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Syntactic and Global Semigroup Theory, a Synthesis Approach Jorge Almeiday Benjamin Steinbergz November 12, 1998 Abstract

This paper is the culmination of a series of work integrating syntactic and global semigroup theoretical approaches for the purpose of calculating semidirect products of pseudovarieties of semigroups. We introduce various abstract and algorithmic properties that a pseudovariety of semigroups might possibly satisfy. The main theorem states that given a nite collection of pseudovarieties, each satisfying certain properties of the sort alluded to above, any iterated semidirect product of these pseudovarieties is decidable. In particular, the pseudovariety G of nite groups satis es these properties. J. Rhodes has announced a proof, in collaboration with J. McCammond, that the pseudovariety A of nite aperiodic semigroups satis es these properties as well. Thus, our main theorem would imply the decidability of the complexity of a nite semigroup.

1. Introduction In virtually any discipline, there will arise various schools or approaches to the development of that discipline. Finite semigroup theory is no dierent. One approach to nite semigroup theory, initiated by John Rhodes in the mid 1960's after the proof of the Krohn-Rhodes prime decomposition theorem [26], goes under the name of global semigroup theory. We will discuss later a little more about what exactly this is, but generally speaking, this approach has traditionally been concerned with nding ways to structurally \decompose" a semigroup into simpler parts, the eventual goal being to nd an algorithm to determine the Krohn-Rhodes group complexity of a nite semigroup. This approach has essentially been a structural approach in that more often than not, the theorems are proved by constructing new nite algebraic objects with various properties. Although no one book dedicated to this approach exists, the book of Eilenberg [20] and the paper of Tilson [43] give the avor of this approach. Another school of thought, given impetus by Reiterman's theorem [31] in the early 1980's and the introduction of implicit operations and pro nite semigroups to the theory of Address: Faculdade de Ci^encias da Universidade do Porto, 4099-002 Porto, Portugal. The rst author was supported, in part, by the project Praxis/2/2.1/MAT/63/94 and by FCT through the Centro de Matematica da Universidade do Porto. z The second author was supported in part by Praxis XXI scholarship BPD 16306 98.  y

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nite semigroups, prefers the syntactical approach. This approach attempts to solve problems in nite semigroup theory by solving word problems (and generalizations of this to implicit operations) and nding bases of (pseudo)identities for pseudovarieties. This school has roots in the work of Simon, Pin, and the rst author. For an introduction to these techniques, see the rst author's [3]. A synthesis of these two approaches, of which in some sense this paper is a culmination, has long been in the making. Eilenberg introduced in [20] the notion of a pseudovariety. Originally his book, written closely with Tilson (who wrote the nal sections which are a good summary of global semigroup theory to that date) was an attempt to add a uniformity of notation and a sense of structure to a relatively new subject. This notion of a pseudovariety, a class of nite semigroups closed under forming nite products, taking subsemigroups, and taking homomorphic images, had a strong resemblance to notions of universal algebra. Eilenberg and Schutzenberger then proved the important theorem that pseudovarieties were eventually de ned by identities, a rst intimation of a syntactical aspect to nite semigroup theory. In addition, this book also formalized the relationship between the study of classes of rational languages and the study of pseudovarieties of semigroups, although we recommend the book of Pin [30] for a more concise, modern version of this approach. Typically, the syntactical approach involves the study of suitable free algebras and often at least a partial solution of the corresponding word problem. Since every pseudovariety is the union of a chain of subpseudovarieties which consist of the nite members of corresponding varieties, a possible approach is to study the free algebras in those varieties. This method has been applied successfully in many instances among others by Simon [37] and by the rst author [3]. It requires however a judicious choice of the chain of subpseudovarieties and often leads to very involved combinatorial work. Moreover, results at the variety level often break down when considering only nite algebras, partly due to incompleteness of equational logic in this setting. Free pro nite algebras, which have long played a role in group theory, number theory, and logic [22], turn out to be a suitable replacement of free algebras for pseudovarieties. See the paper [12] by the rst author and Weil for an introduction. Just as free algebras in varieties encode common algebraic properties of their members, free pro nite algebras relative to pseudovarieties encode common algebraic and combinatorial properties of their members. In general, it is very hard to describe free pro nite algebras and for instance very little is known about (absolutely) free pro nite semigroups. Also, free pro nite algebras are often uncountable and so not very amenable to algorithmic questions. This dif culty has occasionally been overcome in speci c instances while in this paper a new systematic method is proposed to deal with it. The meeting point between the global and syntactic approaches came about in the study of decidability questions for semidirect products of pseudovarieties. Besides the pioneer work by Eilenberg, Schutzenberger and Tilson mentioned above, Tilson's seminal paper [43] already laid the foundations of this synthesis but, in syntactical terms, rested at the level of ultimate equational descriptions, lacking the power of free pro nite algebras. With the aim of calculating some speci c semidirect products, the rst author and Weil [14] added this ingredi2

ent leading to the Basis Theorem (Theorem 6.1 below). See the rst author's paper [6] for the underlying motivation. The exploration of the full power of this approach eventually led the rst author to the notion of hyperdecidability [5] which turned out to be a unifying concept joining together some of the most important work in nite semigroup theory, including the proofs by Ash [16] and Ribes and Zalesski [35] of the Rhodes type II conjecture through the work of Delgado [19]. At the same time and independently, the second author was also developing his own synthesis of the global and syntactic approaches in connection with the calculation of certain joins [38] which turned out to be very similar to the approach followed by the rst author, Azevedo and Zeitoun [8] but with with a signi cant and key dierence. While the latter authors aimed at practical algorithms, requiring more involved combinatorial work, the rst author only aimed to prove decidability, thus paving the way to many \theoretical" decidability results such as those surveyed in this paper. The basic and rather classical idea is to prove that a set is recursive by recursively enumerating both it and its complement in a suitable universe. This paper is a survey of recent work in this area. The central work on which it is based is reported in the authors' paper [11]. For further developments, see the bibliographic references and forthcoming papers.

