morphism followed by a faithful morphismvia the natural congruence associated with the morphism. If ϕ is a morphism, theassociated congruence will be denoted ≡ϕ .

A relational morphism ϕ : C → D of categories consists of a function ϕ :V (C)→ V (D) and a relation ϕ : E(C)→ E(D) (again we are abusing notation) suchthat:

• f ∈ eϕ =⇒ eαϕ = fα, eωϕ = fω ;

• e1ϕe2ϕ ⊆ (e1e2)ϕ for e1 , e2 composable;

• 1cϕ ∈ 1cϕ ;

• eϕ 6= Ø for e ∈ E(C).

Alternatively, if we denote by #ϕ the subcategory of C × D with vertices of theform (c, cϕ) and edge set the graph of the edge relation, then the above conditionsare equivalent to stating #ϕ is a subcategory of C × D which projects to C asa quotient morphism. Conversely, any such subcategory gives rise to a relationalmorphism. A relational morphism ϕ is called a division if the projection of #ϕ to Dis faithful or, equivalently, eϕ∩e′ϕ 6= Ø and e, e′ coterminal implies e = e′ . Relationalmorphisms from C to D arise from considering categories C , D , and R such that Rhas morphisms both to C and D , the one to C being a quotient. We call a relationalmorphism ϕ a quotient relational morphism if both projections to the factors of #ϕare quotient morphisms. If there is a division from C to D , we say that C is a divisorof D .

A pseudovariety of categories is a class of finite categories closed under finiteproducts and divisors. It is easy to see as a consequence that every pseudovarietycontains the trivial monoid and that pseudovarieties are closed under finite coproducts(disjoint unions). If V is a pseudovariety of monoids, we denote by gV the pseudo-variety of categories generated by monoids in V viewed as one element categories. Itis easy to see that g1 is the smallest pseudovariety of categories. We denote by `Vthe pseudovariety of categories whose local monoids are all in V . It is not true thatgV = `V in general. It is an important theorem of Tilson [25] that `1 is generatedby a single category and is contained in every non-trivial pseudovariety of categories.

3. Pointlikes for Categories

Recall that if S is a finite semigroup (monoid) and V a pseudovariety ofsemigroups (monoids), then a subset X ⊆ S is called V -pointlike if for every relationalmorphism of S with a semigroup (monoid) V ∈ V , there exists an element v ∈ V towhich all of X relates. The collection PlV(S) ⊆ P(S) of V -pointlike subsets of Sforms a subsemigroup (monoid) under multiplication of subsets. We generalize theseideas to pseudovarieties of categories.

Let V be a pseudovariety of categories and C a finite category. Let X ⊆C(c1, c2). The author defined in [19] X to be V -pointlike if, for all relational mor-phisms ϕ : C → V ∈ V , there exists v ∈ V (c1ϕ, c2ϕ) such that X ⊆ vϕ−1 . Forexample, since relational morphisms are fully defined, singleton sets are always point-like. We say that V has decidable pointlikes if there is an algorithm which, for anyfinite category C and subset X of coterminal edges, determines whether X is V -pointlike.

The following propositions are key to working with pointlike sets. Fix apseudovariety V of categories. If C and D are A-generated categories, then there is

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a natural quotient relational morphism diagrammed by

A∗ � D

C

↓↓.

We call such a relational morphism canonical. A relational morphism ϕ : C → V ∈ Vis said to compute V -pointlikes for C if, for a subset X ⊆ C(c1, c2), X is V -pointlikeif and only if there exists v ∈ V (c1ϕ, c2ϕ) with X ⊆ vϕ−1 .

Proposition 3.1. Let C be a finite category generated by a graph A . Then thereexists an A-generated category V ∈ V such that the canonical relational morphismfrom C to V computes V -pointlikes for C .

Proof. Suppose we have a relational morphism from C to V ∈ V diagrammed by

R - V

C

τ

↓↓.

Then, by lifting the map from A (which we can do since τ is a quotient map), weobtain a new relational morphism

A∗η - V

C

↓↓

and if X relates to a point for this relational morphism, then it did for the originalone. But X relates to a point under this relational morphism if and only if it does forthe relational morphism

A∗ � A∗/≡η

C

↓↓

and A∗/≡η∈ V . But this relational morphism is canonical. Thus X is V -pointlike ifand only if it relates to a point under all canonical relational morphisms to categoriesin V .

Since E(C) has only finitely many subsets consisting of coterminal arrows,we can find, for each such subset Y of E(C) which is not V -pointlike, a canonicalrelational morphism to a category in V so that Y does not relate to a point. If we takethe intersection of the associated congruences on A∗ , we obtain a finite A-generatedcategory V ∈ V with the desired property.

There is, of course, the analogous, well-known, proposition for pseudovarietiesof semigroups (monoids):

Proposition 3.2. Let V be a pseudovariety of categories and V be a collectionof members of V such that if V ∈ V , then V < M for some M ∈ V . Then if Cis a finite category, there is a relational morphism ψ : C → M with M ∈ V whichcomputes V -pointlikes for C .

Proof. First take a quotient relational morphism ϕ : C → V ∈ V which computesV -pointlikes as above. Then choose a division τ : V → M with M ∈ V . Since τ isinjective, it is easy to see that a subset X ⊆ C(c1, c2) relates to a point under ϕ ifand only if it relates to a point under ψ = ϕτ . So ψ computes V -pointlikes.

6 Steinberg

Again, there is an analogous proposition for pseudovarieties of semigroups ormonoids.

Corollary 3.3. Let V is a pseudovariety of monoids and C a finite category. Thenthere is a relational morphism ψ : C →M ∈ V which computes gV -pointlikes for C .

