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Let W denote the intersection with the pseudovariety of completely regular semigroups of the Mal'cev product B fl m V of the pseudovariety of bands with a pseudovariety of completely regular semigroups. It is shown that the (pseudo)word problem for W is reduced to that for V in such a way that decidability is preserved in case say only -terms (i.e., terms involving only multiplication and the (! \Gamma 1)-power) are considered. It is also shown that, if V is a hyperdecidable (respectively -reducible) pseudovariety of groups, then so is W. 1 Introduction Motivated by the Krohn-Rhodes complexity problem [22], the search for uniform algorithms for computing semidirect products of pseudovarieties has led to substantial research in the theory of finite semigroups. Even though there is no universal solution, since the semidirect product of decidable pseudovarieties is not necessarily decidable [1], under suitable assumptions on the factors, the semidirect product might be decidable. The no...

Within the usual semidirect product S T of regular semigroups S and T lies the set Reg (S T ) of its regular elements. Whenever S or T is completely simple, Reg (S T ) is a (regular) subsemigroup. It is this `product' that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes, U and V , the e-variety U V generated by fReg (S T ) : S 2 U; T 2 V g, is well defined if and only if either U or V is contained within the e-variety CS of completely simple semigroups. General properties of this product, together with decompositions of many important e-varieties, are obtained. For instance, as special cases of general results the e-variety LI of locally inverse semigroups is decomposed as I RZ, where I is the variety of inverse semigroups and RZ is that of right zero semigroups; and the e-variety ES of E-solid semigroups is decomposed as CR G, where CR is the variety of completely regular ...

We prove that the e-variety CR(H), of all completely regular semigroups whose subgroups belong to some group variety H , is e-local; that is, every regular, locally completely regular semigroupoid [with subgroups from H ] divides a completely regular semigroup [with subgroups from H ], in a `regular' way. In a future paper with P.G. Trotter, this theorem will be applied to semidirect products of e-varieties and to e-free E-solid regular semigroups. A key role in the proof is played by the e-free semigroups in the e-variety CR(H) . We provide a solution to the `word problem' in these semigroups, in the style of that for free completely regular semigroups given by Ka dourek and Pol`ak. The solution is derived from the author's work on free products of completely regular semigroups. In an earlier paper [3] the author proved that any category whose local (or `loop') monoids satisfy the identity x n+1 = x divides a monoid satisfying the same identity. This result is easily tra...

The free product CR S i of an arbitrary family of disjoint completely simple semigroups fS i g i2I , within the variety CR of completely regular semigroups, is described by means of a theorem generalizing that of Ka dourek and Pol'ak for free completely regular semigroups. A notable consequence of the description is that all maximal subgroups of CR S i are free, except for those in the factors S i themselves. The general theorem simplifies in the case of free CR-products of groups and, in particular, free idempotent-generated completely regular semigroups. 1980 Mathematics Subject Classification (Amer. Math. Soc.) (1985 Revision): Primary 20MO5; secondary 08A5O. Based on fundamental insights of A. H. Clifford [1], J. A. Gerhard [3], P. G. Trotter [9] and J. Ka dourek and L. Pol'ak [8] have offered solutions to the word problem for the free completely regular semigroup FCRX on a countably infinite set X. This semigroup is clearly the free product, in the variety CR of co...