2. Preliminary Notions We take familiarity with the concepts of a nite semigroup, a nite state automaton, the word problem for an algebra, and the decidability of an algorithmic problem for granted in this paper, see [3] for background. One of the novelties of this paper is the use of universal algebras to solve problems in nite semigroup theory. Hence, we will include here a basic introduction to the necessary universal algebra as well as, set our notation. An algebraic signature  consists of a collection of symbols, each with an associated arity (which we will take to be nite). For instance, the signature could be  = where has arity 2. A -algebra A consists of a non-empty set, which by abuse of notation we also call A, and for each symbol  of , with arity n , an interpretation A : An A of  as an n-ary operation on A. A homomorphism of -algebras is then a function which preserves all the operations of . For instance, a semigroup is a -algebra which satis es the identity of associativity. Note that a 0-ary operation is really a choice of an element of A. Thus, the signature for a monoid would be ; 1 where, as before, is binary and 1 is 0-ary. See the book of Burris and Sankappanavar [18] for more on universal algebra. The notions of division and relational morphism, introduced by Rhodes and Tilson respectively, are key to what follows. A relational morphism of algebras S and T is a relation ' : S T which is a sub--algebra of S T that projects onto S . A division is a relational morphism such that the projection to T is injective. If there is a division ' : S T , then we say S is a divisor of T. A variety of -algebras is then a class of -algebras closed under arbitrary f
g




!

f


!

g






!

3

direct products and taking divisors. For example, if  is the signature above, the collection of all semigroups forms a variety of -algebras while the collection of all groups does not (one can have a subsemigroup of a group, which is not a group). If A is a -algebra and Y A, then we write Y for the smallest sub--algebra of A containing Y . If X is a set and A a -algebra, one says that X generates A if there is a map  : X A such that A = X . A well known theorem of Birkho states that every variety has free algebras generated by any non-empty set. If V is a variety, we will use X V to denote the free object generated by X in V. If W is the variety of all -algebras, then an element of X W is called a -term. The word problem for V is then to determine algorithmically whether two -terms are equal under the canonical projection to X V. A V-identity in a set of variables X is a formal equality v = w of elements of X V. An algebra A of V satis es this identity if, for every homomorphism  : X V A, v = w . For instance, any semigroup satis es the identity (x y) z = x (y z ). Given a set E of V-identities, the collection of all algebras in V, satisfying E , is a variety, called the variety de ned by E . It is also a theorem of Birkho that if V and W are varieties of -algebras and W V, then W is de ned by V-identities (in nite variable sets). So for example, within the variety of all -algebras, the variety of semigroups is de ned by the identity x(yz ) = (xy)z , where as usual, we use concatenation to denote the binary operation. We will normally write identity instead of Videntity if V is understood. So for example, within the context of the variety of semigroups, the variety of commutative semigroups is de ned by the identity xy = yx. If V is the pseudovariety of all -algebras, we will call a V-identity a -identity. A pseudovariety of -algebras is a collection of nite -algebras closed under nite direct products and taking divisors. We mostly will be interested in pseudovarieties of semigroups. If V and W are pseudovarieties, then the join V W of V and W is the pseudovariety generated by V and W. If V and W are pseudovarieties of semigroups, then V W denotes the pseudovariety of semigroups generated by semidirect or wreath products of semigroups in V with those in W. The semidirect product is an associative operation on the lattice of pseudovarieties. We will be primarily interested in deciding the membership problem for iterated semidirect products of the form 

h

!

h

i

i

!















_



V1 V 2 






Vn :

We place in the following table, a list of the pseudovarieties which will appear in this paper. The problem of deciding the Krohn-Rhodes group complexity can then be phrased as determining membership in iterated semidirect products of the form

A G A 








A = f nite aperiodic semigroupsg Ab = f nite Abelian groupsg BG = f nite block groupsg 4

G A: 

Com = f nite commutative semigroupsg CR = f nite completely regular semigroupsg CS = f nite (completely) simple semigroupsg D = f nite semigroups in which idempotents are right zerosg EV = f nite semigroups whose idempotent generated subsemigroup is in Vg G = f nite groupsg Gp = f nite -groupsg ( prime) H = f nite semigroups all of whose subgroups are in Hg J = f nite J-trivial semigroupsg L = f nite L-trivial semigroupsg N = f nite nilpotent semigroupsg PG = fpseudovariety generated by nite power groupsg R = f nite R-trivial semigroupsg S = f nite semigroupsg Sd = f nite semigroupoidsg Sl = f nite semilatticesg p

p

:

3. Implicit Operations After this discussion of varieties, identities, and Birkho's theorem, it seems natural to turn toward implicit operations which indeed, were created to extend Birkho's theorem to the context of pseudovarieties. We will restrict ourselves to pseudovarieties of semigroups for this discussion. These ideas are key to the syntactical approach to nite semigroup theory. The rst author's [3] is a good source for more on these ideas. If one considers the variety of semigroups, one has the binary operation of multiplication de ned on every semigroup. However, there are many other operations of various arities, de ned on all semigroups, derivable from compositions of the projections and the multiplication operation. For instance, for every semigroup S , there is the unary operation which takes a given element s to s2. This operation is automatically preserved by semigroup homomorphisms. It can be described as \evaluation" of the word x2 of the free semigroup generated by x , at s in the semigroup S . In general, we will write the free semigroup on a set X as X + . Similarly, any word in X + , where X = n, gives rise to an n-ary operation on each semigroup, whose interpretation is evaluation, which is preserved by semigroup homomorphisms. Such an operation is called an explicit operation. In general, if one has for each semigroup S , an n-ary operation S with the property that this operation is preserved by all semigroup homomorphisms, then it is an elementary application of Yoneda's lemma [28] to show that the n-ary operation  obtained from all the S is actually the explicit operation corresponding to w = (x1 ; : : : ; xn )X + , where X = x1 ; : : : ; xn . It then follows that the explicit operations are precisely those operations we can add to the signature  without losing any homomorphisms between semigroups f