We now study how pointlike sets behave under relational morphisms. Thefollowing proposition shows that pointlike sets “push” under homomorphisms.

Proposition 3.4. Let ψ : B → C be a morphism of categories. Let X ⊆ B(b1, b2)be V -pointlike. Then Xψ ⊆ C(b1ψ, b2ψ) is V -pointlike.

Proof. Let τ : C → V ∈ V be a relational morphism computing V -pointlikes forC . Then there exists v such that X ⊆ v(ψτ)−1 . Since ψ is a morphism, we see thatXψ ⊆ vτ−1 whence, by choice of V , we can conclude Xψ is V -pointlike.

We also have that pointlike sets “lift:”

Proposition 3.5. Let ψ : B � C be a quotient map. Then X ⊆ C(c1, c2) isV -pointlike if and only there exists X ′ ⊆ B(c1, c2) such that X ′ is V -pointlike andX ′ψ = X .

Proof. The above proposition tells us that if such an X ′ exists, X is V -pointlike.Suppose, on the other hand, that X is V -pointlike. Let τ : B → V ∈ V computeV -pointlikes for B . Then ψ−1τ : C → V is a relational morphism, so there existsv ∈ V (c1τ, c2τ) such that X ⊆ v(ψ−1τ)−1 . Hence, for each x ∈ X , there existsx ∈ B(c1, c2) such that xψ = x and v ∈ xτ . Then setting X ′ = {x|x ∈ X} , we seeX ′ψ = X and X ′ ⊆ vτ−1 which implies that X ′ is V -pointlike by choice of τ .

Putting these together we get:

Corollary 3.6. Let ψ : B → C be a relational morphism of categories. Let X ⊆B(b1, b2) be a V -pointlike set. Then we can choose, for each x ∈ X , x ∈ C(b1ψ, b2ψ)such that x ∈ xψ and X ′ = {x|x ∈ X} is a V -pointlike set.

Proof. Factor ψ = α−1β with α a quotient map. It follows we can “lift and push”X by the above propositions.

We state the following observation as a proposition.

Proposition 3.7. If X1 ⊆ C(c1, c2) and X2 ⊆ C(c2, c3) are V -pointlike subsetsof C , then X1X2 ⊆ C(c1, c3) (the set-wise product) is a V -pointlike subset.

Let C be a finite category and V a pseudovariety of categories. Then wedefine a category PlV(C) by

V (PlV(C)) = V (C),

PlV(C)(c1, c2) = {X ⊆ C(c1, c2)|X is V-pointlike}.Composition is set-wise multiplication and the local identity at c is the singleton {1c} .This is a well-defined category by the above proposition and, in fact, if we consider thecategory of finite categories with morphisms, Proposition 3.4 shows this is a functor.There is a natural embedding ηC : C → PlV(C) defined by the identity on verticesand on edges by eηC = {e} . This embedding commutes with morphisms, so weobtain a natural transformation η from the identity functor to PlV(). The followingproposition easily follows from Proposition 3.1.

Proposition 3.8. If V is a pseudovariety of categories and C a finite category,then C ∈ V if and only if ηC is an isomorphism, that is, if and only if the onlyV -pointlike sets of C are singletons.

Proof. If C ∈ V , then, by considering the identity map, we see that the only V -pointlike sets are the singletons. Conversely, if the only V -pointlikes are singletons,then a relational morphism ϕ : C → V with V ∈ V which computes V -pointlikes isactually a division.

Thus members of V are precisely, the fixed points of the functor PlV().We add, for completeness, the following generalization of a result of Henckell andRhodes [12]. As we do not use it, we omit the proof. Consider the map

⋃

C

: PlV(PlV(C))→ PlV(C)

defined by the identity on vertices and by X 7→ ⋃X on edges.

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Proposition 3.9. The map⋃C defined above is a well-defined morphism and,

in fact, induces a natural transformation⋃

: PlV()2 → PlV() . Furthermore,(PlV(),

⋃, η) forms a monad.

We now wish to understand pointlikes for gV in terms of pointlikes for V .If C is a category, Tilson defined [25] the consolidation of C to be the monoidCcd = E(C) ∪ 0 ∪ 1 where 1 is an identity, 0 a zero, and we use the compositionof C , declaring all undefined products to be 0. Then there is a natural faithfulmorphism ι : C → Ccd . The following theorem, due to Almeida [3], was observed inseveral important cases by the author in [19].

Theorem 3.10. Let V be a pseudovariety of monoids. Then X ⊆ C(c1, c2) isgV -pointlike if and only if Xι is a V -pointlike subset of Ccd .

Corollary 3.11. A pseudovariety of monoids V has decidable pointlikes if andonly if gV has decidable pointlikes.

We end this section with the following important result, whose proof we deferuntil Section 6..

Theorem 3.12. (Bonded Normal Form for Pointlikes) Let V be a non-trivial pseu-dovariety of categories and C a finite category generated by a finite graph A . ThenX ⊆ C(c1, c2) is a V -pointlike set if and only if X can be factored as X0t1X1 · · · trXr

where the ti are transition edges of A and the Xj are V -pointlike subsets of 〈Yj〉 withthe Yj strongly connected components of A .

This theorem can be viewed as a factorization result for PlV(C).

4. A Generalized Derived Category Theorem

In this section, we give a generalization of the Derived Category Theoremof [25] for computing pointlikes with respect to semidirect products. A slightly differentversion is given in the author’s [19] while a semigroupoid version appears in theauthor’s PhD thesis [18]. Recall that if V is a pseudovariety of monoids and Wis a pseudovariety of semigroups, then V ∗W is the pseudovariety generated by all(unitary) semidirect or wreath products of monoids in V with semigroups in W . Itconsists precisely of all divisors of such semidirect products. In [7], Almeida and Weildefine a similar semidirect product, but where V is allowed to be a pseudovariety ofcategories and they show gV ∗W = V ∗W . All the results of this section apply, withonly slight modifications in the proofs, to semidirect products of this form. However,due to the applications we have in mind, we will only handle the case where V is apseudovariety of monoids.

