Irreducibility of certain pseudovarieties 1

We prove that the pseudovarieties of all finite semigroups, and of all aperiodic fin1 e sernigroups are irreducible for join, for semidirect product and for hlal 'ce~. product. In particular, these pseudovarieties do not admit maximal proper suhpseudovariet~es. More generally, analogous results are proved for the pseudovar~ety of all finite serrigroups all of whose subgroups are in a fixed pseudovariety of groups H , provided ti1 ~t H is closed under semid~rect product. R & u m 6 Nous prouvons que la pseudovaridtl de tous les semigroupes fin~s, et celle de tous 1.s semigroupes apdriodiques finis sont irrlductibles pour le sup, pour ie prodult sern ( , I rect et pour le produit de Mal'cev. En particulter, ces pseudovariktls n'adrnettent pas de sous-pseudovaridtl maximale propre. Des rlsultats analogues sent ktablis ~ l d s gendralement pour la pseudovariltd de tous les semigroupes finis dont les sous-groupes sont dans une pseudovariltl de groupes fixee H , pourvu que H sott fermCe par procatlit semidirect 'The two first authors were partially supported by NSF Grant DMS920381 Part of this v ork was done while the second author was an invited Professor at the Universit.4 Paris-\'I All three author!, ~vere partiall) supported by the Center for Communication and Information Science of the Univers~ty of Uebraska. C o p y ~ ~ g h r a; 1998 by Marcel Dekku. Inc D ow nl oa de d by [ H eb re w U ni ve rs ity ] at 0 1: 38 0 8 M ay 2 01 3 780 MARGOLIS, SAPIR, AND WEIL Within the context of the general study of the structure of the lattice of pseudovarieties of finite semigroups, the question of describing the irreducible pseudovarieties is both a natural and an old problem. For instance, only a handful of the "classical" pseudovarieties are known to be join-irreducible or join-reducible (see [ l , chap. 91). In particular, the ~seudovariety of nilpotent semigroups is join-irreducible, whereas J, the pseudovariety of 2-trivial semigroups, is join-reducible (Almeida 121). In this paper, we show that if H is a pseudovariety of groups closed under semidirect product, then the pseudovariety of all semigroups all of whose subgroups are in H is irreducible for join, for semidirect product and for Mal'cev product. The particular cases where H is the pseudovariety of all groups. and where H is trivial yield the irreducibility of S, the pseudovariety of all finite semigroups, and of A , the pseudovariety of aperiodic semigroups. .4s a consequence, it follows that these pseudovarieties do not contain maximal proper subpseudovarieties, a fact which generalizes a result of Margolis [ l l ] . If we consider the analogous problem for S and E, respectively the variety of all semigroups and the variety of all groups, it is known that S is join irreducible (Evans [6]) but that S = Corn @Corn = Corn i Corn where Corn is the variety of all commutative semigroups: the first equality is immediate when one considers the projection from the free semigroup onto the free commutative semigroup; the second one follows from a result of Mal'cev stating tha t the free semigroup on 2 generators is embedded in the free metabelian group [lo]. It is also known that G is irreducible for join, semidirect product and Mal'cev product [12]. The proof of the join irreducibility of S is based on the manipulation of identities, and cannot be used directly for pseudovarieties. However, it is known that each subpseudovariety of a pseudovariety V is defined by a set of formal equalities between elements of certain relatively free profinite structures (Reiterman's theorem, see [l, 17, 20)). We call these formal equalities pro-V-identities, They are also called pseudoidentities [I]. Formal definitions are given in Section 1.1. In order to prove that a pseudovariety V is join (resp. semidirectly, Mal'cev) irreducible, we use an idea inspired by Evans's proof. It is enough to prove the following: From every pair of pro-V-identities ul = vl and uz = vz which are non trivial, i.e. which define proper subpseudovarieties V 1 and V z of V . we can construct a non trivial pro-V-identity which holds in V 1 V V 2 (resp. V 1 * V z , V 1 @ V z ) . This is done in several steps, each of which consists in constructing non trivial consequences of u; = vl and uz = vz with some special properties. These consequences are obtained by encoding ul = vl and u 2 = z.2. That is, we substitute given values for the variables of the given pro-V-identities in such a way tha t the resulting pro-V-identities are again non trivial. The main result is proved in Section 3.