g

j

j

f

5

g

(now viewed as algebras with this expanded signature). One can now consider such operations on nite semigroups. An n-ary implicit operation  on the set of all nite semigroups is a collection of n-ary operations S , where S is a nite semigroup, which is preserved by homomorphisms. For example, any explicit operation (where we now restrict it to nite semigroups) is an implicit operation. However, there are many more. For example consider, the unary operation x! which takes an element s of a nite semigroup S to the unique idempotent, written s! , in s . This operation is preserved by homomorphisms. By considering monogenic semigroups of the form x xn = xn+1 for varying n, one sees that this operation cannot be obtained from any word in x + . Another important example is the unary operation x!?1 which takes an element s of a nite semigroup to the inverse of s! s, denoted s!?1 , in the cyclic subgroup generated by s! s. One can again check that this operation is preserved by all homomorphisms of nite semigroups. By considering cyclic groups of arbitrarily large order, one can verify that this operation cannot be obtained from a word. One can compose implicit operations in the natural way, so for instance one can form the binary implicit operation x! y. It is then easy to see that for a nite set X , the set of X ary implicit operations forms a semigroup. Actually, as we will see later, they form a free pro nite semigroup, topologically generated by X , denoted X S. A pseudoidentity in a nite set of variables X is then a formal equality  =  of elements of X S. A semigroup S satis es this pseudoidentity if S = S . Notice that an identity for nite semigroups can be viewed as a pseudoidentity where both sides are explicit. If E is a set of pseudoidentities, then the collection of all nite semigroups satisfying E is a pseudovariety, called the pseudovariety de ned by E . Reiterman's theorem [31] says that every pseudovariety is de ned by some set of pseudoidentities. For example, the pseudovariety of nite groups is de ned by the pseudoidentities. x! y = y = yx! . In analogy to the situation with explicit operations for arbitrary semigroups, if we restrict our attention to only nite semigroups, implicit operations are precisely those operations which we can add to our signature , without losing any homomorphisms between nite semigroups. The philosophy of this paper is then the following. Given a pseudovariety V, try to nd a \nice" collection of implicit operations so that if we add the operations of this collection to the signature , obtaining a new signature , take the variety V of -algebras generated by nite semigroups in V, viewed in the natural way as -algebras, then all the relevant algorithmic problems concerning V are reduced to the word problem for V . This paper will describe what we mean by \nice" in order to solve the membership for iterated semidirect products. h i

h

j

i

f

g

j

j

4. Pro nite Semigroups A pro nite semigroup is a semigroup S which is a projective (or inverse) limit of nite semigroups, or equivalently, it is a compact, Hausdor, and totally disconnected topological semigroup. Another equivalent formulation is that of a compact residually nite semigroup. If S is a pro nite semigroup and V 6

a pseudovariety of semigroups, we say S is pro-V if S is a projective limit of semigroups in V, or equivalently, points of S can be separated by continuous homomorphisms into semigroups in V (viewed as discrete topological semigroups), that is, S is residually in V. A pro nite semigroup S is said to be topologically generated by a pro nite set X if there is a continuous map  : X S such that X = S . One can prove that given a pseudovariety V, there exists a (relatively) free pro-V semigroup, topologically generated by any pro nite set X , denoted X V. If S is a nite X -generated semigroup, then S V if and only if S is a continuous homomorphic image of X V, and hence these pro nite semigroups behave as free objects for V. Given an element  X S, where X = n, one obtains an n-ary implicit operation by evaluating  as follows. An element of S n determines a map from X to S . The image of  in S under the continuous extension to X S is then the value of S on this n-tuple. On the other hand, given an n-ary implicit operation , there is a natural way to extend it to an n-ary operation de ned on the set of all pro nite semigroups, preserved by all continuous homomorphisms. Then an application of Yoneda's lemma shows that  is obtained by evaluation of  = (x1 ; : : : ; xn ) X S where X = x1 ; : : : ; xn . Thus in analogy to the varietal setting, implicit operations correspond to elements of a free pro nite object. Note this also shows that every implicit operation is a limit of explicit operations. In general, relatively free pro nite semigroups are uncountable and hence, not receptive to algorithmic approaches. However, we will see below that they encode certain important algorithmic problems. Viewed in this manner, the philosophy of this paper is to restrict our attention to a nice enough set of implicit operations, so that these important algorithmic problems which are encoded in continuous morphisms (and if you like, continuous relational morphisms) with uncountable free pro nite semigroups, are actually encoded in algebraic morphisms and relational morphisms with countable free algebras in expanded signatures. Then to solve these algorithmic problems, we are instead reduced to solving a word problem. A rst attempt at trying to compute semidirect products by syntactic means was to give a representation theorem for X (V W) in terms of relatively free pro nite semigroups for each of the factors. This was done by the rst author and Weil in [13] and the result is the following. If S is a pro nite semigroup topologically generated by X , then we use ?X (S ) to denote the pro nite Cayley graph of S . It has vertices S 1 (that is, add an identity if S is not a monoid) and edge set S X where an edge (s; x) goes from s to sx. There is a natural continuous left action of S on its Cayley graph. We use the notation E (?) to denote the edge set of a graph ?. The representation theorem is then in terms of the semidirect product of E (?X ( X W)) V with X W, the action being induced by the left action of X W on its Cayley graph. !

h

i

2

2

j

j

f

g





Theorem 4.1. Let V and W be pseudovarieties and X a nite set. Then the continuous map a ((1; a); a) induces a continuous embedding of X (V W) into E (?X ( X W)) V X W: 7!