Let ϕ : M → N be a relational morphism of finite monoids. Define De r(ϕ)to be the category:

V (De r(ϕ)) = N ;

E(De r(ϕ)) = V (De r(ϕ))×#ϕ;

(nL, (m,n))α = nL, (nL, (m,n))ω = nLn;

composition (nL, (m,n))(nnL, (m′, n′)) = (nL, (mm

′, nn′));

and local identities (nL, (1, 1)).

This clearly is a finite category; for an exposition of the properties of this version ofthe derived category, see Tilson [26].

Given a semigroup S , we let

S1 =

{S if S is a monoid,

S ∪ {1} otherwise.

Then if ϕ : S → T is a relational morphism, we obtain a relational morphismϕ1 : S1 → T 1 by letting #ϕ1 = (#ϕ)1 .

For semigroups V and W , we write V ◦W for the wreath product of V andW ; see [11]. Recall that

V ◦W = VW1 ×W

with multiplication given by

(f, w)(f ′, w′) = (f + wf ′, ww′),

8 Steinberg

where we write V and V W1

additively, and where xwf = (xw)f . If V and W aremonoids, then V ◦W is also a monoid. Note that (V ◦W )1 ⊆ V ◦W 1 in a naturalway by viewing (f0, 1) as the identity of V ◦W (where f0 is the function which isconstantly the identity of V ).

The following is then the Generalized Derived Category Theorem; the originaltheorem [25] can be proved from this one by asking when does a semigroup have onlysingleton pointlikes sets for V ∗W .

Theorem 4.1. (Generalized Derived Category Theorem) Let S be a finite semi-group, V be a pseudovariety of monoids, and W be a pseudovariety of semigroups.Then X ⊆ S is pointlike for V ∗ W if and only if for all relational morphismsϕ : S → W ∈ W , there exists w0 ∈ W such that X ⊆ w0ϕ

−1 and the set(1, (X,w0)) = {(1, (s, w0))|s ∈ X} ⊆ De r(ϕ1) is pointlike with respect to gV .

Proof. Suppose X is pointlike for V ∗W , ϕ : S → W ∈ W , is a relationalmorphism, and ψ : De r(ϕ1)→ V ∈ V computes gV -pointlikes (as per Corollary 3.3).Define τ : S → V ◦W by

sτ = {(f, w)|w ∈ sϕ, wf ∈ (w, (s, w))ψ, ∀w ∈W 1}.

It is easy to see that τ is a relational morphism. Indeed, it’s clearly fully defined. If(f, w) ∈ sτ , (f ′, w′) ∈ s′τ , then ww′ ∈ (ss′)ϕ . Also, for w ∈W 1 ,

w(f + wf ′) = wf + (ww)f ′ ∈ (w, (s, w))ψ + (ww, (s′, w′))ψ ⊆ (w, (ss′, ww′))ψ.

Thus (f, w)(f ′, w′) ∈ (ss′)τ . We then have the following diagram

Sτ- V ◦W

W.

π

↓↓

ϕ

-

Since X is pointlike for V∗W , there exists (f, w0) ∈ V ◦W , such that X ⊆ (f, w0)τ−1 .But then X ⊆ w0ϕ

−1 and (1, (X,w0)) ⊆ (1f)ψ−1 . Thus the set (1, (X,w0)) ispointlike (relates to a point) for this relational morphism and hence is pointlike forgV .

Conversely, suppose that, for every relational morphism ϕ : S → W ∈ W ,there exists w0 ∈W such that X ⊆ w0ϕ

−1 and the subset (1, (X,w0)) ⊆ De r(ϕ1) isgV -pointlike. Let τ : S → V ◦W be a relational morphism with V ∈ V , W ∈ Wwhich computes V ∗ W -pointlikes (such a relational morphism exists since everysemigroup in V ∗W divides such a wreath product, see Proposition 3.2). Then, bycomposing τ with the projection π to W , we get a relational morphism ϕ : S → Was in

Sτ- V ◦W

W.

π

↓↓

ϕ

-

Define ψ : De r(ϕ1)→ VW1

by

(w, (s, w))ψ = {wf |(f, w) ∈ sτ1}.

Note that ϕ1 is τ1 composed with the projection from V ◦ W 1 . We check thatψ is a relational morphism. Clearly it’s fully defined. If g ∈ (w, (s, w))ψ , g ′ ∈(ww, (s′, w′))ψ , then there exist f , f ′ ∈ VW

1

such that g = wf , g′ = wwf ′ , and(f, w) ∈ sτ1 , (f ′, w′) ∈ s′τ1 . Then we have

g + g′ = wf + wwf ′ = w(f + wf ′),

(f + wf ′, ww′) ∈ (ss′)τ1.

Steinberg 9

So g + g′ ∈ (w, (ss′, ww′))ψ . Finally, since (f0, 1) ∈ 1τ1 and wf0 = f0 , we see that

f0 ∈ (w, (1, 1))ψ . But f0 is the identity of V W1

.

Now, by assumption, there exists w0 ∈ W such that X ⊆ w0ϕ−1 and

(1, (X,w0)) is a gV -pointlike subset of De r(ϕ1). Since V W1 ∈ V , there exists f

such that (1, (X,w0)) ⊆ fψ−1 . But then, by definition of ψ and the observation that1g = g for all g ∈ V W 1

, we see that X ⊆ (f, w0)τ−1 . Thus X is pointlike for τ andhence for V ∗W .

