ON MAXIMAL SUBGROUPS OF FREE OBJECTS OF CERTAIN COMPLETELY REGULAR SEMIGROUP VARIETIES

By adjusting a method of Kadourek and Polák developed for free semigroups satisfying xr ≏ x, we prove that if [Formula: see text] is a periodic group variety, then any maximal subgroup of the free object in the completely regular semigroup variety of the form [Formula: see text] is a relatively free group in [Formula: see text] over a suitable set of free generators. When [Formula: see text] is locally finite, we provide some bounds for the sizes of its finitely generated members.

Polyhedral convex cones and the equational theory of the bicyclic semigroup

To any given balanced semigroup identity U ≈ W a number of polyhedral convex cones are associated. In this setting an algorithm is proposed which determines whether the given identity is satisfied in the bicylic semigroup or in the semigroup . The semigroups BC and E deserve our attention because a semigroup variety contains a simple semigroup which is not completely simple (respectively, which is idempotent free) if and only if this variety contains BC (respectively, E). Therefore, for a given identity U ≈ W it is decidable whether or not the variety determined by U ≈ W contains a simple semigroup which is not completely simple (respectively, which is idempotent free).

ON THE SUPERDIMENSIONS OF RELATIVELY FREE NILPOTENT SEMIGROUPS

We prove that in the variety of nilpotent semigroups of class ≤2, which is defined by the Neumann–Taylor identity xyzyx=yxzxy, the sequence of the superdimensions for relatively free semigroups is convergent to 1 and at the same time every element of the sequence is strictly less than 1. This gives the first example of a semigroup variety for which the set of superdimensions for the free objects is infinite.

THE IDEMPOTENTS IN A PERIODIC SEMIGROUP

Let [Formula: see text] be the semigroup variety determined by the identity xm=xm+k. For [Formula: see text] we define operations on the set E(S) of idempotents of S and thus obtain the idempotent algebra of S. For any subvariety [Formula: see text] of [Formula: see text] the idempotent algebras of the members of [Formula: see text] form a variety [Formula: see text] and [Formula: see text] yields a complete homomorphism of the lattice [Formula: see text] of subvarieties of [Formula: see text] onto the lattice [Formula: see text] of subvarieties of [Formula: see text]. The lattice [Formula: see text] contains a ∩-semilattice isomorphic to the ∩-semilattice [Formula: see text] of group varieties of exponent dividing k for every m≥1. In particular, for appropriate k, the lattice of subvarieties of the variety of all idempotent algebras of the completely regular semigroups over groups that belong to [Formula: see text] is of the power of the continuum. For any [Formula: see text], ρ→ρ|E(S) yields a complete homomorphism of the congruence lattice of S into the lattice of equivalence relations on E(S).

Semigroup varieties closed for the Bruck extension

We shall show that there exists a chain, order isomorphic to the chain of real numbers, of semigroup varieties closed for the Bruck extension. The least semigroup variety closed for the Bruck extension will be obtained as the union of varieties in an infinite chain of semigroup varieties.

PERMUTATION IDENTITIES AND VARIETIES OF NILSEMIGROUPS

For any variety V of semigroups there exists a smallest semigroup variety PV containing V and closed for the construction of power semigroups. These varieties PV form a countably infinite subset PL(S) of the lattice L(S) of semigroup varieties. Though (PL(S), ⊆) is a complete lattice, it is not a complete sublattice of L(S). There exists however an interval in L(S) consisting of varieties of nilsemigroups which is isomorphic to (PL(S), ⊆). It will be shown that the equivalence classes of the equivalence relation induced by P: L(S)→PL(S), V↦PV, each contain a unique minimal variety consisting of nilsemigroups.

Saturated and epimorphically closed varieties of semigroups

We establish a necessary condition (E) for a semigroup variety to be closed under the taking of epimorphisms and a necessary condition (S) for a variety to consist entirely of saturated semigroups. Condition (S) is shown to be sufficient for heterotypical varieties and a stronger condition (S′) is shown to be sufficient for homotypical varieties.

Universal varieties of semigroups

A category V is called universal (or binding) if every category of algebras is isomorphic to a full subcategory of V. The main result states that a semigroup variety V is universal if and only if it contains all commutative semigroups and fails the identity xnyn = (xy)n for every n ≥ 1. Further-more, the universality of a semigroup variety V is equivalent to the existence in V of a nontrivial semigroup whose endomorphism monoid is trivial, and also to the representability of every monoid as the monoid of all endomorphisms of some semigroup in V. Every universal semigroup variety contains a minimal one with this property while there is no smallest universal semigroup variety.

On product varieties of inverse semigroups

This paper extends results on product varieties of groups to inverse semigroups. We show that if is a variety of groups and any inverse semigroup variety, then ∘ is a variety. We give a characterization of the identities of∘in terms of the identities of and ofWe show that if does not contain the variety of all groups then it has uncountably many supervarieties. Finally we show that ifis another variety of groups then Subject classifiaction (Amer. Math. Soc. (MOS) 1970): 20 M 05.