7

5. Global Semigroup Theory In some sense, global semigroup theory is the study of relational morphisms of nite semigroups. The general recipe of global semigroup theory is the following. Starting with a nite semigroup S , one expands S (see [17] for a precise de nition of expansion), to improve certain properties of S , without losing those properties important for the problem at hand. For instance, when studying complexity it is traditional to use aperiodic expansions as they preserve complexity. Then one tries to nd a relational morphism with a nite semigroup in a particular pseudovariety which is optimal in some sense, either in that it has a small derived or kernel category, discussed below, or it veri es a proposed solution to some sort of algorithmic problem. Consider for instance, the following important example. If V is a pseudovariety and S is a nite semigroup, a subset A of S is said to be V-pointlike if for every relational morphism ' : S T , with T V, there is a t T such that A t'?1 . Henckell and Rhodes de ned a pseudovariety to have decidable pointlikes (called more concisely by the rst author, strongly decidable, as this implies decidability) if there is an algorithm to determine if a subset of a nite semigroup is V-pointlike. The general approach, used successfully by Henckell [23] to prove A strongly decidable, was to nd some reasonable criterion that was sucient for a set to be V-pointlike and then using expansions and various other methods, nd a relational morphism with a nite semigroup in V with the property that only sets satisfying this criterion related to a single point. We will see below that the syntactical approach to this problem of pointlike sets is quite dierent. It was essentially by syntactic means, that the second author in joint work with Rhodes [34] proved that there exist decidable pseudovarieties which are not strongly decidable. !

2

2



5.1. Categories and Semigroupoids

Essential to the global semigroup theory approach is Tilson's derived category theorem. Having roots in the work of Simon, Knast, Therien, Margolis, Rhodes and others, Tilson, in his ground breaking paper [43], introduced the idea of using small categories (and their ideals, called semigroupoids) not as classifying objects, but rather as algebraic objects. He de ned notions of relational morphisms, divisions, varieties, and pseudovarieties for categories. Then he proposed a derived category (semigroupoid) which plays a role in the wreath product decomposition theory of monoids (semigroups) analogous to that played by the kernel of a group homomorphism in the wreath product decomposition theory of groups. We will call functors, morphisms of categories, or more succinctly morphisms. We refer the reader to Maclane's book [28] for terminology involving categories, including the notions of a quotient morphism and a faithful morphism. A relational morphism ' : C D of categories is then a subcategory of C D which projects to C as a quotient morphism. The relational morphism is called a division if the projection to D is faithful. A variety of categories is a class of small categories closed under arbitrary direct products and taking divisors. A pseudovariety of categories is a class of nite categories closed un!



8

der nite direct products and taking divisors. The motivating drive behind this generalization to categories was the idea that a monoid is nothing more than a one object category. Hence, if V is a pseudovariety of monoids, one can consider the pseudovariety of categories generated by V. We denote this gV, the global of V. On the opposite extreme, there is the pseudovariety `V consisting of all categories whose local monoids are in V. In general, these pseudovarieties are dierent, perhaps J is the best known example. When gV = `V, V is called local. Many important pseudovarieties are local including Sl, G, and A, to name a few. A semigroupoid is like a category, but without the requirement of local identities. Semigroupoids play the same role in semigroup theory that categories do in monoid theory. Note for instance, a semigroup is a one vertex semigroupoid. To each relational morphism ' : S T of semigroups, Tilson assigns a semigroupoid D' called the derived semigroupoid. He then proves the derived semigroupoid theorem, dubbed by Rhodes, the covering lemma. !

Theorem 5.1. Let S be a semigroup and V and W pseudovarieties of semigroups. Then S 2 V  W if and only if there is a relational morphism ' : S ! T with T 2 W such that D' 2 gV. A pseudovariety V is called locally nite if X V is nite for X a nite set. If in addition there is a computable bound on X V in terms of X , V is called order computable. An immediate consequence of the work of Tilson is the following.

Proposition 5.2. If W is an order computable, locally nite pseudovariety and gV is decidable, then V W. 

Another triumph of Tilson's theory was the delay theorem which in a weak form can be phrased as the following.

Theorem 5.3. If V is a pseudovariety of monoids, then gV is decidable if and only if V D is decidable. 

Finally, we note that the second author has de ned notions of semidirect products of categories (and semigroupoids) and semidirect products of pseudovarieties of categories (and semigroupoids). He then proved the following theorem [40], which will be useful in the sequel. Theorem 5.4. Let V and W be pseudovarieties of semigroups. Then g(V W) = g V g W. The proof was syntactic in nature. However, recent joint work of the second author with Tilson has re ned the de nition of a semidirect product of categories (without changing the de nition on the pseudovariety level) allowing a simpler proof of the above theorem in the monoid case via a structural rather than syntactic approach. 



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5.2. The Type II Theorem

Another success of global semigroup theory is the type II theorem of Ash [16], although Ash's proof already starts to blend in syntactic methods in that he makes use of free objects. Rhodes and Tilson de ned an element s of a nite semigroup S to be type II if under every relational morphism of S with a nite group, s relates to 1. The collection of all such elements forms a subsemigroup called the type II subsemigroup (or the group kernel) of S . Rhodes and Tilson conjectured an algorithm to compute the type II semigroup. Ash proved the validity of this conjecture in his landmark paper [16] which as we will shall shortly see, proved much more. A simple consequence of the derived semigroupoid theorem is the following.

Theorem 5.5. Let V be a local pseudovariety of semigroups. Then S V G if and only if the type II semigroup of S is in V. 2



Consequences include the determination of the pseudovarieties generated by nite inverse semigroups, nite orthodox semigroups, and also a proof of the equality PG = BG [25].

6. Pro nite Categories and the Basis Theorem The rst author and Weil attempted to integrate Tilson's theory to obtain a new syntactical approach to computing semidirect products [14]. First, they developed an analog of Reiterman's theorem for pseudovarieties of semigroupoids (done independently by Jones [27] at the same time). De nitions and results for pro nite semigroupoids with nitely many vertices are similar to the semigroup case, we refer the interested reader to [14, 27]. Using these ideas, the rst author and Weil obtained a basis theorem giving a basis of pseudoidentities for V W in terms of pseudoidentities for gV and W. While this approach already allowed the calculation of many semidirect products through a case by case reduction of this basis, it fell short of proving general decidability results as in most cases, this basis is uncountable. In an attempt to rectify this, the rst author developed the notion of hyperdecidability. To even state the basis theorem, we need the notion of directed graphs labeled by semigroups. A directed graph ? consists of two sets V (?), E (?) and maps ; ! : E (?) V (?) selecting the initial and terminal vertices respectively of each edge. Since a semigroupoid can be viewed as a directed graph with a composition operation, we will also use the notation  and ! for semigroupoids. A labeling  of ? over a semigroup S consists of two functions, which by abuse of notation we both denote by ,  : V (?) S 1 ,  : E (?) S . A labeling is said to be consistent if for each edge e, 

!