Proposition 4.2. In Theorem 4.1, if S is generated by A , one need only checkcanonical relational morphisms with A-generated W ∈W .

Proof. Suppose we have a relational morphism ϕ : S → V ◦W . Then, by liftingeach generator of S to V ◦ W and closing under multiplication, we can changethe relational morphism so that Sϕ is A-generated and the relational morphism,canonical. Furthermore, if a subset relates to a point for this new relation, it relatedto a point for the original one. Let Z ⊆ W be the image of Sϕ under the projectionto W . Then Z is A-generated and the composition of ϕ and the projection is thecanonical relational morphism. We now embed Sϕ in V W 1 ◦ Z . The result willthen follow. Define a map τ : Sϕ → V W 1 ◦ Z by (f, z) 7→ (f, z) where z′f = z′ffor z′ ∈ Z1 . This map is clearly injective since 1f = f . It’s a morphism since(f, z)(g, z′) = (f + zg, zz′) = (f + zg, zz′).

The following theorem was first proved by the author in [19] and later, inde-pendently, by Almeida and Silva [5] as part of a more general result.

Theorem 4.3. Let V be a pseudovariety of monoids with decidable pointlikes andW a locally finite pseudovariety of semigroups with a computable bound on the orderof the free object on any finite set. Then V ∗W has decidable pointlikes.

Proof. Let S be a finite semigroup generated by a finite set A . By Proposition 4.2,we just need to check pointlikes of De r(ϕ1) for canonical relational morphisms ϕ :S → W with W ∈ W generated by A . Let τ : S → FW(A) be the canonicalrelational morphism and ψ : FW(A) � W the canonical surmorphism. If FW(A) isa monoid, then ψ is a monoid morphism while, otherwise, ψ1 is a monoid morphism.Hence we have a morphism ρ : De r(τ 1)→ De r(ϕ1) defined by

(w, (s, w′))ρ = (wψ1, (s, w′ψ1)).

It follows that if (1, (X,w0)) is gV -pointlike in De r(τ 1), then (1, (X,w0ψ)) is gV -pointlike in De r(ϕ1) by Proposition 3.4. Thus X is pointlike for V ∗W if and onlyif there exists w0 ∈ FW(A) such that X ⊆ w0τ

−1 and (1, (X,w0)) ⊆ De r(τ1) is gV -pointlike. But, by assumption on W , we can compute #τ and so we can computeDe r(τ1). Since gV has decidable pointlikes if and only if V does, we can verifywhether the above condition is satisfied.

By considering the case where |A| = 2, we see that if we merely assume in thehypotheses of the above theorem that V has decidable pointlike pairs (that is one candecide if a pair of elements of a finite semigroup is V -pointlike), we can still concludethat V ∗W has decidable pointlike pairs.

5. The Generalized Delay Theorem

In this section, we use the Generalized Derived Category Theorem to generalizethe Delay Theorem of [25] from a result about membership, to a result about pointlikes.

We prove that if a pseudovariety V of monoids has decidable pointlikes, thenso does V ∗D ; furthermore, if a semigroup S has delay n , then its V ∗D-pointlikesare exactly the same as its V ∗ Dn -pointlikes. In the case where V is the trivialpseudovariety, this is already known; see [19]. Thus we will be concerned with thecase where V is non-trivial and so we can make use of the BNFP (Theorem 3.12).All of our results can easily be shown to hold in the case where V is a pseudovarietyof categories as well. The proof will actually give a nice description of the V ∗ Dpointlike sets of S in terms of the V -pointlike sets of a certain category formed fromthe idempotents of S .

10 Steinberg

Let S be a finite semigroup and E(S) = {e ∈ S|e2 = e} , the set of idempotentsof S . Then we define a category SE by

V (SE) = E(S),

SE(e1, e2) = e1Se2

with the multiplication of S as the composition, and with e as the local identity ate . This construction is functorial for morphisms and there is a natural transformationψ back to the identity functor where ψS : SE → S is the map taking each elementto itself, viewed once again as an element of S . Clearly ψS is a faithful morphism.We will often blur the difference between an edge s of SE and its image sψS in S .This construction is almost functorial for relational morphisms as we now show. Letϕ : S → T be a relational morphism of finite semigroups. If e ∈ E(S), then eϕis a subsemigroup of T and hence there is an idempotent fe of T in eϕ . Define arelational morphism ϕ : SE → TE by

eϕ = fe,

sϕ = fesϕfe′ for s ∈ SE(e, e′).

Lemma 5.1. One has that ϕ is a relational morphism and, furthermore, if X ⊆SE(e, e′) is such that X ⊆ tϕ−1 , then XψS ⊆ tϕ−1 . In particular, if ϕ is a division,so is ϕ .

Proof. The second statement follows from the first. Clearly, ϕ is fully definedon each hom set, since ϕ is fully defined. Also, if s ∈ e1Se2 , s′ ∈ e2Se3 , withei ∈ E(S), and t ∈ sϕ , t′ ∈ s′ϕ , then tfe2t

′ ∈ (se2s′)ϕ = (ss′)ϕ so, fe1tfe2fe2t

′fe3 =fe1tfe2t

′fe3 ∈ fe1(ss′)ϕfe3 . Also, fe ∈ feeϕfe . It follows ϕ is a relational morphism.

Suppose X and t are as in the hypothesis. Then for s ∈ X , t ∈ fesϕfe′ , sot = fetsfe′ with ts ∈ sϕ . But, since s = ese′ , t = fetsfe′ ∈ (ese′)ϕ = sϕ . It followsXψS ⊆ tϕ−1 .

As a consequence, we have:

Corollary 5.2. Suppose S is a finite semigroup and V a pseudovariety of mon-oids. Suppose X ⊆ S can be factored as uY v with u, v ∈ S and Y = Y ′ψS with Y ′

a gV -pointlike subset of SE(e, e′) . Then X is a V ∗D-pointlike subset of S .

Proof. Since the product of pointlike sets is pointlike, it suffices to show Y isV ∗D-pointlike. Let ϕ : S → T with T ∈ V ∗W be a relational morphism. Then, byProposition 18.4 of [25] (or see [7] for another proof), TE ∈ gV . Thus, since Y ′ is gV -pointlike, Y ′ ⊆ tϕ−1 for some t ∈ T and so, by the above lemma, Y = Y ′ψS ⊆ tϕ−1 .It now follows that Y is a V ∗D-pointlike set.

Our next goal is to prove the converse of the above corollary. First recall thatif S is an A-generated finite semigroup, where we write w 7→ w for the image of win S , then, for n = |S| , we have that:

1. Every word w ∈ A∗ of length n has a factorization w = uv , with u ∈ A+, v ∈A∗ , such that ue = u for some idempotent e in S;

2. Every word w ∈ A∗ of length n has a factorization w = uv , with u ∈ A∗, v ∈A+ , such that ev = v for some idempotent e in S.

One then defines rdel(S) to be the smallest integer satisfying i for S andldel(S) to be the smallest integer satisfying ii for S . Finally, define the delay index ofS , written del(S), as follows:

del(S) =

{0 if S is a monoid,

min(rdel(S), ldel(S)) otherwise.

It is easy to verify that Dn = {S ∈ D|del(S) ≤ n} . We assume, from now on, that Vis a non-trivial pseudovariety of monoids.

Lemma 5.3. Let S be a finite semigroup with del(S) = n . Suppose X ⊆ S ,|X| > 1 , is V ∗Dn -pointlike. Then X = uY v where Y = Y ′ψS with Y ′ ⊆ SE(e, e′)a gV -pointlike set.

Steinberg 11

Proof. Suppose S is A-generated as a semigroup. Write α : A+ → S for thecanonical morphism. Then we note that Dn is locally finite and that the relativelyfree semigroup on A is the semigroup An = A/≡n where ≡n is the congruencewhich sets two words of length greater than or equal to n equivalent if they have thesame suffix of length n . Let ϕ : S → An be the canonical relational morphism andϕ1 : S1 → A1

n be the extension. We use the empty word to represent the identity ofA1n . Let α1 : A∗ → S1 be the extension of the canonical projection. We will write w

for wα1 . Then Theorem 4.1 says that there exists w0 ∈ An such that X ⊆ w0ϕ−1

and (1, (X,w0)) ⊆ De r(ϕ1)(1, w0) is gV -pointlike. Since |X| > 1 and only words oflength n in An can relate to more than one point of S , |w0| = n .

Let ρn : A∗ → A1n be the canonical morphism (it is identity on words of length

less than n and otherwise, takes a word to its suffix of length n). First note thatE(De r(ρn)) can be viewed as A1

n × A∗ since ρn a morphism implies that edges looklike (w′, (w,wρn)) and so the wρn coordinate is superfluous. Then there is a naturalquotient map η : De r(ρn) � De r(ϕ1) which is the identity on vertices and maps anedge (w′, w) to (w′, (w,wρn)). Let Γ be the Cayley graph of An , so V (Γ) = A1

n andE(Γ) = A1

n × A . Then Γ clearly embeds in De r(ρn) and generates it (since an edge(w′, w) has an obvious factorization as a path in Γ by reading from w′ the letters inw ). In fact, it is easy to see that De r(ρn) ∼= Γ∗ by the map which is the identity onvertices and takes an edge (w0, w) to the path from w0 obtained by reading w in Γ.It follows that De r(ϕ1) is generated by Γ.

The transition edges of Γ are precisely those edges that are part of a path oflength ≤ n from 1. All of the vertices corresponding to words w of length n are in thesame strongly connected component of Γ (to get a path from w to w′ , just read w′

from w ). This strongly connected component, which we will denote ∆n , is the uniquestrongly connected component containing more than one vertex. Furthermore, thesingleton strongly connected components of Γ have no edges. Also, the subcategoryof De r(ϕ1) generated by ∆n , which we shall denote by Cn , is the only stronglyconnected component of De r(ϕ1) to have more edges than just a local identity. Itnow follows, by Theorem 3.12, that (1, (X,w0)) = (1, (w1, w1))B where w1 is word oflength n and B ⊆ De r(ϕ1)(w1, w0) is a gV -pointlike subset of Cn (w1 is the labelof the transition edges read from 1 in the bonded normal form for (1, (X,w0))).

We observe that ∆∗n is the strongly connected component of Γ∗ whose verticesare the words of length n . Now we define a relational morphism θ : ∆∗n → SE . Wewill use An to denote the words of A∗ of length n ; note that V (∆n) = An . Wefirst assume that del(S) = rdel(S). For each word wk ∈ An , let wk = ukvk be thefactorization with smallest |uk| such that uke = uk for some idempotent e of S . Letek be such an idempotent. Define on vertices θ : An → E(S) by wkθ = ek . Todefine the relation on edges, we once again use that Γ∗ = De r(ρn) and ∆∗n = 〈∆n〉 inDe r(ρn). So if wi ∈ An , w ∈ A∗ , and wj = (wiw)ρn , we define

(wi, w)θ = {s ∈ eiSej |uisvj = wiw and uis ∈ S1uj}.