!

!

(e)(e) = e!: For instance, if S is a semigroup, then the Cayley graph of S has a natural, consistent labeling over S . If  is a labeling of a nite graph ? over a pro nite 10

semigroup S , then we will, in one nal abuse of notation, also use  for the induced homomorphism  : ? Sd S . We note a pseudoidentity for a pseudovariety of semigroupoids is actually a pseudoidentity over a nite graph and so, we write (u = v; ?) for such a pseudoidentity. The basis theorem then has the following statement. !

Theorem 6.1. Let V and W be pseudovarieties of semigroups and let (uj = vj ; ?j ) j J be a basis of pseudoidentities for gV. For a nite set X , let pW denote the canonical projection pW : X S X W. Then V W is de ned by f

j

2

g

!

pseudoidentities of the following form



(uj )(uj ) = (vj )(vj ) where  is a labeling of ?j over X S (X nite) such that the labeling pW of ?j over X W is consistent.

The goal of this paper is then to nd conditions on V and W so that we may trim this basis down to a more manageable level. It turns out that the more stringent conditions must be placed on W.

7. Hyperdecidability In an attempt to turn the above basis theorem into a decidability algorithm, the rst author created the notion of hyperdecidability. Though not motivated in this fashion, it then turned out to be a generalization of the notion of strong decidability as well as of a notion introduced by Ash in his proof of the type II conjecture. Let V be a pseudovariety, S a nite semigroup, ? a nite graph, and  a labeling of ? over S . Let ' : S T be a relational morphism. We say that a labeling of ? over T is '-related to  if for any edge or vertex x of ?, x '-relates to x . The labeling  is said to be V-inevitable if and only if for every relational morphism ' : S V with V V, there exists a consistent labeling of ? over V which is '-related to . If there is an algorithm to determine V-inevitability of nite graphs labeled over nite semigroups, V is said to be hyperdecidable. Hyperdecidability implies strong decidability. Examples will be provided later in the paper. The following important relationship then exists between inevitable labelings and free pro nite objects, see [5]. !

!

2

Proposition 7.1. Let S be a nite X -generated semigroup,  : X S ! S the canonical projection, and pV : X S ! X V the canonical projection. We denote by 'V the relational morphism  ?1 pV . Then a labeling  of a nite graph ? over S is V-inevitable if and only if there is a consistent labeling over X V, '-related to . We call 'V the canonical relational morphism. Using this theorem and the basis theorem, the rst author could prove the following result [5]. 11

Theorem 7.2. Let V and W be pseudovarieties of semigroups such that W is hyperdecidable, gV is decidable and has a basis  of pseudoidentities with a computable bound on the number of vertices of any graph in . Then V W is 

decidable.

One can also de ne such notions for pseudovarieties of semigroupoids. The key result, due to the rst author [5] (although discovered independently to hold in many cases for the question of strong decidability by the second author [38]), is that for a pseudovariety of semigroups, the hyperdecidability of V and gV are equivalent. This led to the question of nding methods to prove hyperdecidability.

8. Ash's Theorem and Weak Reducibility

The rst known hyperdecidability result was the hyperdecidability of G, proved by Ash in his proof of the type II conjecture [16], we will speak more on his proof technique below. After developing the notion of hyperdecidability, the rst author with Zeitoun proved the hyperdecidability of J [15] and with Silva, the strongly connected hyperdecidability of R (hyperdecidability for strongly connected graphs) [10]. These arguments were rather technical and automata theoretic in nature, rather than exploiting free pro nite objects, but were none the less a syntactic approach. At the same time, the second author proved that J was strongly decidable using the rst author's solution to the word problem for X J and in fact proved that J G was strongly decidable by using Ash's theorem as well [39] (the decidability of pointlike pairs was proven independently by the rst author, Azevedo, and Zeitoun [8]). When the second author learned of hyperdecidability at the International Conference on Algebraic Engineering in Japan in March 1997, he could easily generalize these results to hyperdecidability. His idea was that since X J is recursively enumerable with decidable word problem, one could use proposition 7.1 to recursively enumerate all J-inevitable labelings. On the other hand since J is decidable, one can recursively enumerate all J-inevitable labelings which are not inevitable. Ash proved the hyperdecidability of G by showing that the properties of proposition 7.1 were enjoyed by a certain relational morphism of a nite X generated semigroup with the free group FG(X ) generated by X , again a countable semigroup with decidable word problem. Work by Delgado to prove that Ab is hyperdecidable [19], showed that this relational morphism considered by Ash was precisely the restriction of the canonical relational morphism 'G to FG(X ). Delgado proved a similar result for Ab. These ideas led the rst author to formulate, in a talk at Braga in June 1997, the notion which we now call weak -reducibility. We de ne an implicit signature  to be a signature containing  (recall  = ), all of whose symbols represent implicit operations of nite arity. Any nite or pro nite semigroup can be made a -algebra by giving the implicit operations in the signature their normal interpretation. A -algebra for which is associative is called a -semigroup. If V is a pseudovariety, we denote by V the variety of -algebras generated by nite semigroups in V where the implicit operations have their _

f
g




12

standard interpretation. Note that V consists of -semigroups. We use X V to denote the free object generated by X in this variety. One can easily show that X V is exactly the sub--algebra of X V generated by X . So for instance if  = ; ()!?1 , then X G = FG(X ). For J, X J = X J. We say that an implicit operation  for semigroups of arity n is computable if there is an algorithm which given an n-tuple of elements from a nite semigroup S and the multiplication table of S , returns the value of S on this n-tuple. It is easy to see that the composition of computable implicit operations is again computable. We call an implicit signature highly computable if it is recursively enumerable and consists of computable implicit operations. A pseudovariety V is said to be weakly -reducible for a highly computable implicit signature  if the statement of proposition 7.1 remains true for the restriction of 'V to

X V, that is the relation obtained by taking those pairs in 'V with second coordinate in X V. This is an abstract property involving  and V rather than an algorithmic property. But if we place some algorithmic assumptions on V, we obtain some important decision procedures. We use  : X + X V for the natural map. Note that X V is a topological semigroup in the subspace topology. One then has the following result. f


g

!