It is shown in the proof of [25, Theorem 18.2] that this is a relational morphism.Recall that η : Γ∗ � De r(ϕ1) denotes the canonical quotient morphism. Then(η|∆n

)−1θ : Cn → SE is a relational morphism. It follows, by Corollary 3.6, that thereis a subset Y ′ ⊆ SE(e1, e0) with Y ′ ⊆ B(η|∆n

)−1θ which is gV -pointlike and suchthat each element of B relates by (η|∆n

)−1θ to some element of Y ′ . Let Y = Y ′ψS ,we now show X = u1Y v0 .

Suppose s ∈ X . Then, in De r(ϕ1), we have

(1, (s, w0)) = (1, (w1, w1))(w1, (w,wρn))

with w ∈ A∗ and (w1, (w,wρn)) ∈ B . So, by choice of Y ′ , there exists t ∈ Y andw′ ∈ A∗ such that t ∈ (w1, w

′)θ and (w1, (w,wρn)) = (w1, (w′, w′ρn)). It follows that

u1tv0 = w1w′ = w1w = s.

So s ∈ u1Y v0 as desired. Thus X ⊆ u1Y v0 .

Suppose s ∈ u1Y v0 . Then s = u1tv0 with t ∈ e1Se0 such that t ∈(w1, w)θ for some w with (w1, (w,wρn)) ∈ B . Then s = u1tv0 = w1w . But, since

12 Steinberg

(1, (X,w0)) = (1, (w1, w1))B , it follows w1w ∈ X . So s ∈ X and we have completedthe proof that X = u1Y v0 .

The proof for del(S) = ldel(S) is dual, one applies the above arguments tothe mirror image relational morphism in [25].

Corollary 5.4. Let V be a pseudovariety of monoids and S a finite semigroupwith del(S) = n . Then a subset X ⊆ S is V ∗D-pointlike if and only if it is V ∗Dn -pointlike.

Proof. As remarked earlier, we can assume V is non-trivial. Clearly a V ∗ D-pointlike set is V ∗ Dn -pointlike. On the other hand, if X ⊆ S , not a singleton,is V ∗ Dn -pointlike, then, by Lemma 5.3, it is of the form uY v where Y = Y ′ψSwith Y ′ ⊆ SE(e, e′) a gV -pointlike set. But Corollary 5.2 shows that such a set isV ∗D-pointlike.

Corollary 5.5. Let V be a pseudovariety of monoids and S a finite semigroup.Then PlV∗D(S) is generated by singletons and sets of the form Y ′ψS with Y ′ ⊆SE(e, e′) a gV -pointlike set.

We combine the above results into what we call the Generalized Delay Theo-rem:

Theorem 5.6. (Generalized Delay Theorem) Let V be a non-trivial pseudovarietyof monoids and S a finite semigroup with del(S) = n . Then a subset X ⊆ S isV ∗D-pointlike if and only if it is V ∗Dn -pointlike if and only if it is of the form sY twith Y = Y ′ψS where Y ′ ⊆ SE(e, e′) is a gV -pointlike set.

Corollary 5.7. If V is a pseudovariety of monoids with decidable pointlikes, thenV ∗D has decidable pointlikes.

Proof. Since del(S) ≤ |S| , one can compute del(S) and so Theorem 5.6 impliesthat V ∗ D has decidable pointlikes if V ∗ Dn does for every n . But, since Dn islocally finite with a computable bound on the order of the free object on any finiteset, this follows from Theorem 4.3.

Alternatively, one can find the subsemigroup of P(S) generated by singletonsand sets of the form Y ′ψS , where Y ′ ⊆ SE(e, e′) is a gV -pointlike set, since, byTheorem 3.10, gV has decidable pointlikes if V does. Thus, by Theorem 5.6, we candecide pointlikes for V ∗D .

Our remarks after Theorem 4.3 shows that if, in the above corollary, we onlyassume that V has decidable pointlike pairs, then we may conclude that V ∗D hasdecidable pointlike pairs.

We now obtain the original Delay Theorem [25] as a corollary.

Corollary 5.8. Let V be a pseudovariety of monoids and S a finite semigroup,del(S) = n . Then S ∈ V ∗D if and only if SE ∈ gV if and only if, S ∈ V ∗Dn .

Proof. This follows immediately from Theorem 5.6 upon recalling that a semigroup(or a category) is in a pseudovariety if and only if it has only singleton pointlike setsfor that pseudovariety.

We remark that that Theorem 5.6 actually allows us to get a handle on whatthe pointlike sets for V ∗D “look like” if we have a nice form for the pointlike sets ofV , something true for G , Sl , and J amongst others.

Now we aim to prove the converse to Corollary 5.7; first a lemma.

Lemma 5.9. Let W be a pseudovariety of semigroups, and V = W ∩M (M isthe pseudovariety of finite monoids). Let M be a monoid and X ⊆ M a set. ThenX is W -pointlike if and only if it is V -pointlike.

Proof. Clearly, if X is W -pointlike, it is V -pointlike. For the converse, supposewe have a diagram

Rβ - W

M

τ

↓↓

Steinberg 13

with W ∈W , τ onto, and R finite. Let e ∈ 1τ−1 be an idempotent and N = eRe .Then N is a monoid and τ |N : N →M is a surmorphism of monoids. Let V = Nβ .Then V ∈ V , being a monoid in W . Hence we have a relational morphism of monoids

Nβ|N - V

M

τ |N↓↓

.

Thus if X is V -pointlike, we see that, under our original relational morphism (whichcontains this new one as a relation), X relates to a point.

We now prove Theorem 1.2.

Proof. By Corollary 5.7, if V has decidable pointlikes, so does V ∗ D . On theother hand, it is well-known, see [25] for instance, that V ∗D ⊆ LV . This implies,(V ∗D)∩M = V . Hence if X ⊆M , with M a finite monoid, we see, by Lemma 5.9,that X is V -pointlike if and only if it is V ∗D-pointlike. Thus we see that if V ∗Dhas decidable pointlikes, so does V .

This has some rather interesting consequences. A personal communicationwith J. Rhodes has informed us that there exists a pseudovariety of monoids V ,defined by a finite set of equations E , such that one can not solve the word problemfor the monoid of explicit operations; see [2] for the definition of an explicit operation.A similar result was proved in [1] for pseudovarieties of semigroups. This implies,by a slight modification of the argument used in the author’s and Rhodes’ [17], thatthere exists a decidable pseudovariety of monoids W , which does not have decidablepointlikes. Hence W ∗D does not have decidable pointlikes. However, W ∗D doeshave decidable word problem for its semigroup of explicit operations (since any twoexplicit operations are distinct when restricted to D). Now, by the Delay Theoremand its converse, W ∗D has decidable membership if and only if gW has decidablemembership. It is open whether V decidable, implies gV is decidable. It is also openwhether there is a decidable pseudovariety of semigroups without decidable pointlikes,but for which one can decide the word problem for explicit operations. We then seethat W ∗D is an example of one of these phenomena, but we do not yet know whichone.

In Lemma 5.3, the BNFP and the Generalized Derived Category Theorem areused to show that there is a word w0 , a gV -pointlike subset B of Cn , and a wordw1 such that (1, (X,w0)) = (1, (w1, w1))B . We now give a sketch of a proof, based onan idea suggested by the referee, on how to prove this without the BNFP. We decidedto keep our original proof as we feel the BNFP is important in its own right; also,it will be used in a sequel paper. The idea (retaining the notation from the proof ofLemma 5.3) is as follows: let ψ : Cn → M ∈ V be a relational morphism computinggV -pointlikes. Recall that Cn is generated by ∆n , call the canonical projection from∆∗n : η . Also we have that E(∆n) = An × A . Define a graph morphism from ∆n

to M by the unique map on vertices and sending an edge (w, a) to some element of(w, (a, a))ψ . Let β : ∆∗n → M be the induced map. Then η−1β : Cn → M stillcomputes gV -pointlikes.

Define a congruence (β, n) on A∗ by letting words of length less than n beequivalent only to themselves and setting two words of length at least n equivalent ifthey have the same prefix of length n , the same suffix of length n , and the sequenceof factors of length n+ 1 of the two words are the same when evaluated in M (wherewe identify E(∆n) with An × A). One can show that N = A∗/(β, n) ∈ V ∗ Dn .Hence X relates to an element r ∈ N corresponding to the equivalence class of aword v of length at least n (since |X| > 1). Let w1 be the prefix of length nof v , w0 its suffix of length n , and B = {(w1, (s, w))|w1s ∈ X and w1w

′ ≡ v mod(β, n) for some word w′ with suffix of length n equal to w} . One can then show thatB ⊆ Cn(w1, w0), (1, (X,w0)) = (1, (w1, w1))B , and B relates to a point under η−1β .The first two statements are clear. To see that B relates to a point, it suffices toobserve that all of B relates to the product of the image, in N , of the sequence offactors of v of length n+ 1. Thus w0 , w1 , and B are as desired.

14 Steinberg

6. Implicit Operations on Categories

A topological category C is a category such that V (C) and E(C) are topologi-cal spaces and all the maps in the definition of a category are continuous. A topologicalcategory is called profinite if it is an inverse limit of finite categories. A congruenceon a topological category C is called open if the equivalence classes are open. Wecall a category a finite vertex category if it has a finite vertex set. For a finite vertexprofinite category, by compactness, C/≡ is finite if and only if ≡ is open If C is afinite vertex profinite category, one can show [13] that C = lim←− i∈IC/≡i where the ≡irun through the collection of open congruences of C .

If A is a finite graph, then there is a free profinite category A∗ topologicallygenerated by A obtained by taking the inverse limit of all finite quotients of A∗ .Note that V (A∗) = V (A). This category has the property that any graph morphism

of A to a profinite category has a unique continuous extension to A∗ . In general,we say that a finite vertex profinite category is topologically generated by A if it isa continuous quotient of A∗ . If V is a pseudovariety of categories, a finite vertexprofinite category C is called pro-V if C/≡ ∈ V for all open congruences ≡ on C .This is equivalent to asking that C be an inverse limit of categories in V [13]. One canshow that, for any finite graph A and pseudovariety V , there is a free pro-V categorytopologically generated by A , denoted FV(A), obtained by taking the intersection of

all open congruences on A∗ with quotient in V . Observe that there is a natural mapηV : A∗ → FV(A) which is a bijection on vertices.

For historical reasons, an element of a free profinite (or pro-V) category iscalled an implicit operation.

The proof of the following technical lemma is based on a proof from [25].

Lemma 6.1. Let A be a finite graph and Y a strongly connected component.Then the inclusion Y ⊆ A induces a full embedding ι : FV(Y ) ↪→ FV(A) for anypseudovariety V of categories.

Proof. Since V (Y ) embeds in V (A) = V (FV(A)), ι is injective on vertices. It is

full since any implicit operation of FV(A) between vertices of Y is a limit of pathsbetween these vertices (since V (A) is a finite set and α , ω are continuous) and hence

is in 〈Y 〉 , the image of FV(Y ) under ι .

We now show ι is faithful. Choose y ∈ V (Y ). Define a morphism ϕ : A →FV(Y ) as follows: on vertices

cϕ =

{c if c ∈ V (Y ),

y otherwise.