Theorem 8.1. Let V be a pseudovariety of semigroups,  a highly computable implicit signature for which V is weakly reducible, and suppose that for every nite set X , X V has a decidable word problem. If there is an algorithm to decide membership in L for elements of X V for any rational language L X  , then V is hyperdecidable. For instance, these hypothesis are ful lled by G (due to Ash's theorem and Delgado's results) for , and are easily seen to hold for J. The second author also showed they hold for J G [39] and Gp , the pseudovariety of p2

_

groups for p a prime [41]. We will discuss this more later. Notice this is a syntactical approach, much dierent from the techniques used by Henckell to prove A strongly decidable. The key dierence is that Henckell looked for a nite aperiodic semigroup which computed pointlikes, while this approach looks for an in nite one.

9. Questions of a Recursive Nature Before continuing, we must address some questions of a recursive nature. We will mostly be interested in the case where V is a recursively enumerable pseudovariety, meaning there is an algorithm to enumerate all members of V in some xed countable universe. Any decidable pseudovariety, that is a pseudovariety with decidable membership probem, is recursively enumerable. It is also not dicult to see that the join or semidirect product of recursively enumerable pseudovarieties are again recursively enumerable. A pseudoidentity  =  is said to be computable if  and  are computable implicit operations. We call V recursively de nable if there is a recursively enumerable list of computable pseudoidentities de ning V. The following observation is key to what follows. 13

Proposition 9.1. Let V be a recursively enumerable, recursively de nable pseudovariety. Then V is decidable. Proof. We just need to show that the complement of V is recursively enumerable. To prove this, it suces to exhibit a procedure which on input of a nite semigroup S (or rather its multiplication table) outputs \no" if S is not in V but which need not halt if S V. Let  be a recursively enumerable basis of computable pseudoidentities for V. Then we can algorithmically enumerate the pseudoidentities in . As we enumerate each pseudoidentity, we verify whether it is valid in S , which we can do by hypothesis since the pseudoidentity is computable. If the pseudoidentity is satis ed by S we continue, otherwise we output \no". If S is not in V, we will eventually nd a pseudoidentity it fails to satisfy, while if S V, we will never nd such a pseudoidentity. 2

2

Let  be an implicit signature. We call V -recursive if V has a recursive basis of -identities (that is identities as a -algebra). An adaptation of a well known theorem of Evans [21] shows the following.

Proposition 9.2. Let  be a highly computable implicit signature. Then for a pseudovariety V, the following are equivalent. i) V has a recursively enumerable basis of -identities;

ii) V is -recursive;

iii) X V has a decidable word problem for every nite set X (in this case we say V has decidable -word problem).

It is not dicult to provide analogs of all these concepts for pseudovarieties of semigroupoids. In fact, if  is a highly computable implicit signature of semigroup operations, there is a canonical way to transform it into a highly computable implicit signature of semigroupoid operations [11]. Furthermore, V will be -recursive if and only if gV is -recursive.

10. Reducibility The authors were able to prove the following theorem, lifting in many cases the vertex bound restriction of theorem 7.2.

Theorem 10.1. Let V and W be recursively enumerable pseudovarieties of semigroups such that W is hyperdecidable and gV is recursively de nable. Then V  W is decidable. This spurred the authors to ask under what hypothesis on V and W will cause g(V W) to be recursively de nable so that we may iterate the process. To this end, they invented the notion of -reducibility which re nes the notion of weak -reducibility by replacing the topological relational morphism, with an algebraic one. 

14

Let  be a highly computable implicit signature. Let S be a nite X generated semigroup (X is assumed to be nite). We denote by ';V the relational morphism of -algebras whose graph is the sub--algebra of S X V generated by pairs (x; x) with x in X . We then say V is -reducible if the property of proposition 7.1 is enjoyed by the relational morphism ';V . That is to say, a labeling  of a nite semigroup S over a nite graph ? is V-inevitable if and only if there is a consistent labeling of ? over X V which is ';V related to . It is easy to verify that this relational morphism is contained in the relational morphism considered in the de nition of weak -reducibility and hence, we have the following proposition. 

Proposition 10.2. Let  be a highly computable implicit signature. Then if V is -reducible, V is weakly -reducible. If these two relational morphisms coincide, we call V -full. The authors

have shown that the relational morphism considered by Ash in his proof of the type II theorem is exactly ';G [11] and hence, G is -reducible. In light of Delgado's results, it follows G is -full. It is easy to verify that J is -full and hence -reducible. It is more dicult to verify that J G is -reducible as well [39]. We say a pseudovariety V is -de nable if it can be de ned by pseudoidentities built up from the composition of operations in . For instance, G is -equational, being de ned by the pseudoidentity x!?1xy = y = yxx!?1. On the other hand, the pseudovariety Gp is not -equational since the free group is residually a p-group and hence, there can be no -identities satis ed by all p-groups which are not satis ed by all nite groups. One can show using that ';V determines which sets are pointlike, that if V is -reducible, then it is -equational [11]. Hence, Gp is not -reducible. The rst author and Volkov announced at this conference that Ab is not -equational and so, cannot be -reducible. We will call a pseudovariety V tame if it is recursively enumerable and there exists a highly computable implicit signature  for which V is -reducible and -recursive. Once again these are abstract properties of the pseudovariety and the signature  rather than algorithmic ones. However, in combination with the right algorithmic properties for  and V, we will obtain decidability results of a very strong sort. A rst result in this direction is the following. _