To define ϕ on edges, choose, for each pair of vertices c, c′ ∈ V (Y ), a path p(c, c′) :c → c′ in Y (we can do this since Y is strongly connected). We always choosep(c, c) = 1c . Now if x : c→ c′ is an edge of A , define

xϕ =

{xηV if x ∈ E(Y ),

p(cϕ, c′ϕ)ηV otherwise

where ηV : Y ∗ → FV(Y ) is the map considered above. Then ϕ is a graph morphism

and ϕ|Y = ηV|Y . So there is an induced continuous map ϕ : FV(A)→ FV(Y ). But,

(ηVιϕ)|Y = ηV|Y and so, by the universal property of FV(Y ), ιϕ = 1cFV(Y ). It follows

that ι is faithful.

The above proposition, together with [13, Proposition 6.6], proves the followingtheorem.

Theorem 6.2. If V is a non-trivial pseudovariety of categories and A a finitegraph, then each u ∈ E(FV(A)) has a unique factorization (called its bonded normalform) u = u0t1u1 · · · trur where the ti are transition edges of A and each ui is an

element of E(FV(Yi)) with Yi a strongly connected component of A .

If C a finite A-generated category, then we can consider the canonical (quo-

Steinberg 15

tient) relational morphism diagrammed by

A∗pV� FV(A)

C

↓↓

(this, of course, is an extension of the notion of a canonical relational morphism).

Theorem 6.3. Let C be a finite A-generated category and ϕ : C → FV(A) thecanonical relational morphism introduced above. Then X ⊆ C(c1, c2) is V -pointlike

if and only if X ⊆ πϕ−1 for some π ∈ FV(A)(c1, c2) .

Proof. To prove sufficiency, suppose that such a π exists and let ψ : C → V ∈ Vbe a canonical relational morphism which computes V -pointlikes. Let ρ : FV(A)� Vbe the canonical surjection; then ϕρ = ψ . So X ⊆ (πρ)ψ−1 and hence is pointlike bychoice of ψ .

For the converse, we use that FV(A) is the inverse limit of its quotients byopen congruences, that is, it is the inverse limit of all A-generated categories Vj in V .

Let pj : FV(A) → Vj be the quotient map. Then we let ϕj = ϕpj . By assumption,there exists πj ∈ Vj(c1, c2) with X ⊆ πjϕ−1

j , so

Cj = {ρ ∈ FV(A)(c1, c2)|X ⊆ (ρpj)ϕ−1j } 6= Ø

(since pj is a quotient map). Also, Cj is closed. Indeed, FV(A)(c1, c2) is closed sincethe initial and terminal vertex functions are continuous and V (A) is Hausdorff. Also,Bj = {v ∈ Vj(c1, c2)|X ⊆ vϕ−1

j } is a closed subset of Vj (since Vj is discrete). Hence,

by continuity, Bjp−1j is closed. But

Cj = FV(A)(c1, c2) ∩Bjp−1j ,

so Cj is closed. Finally, the collection of Cj ’s satisfies the finite intersection conditionsince if Vj maps onto Vj1 , . . . , Vjn , then Cj ⊆

⋂Cjk . Thus

⋂Cj 6= Ø. Let π be in

this intersection, we claim X ⊆ πϕ−1 . Indeed, let x ∈ X . Let ψ : A∗ → C be thecanonical projection and pV denote, as before, the projection of A∗ to FV(A). Thenxϕ = xψ−1pV and hence is a closed set (since ψ is continuous and pV is a closedmap). But πpj ∈ xϕpj for all j , so π ∈ xϕ = xϕ . Since x was arbitrary, X ⊆ πϕ−1 .

We are now in a position to prove Theorem 3.12.

Proof. If X has such a factorization, it clearly is V -pointlike. Indeed, the Xj

must be V -pointlike subsets of A by Proposition 3.4 and pointlike sets are closedunder products.

Conversely, by Theorem 6.3, if X is V -pointlike, then there exists a choice ofimplicit operations πx ∈ A∗ (x ∈ X) such that πx maps to x in C and the πx all have

the same image in FV(A). Since V is non-trivial, `1 ⊆ V . Two implicit operations

in A∗ have the same image in F`1(A) if and only if they have the same transitionedges in their bonded normal form [13, 25]. So the fact that the πx all have the same

image in FV(A) implies that they all have the same transition edges and have bondednormal forms πx = ux,0t1 · · · trux,r with ux,i defined on the same strongly connectedcomponent Yi of A for all x , i = 0, . . . , r . Thus we see that if pV is as before,πxpV has bonded normal form ux,0pVt1 · · · trux,rpV . So, by the uniqueness of bondednormal forms, ux,ipV is the same for all x . Since, by Lemma 6.1, the subcategories

of A∗ and FV(A) topologically generated by Yi are Y ∗i and FV(Yi), respectively,the result follows by another application of Theorem 6.3 (note that pV , the canonical

projection of A∗ to FV(A), restricts on each strongly connected component Yi to

the canonical projection of Y ∗i to FV(Yi) by Lemma 6.1, so there is no confusion interminology).

16 Steinberg

We now obtain the original Bonded Component Theorem of [25] as a corollary.

Corollary 6.4. Let C be a finite category and V a pseudovariety of categories.Then C ∈ V if and only if every strongly connected component of C is in V .

Proof. Clearly if C ∈ V , then so is every strongly connected component. Supposethat every strongly connected component of C is in V . Then the only V -pointlikesubsets of any strongly connected component of C , viewed as a category in its ownright, are singletons. Hence if we have a V -pointlike subset X of the category C andconsider a factorization as in Theorem 3.12, we see that X is a singleton. Thus, byProposition 3.8, C ∈ V .

Acknowledgements

I would like to thank the anonymous referee, both for the alternate proof ofLemma 5.3, and for the many useful suggestions on reorganizing the paper.

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Faculdade de Cienciasda Universidade do Porto4099-002 Porto, Portugalbsteinbg@agc0.fc.up.pt