Theorem 10.3. Let V be a tame pseudovariety. Then V is hyperdecidable. Proof. Since V is recursively enumerable, one can clearly recursively enumer-

ate all labelings of nite graphs over nite semigroups in some xed countable universe which are not V-inevitable. To show the set of V-inevitable labelings is recursively enumerable, one must show there is a procedure which when given a nite semigroup S and a labeling  of a nite graph ? over S , produces the output \yes" if  is V-inevitable, but is not required to halt in the case where the labeling is not V-inevitable. The procedure is as follows, calculate a nite generating set X for S . Let  be a highly computable implicit signature for 15

which V is -recursive and -reducible. Then since  is recursively enumerable, one can recursively enumerate all -terms and hence, since ? is nite, one can recursively enumerate the set
of all labelings of ? by -terms. The -reducibility of V then says that the labeling  is V-inevitable if there is a labeling
, which when evaluated in S (that is, each of the -terms of the labeling is evaluated in S ), is  and when evaluated in X V, is consistent. Since  consists of computable implicit operations, we can check if a labeling in
evaluates to  in S . Since V is -recursive, we can decide the word problem for X V and hence, check whether a labeling in
evaluates in X V to a consistent labeling. Thus, if such a exists, we will eventually nd it and output \yes." Otherwise, the procedure continues inde nitely. 2

Of course, this theorem does not give an ecient algorithm, but then again it does not use very much of the structure of the pseudovariety V. One would hope that in individual cases, one can obtain a faster algorithm. For instance, the hyperdecidability algorithm for J given by the rst author and Zeitoun is more ecient (but also more dicult to prove) than the one implied by the above theorem. We point out that once again, analogous de nitions exist for pseudovarieties of semigroupoids. In particular, if  is a highly computable implicit signature, then gV is -reducible if and only if V is -reducible (where as before  is transformed into a semigroupoid implicit signature in the canonical way).

11. Comparisons and Examples We now pause to compare some of the various concepts which we have de ned, as well as to give examples. The following diagram (Figure 1) goes a long way towards summarizing the relationships between these dierent notions. The right hand side of the diagram refers to abstract properties while the left hand side refers to algorithmic properties. We will rst give some examples for the cases in which we know it is impossible to reverse the implications in the above diagram. If E is a set of semigroup identities, we will write [ E ] for the pseudovariety of semigroups satisfying these identities. In [1], an example is given of a nite set of semigroup identities E for which it is undecidable as to whether a semigroup identity u = v is satis ed by all members of [ E ] . This pseudovariety is clearly decidable, one just has to check if a nite semigroup satis es E . In joint work with Rhodes [34], the second author has shown that one cannot compute pointlikes for this pseudovariety. It is unknown whether there is a pseudovariety for which one can compute pointlikes, but which is not hyperdecidable. We now observe that this pseudovariety [ E ] is recursively de nable, but not -recursive for any highly computable implicit signature. Indeed, since [ E ] is de ned by nitely many semigroup identities, it is clearly recursively de nable. However, it cannot be recursive for any highly computable implicit signature. Indeed, suppose u = v is a semigroup identity in variables X . Then one cannot decide whether u = v in X V (since  as an implicit signature is required to include , u; v X V) and hence, V cannot be -recursive.


16

2

 -recursive

abstract properties concerning 

algorithmic properties concerning 

 -WP decidable

 -full

computable -closures

weakly -reducible  -reducible  -equational

hyperdecidable strongly decidable

recursively de nable algorithmic properties

decidable

Figure 1: Relationships between various properties of a recursively enumerable pseudovariety, where  is a recursively enumerable implicit signature consisting of computable operations, and a (branched) arrow leading to a property means that the property follows from the conjunction of the properties at the other extremities of the arrow.

17

We comment that if V is recursively de nable with recursively enumerable basis  of computable pseudoidentities, then by letting  be the implicit signature consisting of and all the implicit operations appearing on any side of a pseudoidentity in , we obtain an implicit signature  for which V is equational. We now show that a pseudovariety can be -equational, without being -reducible. Consider the implicit signature  = ; ()! . Then G is equational. We show it is not -reducible. It is not hard to see X G is nothing but X  , the free monoid on X . The word problem is solved by merely replacing any term t! with 1. Since this reduces the number of occurrences of the ()! operation, one eventually obtains a term which is a word and which maps in any nite group to the same element as the original term. Of course no two words give the same image in every nite group since the free group is residually nite. Suppose G was -reducible. Then an element S of a nite X -generated semigroup would be in the type II subsemigroup if an only if it is the image of an -term which maps to 1 in X  . We show by induction on the number of occurrences of ()! in a term, that any term t which reduces to 1 is a product u!1 u!n where the ui are -terms. If the term has no occurrences of ()! it cannot map to 1. If t has one such occurrence, it must look like w1 (w2 )! w3 with w1 ; w3 X  ; w2 X + . But then in X  , this term reduces to w1 w3 , so t = (w2 )! . In general, t = u1(u2 )! u3 where u1; u3 are either 1 or -terms with less occurrences of ()! than t and, u2 is an -term with less occurrences of ()! than t. One nds u2 by choosing a term to which ()! is applied, but which is not a subterm of any other term to which ()! is applied. Formally, one inductively de nes a notion of ()! -height on terms and chooses a subterm (u2 )! of maximum height. Since t and u!2 reduce to 1 in X  , we have 1 = red(u1 )red(u3 ) (where for a term u, red(u) is the reduction of u in X  ). So by induction, u1 and u3 have the desired factorizations and hence, t does as well. It follows that any term t which reduces to 1 in X  maps to a product of idempotents in any nite semigroup. Thus if G were -reducible, the type II subsemigroup of a nite semigroup would always be the subsemigroup generated by the idempotents, rather than the smallest subsemigroup generated by idempotents and closed under weak conjugation. But there are semigroups whose type II subsemigroup is dierent from its idempotent generated subsemigroup. For instance, in [32] Rhodes gives an example of a semigroup S of complexity 2 in EA. Hence, the idempotent generated subsemigroup of S is aperiodic. However, since A is local, if the type II subsemigroup of S were aperiodic, then S would be in A G, which would imply S has complexity 1, a contradiction. Alternatively, one may consider the semigroup S presented by S = e; s e2 = e; s3 = s; es2e = ese = 0 : Then ses = ses!?1 is a type II element but not a product of idempotents. Note that in this example, S A G. The second author proved in [41] that Gp is weakly -reducible, but as we have already seen, this pseudovariety is not -equational and hence not reducible. Also, Gp is an example where we can compute the -closure of a rational language, but which is not -full (since it is weakly -reducible, but not -reducible).


f







2

2



h

2

j

i



18

g

The following questions however, are open. Is there a decidable pseudovariety which is not recursively de nable? Is there a pseudovariety which is reducible for some highly computable implicit signature but which is not, full? Is there a pseudovariety which is hyperdecidable, but not tame? Is there a decidable pseudovariety which is -recursive but does not have computable pointlikes? Also is there a decidable pseudovariety for which gV is not decidable? For more on these last two questions see the second author's [42] which amongst other things gives an example of a pseudovariety of monoids V such that either gV is undecidable or V D is decidable, -recursive, but does not have decidable pointlikes. The following pseudovarieties are amongst those known to be -reducible: G, Ab, J, J G (and a good many subpseudovarieties), D, CS, and locally nite pseudovarieties. Work in progress by the rst author and Trotter would indicate that CR is -reducible, work in progress by Rhodes would indicate A is -reducible, while work in progress of the second author would indicate R is -reducible for strongly connected graphs. 

_

12. A New Basis Theorem and Applications The following theorem may be viewed as a basis theorem for -reducible pseudovarieties. Theorem 12.1. Let  be an implicit signature, V and W be pseudovarieties of semigroups, let (uj = vj ; ?j ) j J be a basis of pseudoidentities for gV, and W be -reducible. For a nite set X , let pW denote the restriction of the canonical projection pW : X S

X W. Then V W is de ned by pseudoidentities of the following form (uj )(uj ) = (vj )(vj ) where  is a labeling of ?j over X S (X nite) such that the labeling pW of ?j over X W is consistent. Proof. Clearly, such pseudoidentities are a subset of those given by the Basis Theorem and hence any nite semigroup in V W satis es them. Conversely, let S be a nite semigroup satisfying the above pseudoidentities. Let X be a nite set and  a labeling of a ?j over X S such that pW is consistent. We must show that S satis es (uj )(uj ) = (vj )(vj ): It suces to show that any X -generated subsemigroup of S satis es this identity, and hence without loss of generality, we may assume S is X -generated. Then by proposition 7.1, the evaluation of  in S (in the sense of the proof of Theorem 10.3) gives a W-inevitable labeling 0 of ?j . Hence by the assumption of -reducibility, there is a labeling of ?j over S X which evaluates to 0 in S and so that pW is consistent. Since S satis es the pseudoidentities above, we have that (uj 0 )(uj 0 ) = (vj 0 )(vj 0 ) f

j

2

g

!





19

and so S satis es

(uj )(uj ) = (vj )(vj ):

We then get the following consequence [11]. Corollary 12.2. Let V and W be pseudovarieties such that gV is recursively de nable and W is tame. Then V W is recursively de nable. In addition, if V and W are recursively enumerable, then V W is decidable. 



Proof. The hypotheses are exactly what one needs to make the above basis

recursively enumerable and consist of computable pseudoidentities and so, the result follows from Proposition 9.1.

There is an analogous result for semigroupoid pseudovarieties. In particular, since g(V W) = gV gW and our hypotheses are stable under going to gW, we obtain the following. 



Corollary 12.3. Let V and W be pseudovarieties such that gV is recursively de nable and W is tame. Then g(V W) is recursively de nable. In addition, if V and W are recursively enumerable, then g(V W) is decidable. 



One can then use induction to prove the following important result [11] which is the culmination of this paper, fusing syntactic techniques and ideas from global semigroup theory. As a consequence we obtain a reduction theorem for the decidability of complexity.

Theorem 12.4. Let V1; : : : ; Vn be recursively enumerable pseudovarieties such that gV1 is recursively de nable while for 2  j  n, Vj is tame. Then V1 V 2 






Vn

is decidable.

Corollary 12.5. If A is tame, then the Krohn-Rhodes group complexity of a nite semigroup is decidable.

Proof. By Ash's theorem, G is -reducible. Of course the free group on a nite set has a well known solvable word problem, so G is -recursive. Since

the pseudovariety of complexity n semigroups is an iterated semidirect product of A and G, the result follows. A proof that A is -recursive for the implicit signature  above, is to appear in these proceedings in a paper by McCammond. At the conference itself, Rhodes announced a proof of the -reducibility of A. Some other applications have been the following. The second author used the ideas of this paper to prove that J G is -reducible. As the join of two -recursive pseudovarieties, it follows that J G is -recursive. Hence this pseudovariety has a recursively enumerable basis of -identities. As Trotter and Volkov have shown that this pseudovariety is not nitely based [44], this _

_

20

is perhaps the best one can hope for. In particular this gives the decidability of J G. The rst author in joint work with Escada [9] used that G was reducible and the basis implied by -reducibility for J G to give a syntactic proof that J G = EJ from which PG = BG follows as in [25]. They also used these ideas to explore the question, when does V G = EV. Finally as another application, the rst author and Volkov have announced at this conference that if H is a recursively de nable pseudovariety of groups, then H is recursively de nable. Since this pseudovariety is easily seen to be local, it follows from the basis theorem that H G has a recursively enumerable basis of a particularly nice form. For instance, in the case H = 1, H = A and we obtain the basis consisting of all pseudoidentities u! = u!+1 where u is a -term which reduces to 1 in the free group. Actually, one can re ne this basis some, see [11]. Future work to be done would include the following. First, it would be nice to answer some of the open questions in the previous section. Secondly, it is important to prove more pseudovarieties tame. In particular, since many important pseudovarieties are not de ned by -identities, it is of utmost importance to nd more computable implicit operations to work with. For instance, the fact that Gp is weakly -reducible, implies that there may be a highly computable implicit signature  consisting of implicit operations which when projected to

X Gp , land in FG(X ). We end by pointing out that these stronger versions of hyperdecidability are needed since the rst author and Azevedo have announced at a conference in St. Andrews in 1997 that there do exist decidable pseudovarieties whose global cannot be de ned by pseudoidentities on graphs with a bonded number of vertices. _









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