On extensions of completely simple semigroups by groups

An example of an extension of a completely simple semigroup \(U\) by a group \(H\) is given which cannot be embedded into the wreath product of \(U\) by \(H\) . On the other hand, every central extension of \(U\) by \(H\) is shown to be embeddable in the wreath product of \(U\) by \(H\) , and any extension of \(U\) by \(H\) is proved to be embeddable in a semidirect product of a completely simple semigroup \(V\) by \(H\) where the maximal subgroups of \(V\) are direct powers of those of \(U\) .     

On the Cayley graphs of completely simple semigroups

In this paper, the Cayley graphs of completely simple semigroups are investigated. The basic structure and properties of this kind of Cayley graph are given, and a condition is given for a Cayley graph of a completely simple semigroup to be a disjoint union of complete graphs. We also describe all pairs (S,A) such that S is a completely simple semigroup, A⊆S, and Cay (S,A) is a strongly connected bipartite Cayley graph.     

Archimedean ordered semigroups as ideal extensions

The main result of the paper is a structure theorem concerning the ideal extensions of archimedean ordered semigroups. We prove that an archimedean ordered semigroup which contains an idempotent is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Conversely, if an ordered semigroup S is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then S is archimedean. As a consequence, an ordered semigroup is archimedean and contains an idempotent if and only if it is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup.     

On disjunctions of algebraic sets in completely simple semigroups

A semigroup S is called an equational domain if any finite union of algebraic sets over S is algebraic. We give some necessary and su?cient conditions for a completely simple semigroup to be an equational domain.     

Elementary theories of completely simple semigroups

The connections between first-order formulas over a completely simple semigroupC and corresponding formulas over its structure groupH are found in this paper. For the case of finite sandwich-matrix the criterion of decidability of the elementary theoryT(C) is established in terms of the elementary theory ofH in the enriched signature (Theorem 1). For the general case the criterion is established in terms of two-sorted algebraic systems (Theorem 2). Sufficient conditions in terms ofH for decidability and for undecidability ofT(C) are outlined. Corollaries and examples are presented, among them an example of a completely simple semigroup with a finite structure group and with undecidable elementary theory (Theorem 3).     

Two-sided networks for completely simple semigroups

Let S be a completely simple semigroup represented as a Rees matrix semigroup M(I,G,P) with normalized sandwich matrix P. On the congruence lattice C(S) of S we consider the relations T _i, K and T _r which identify congruences with the same left trace, kernel and right trace, respectively. These are equivalences whose classes are intervals. The upper and lower ends of these intervals induce the following operators on C(S) T_l, K, T_r, t_l, k and t_r .We construct here the semigroup generated by these operators as a homomorphic image of a semigroup given by generators and relations and demonstrate the minimality of the latter.     

Two congruence lattices of completely simple semigroups

For a Rees matrix semigroupS with normalized sandwich matrix and C(S), the congruence lattice ofS, we consider the lattice generated by {itpT_l, pK, pT_r, pt_l, pk, pt_r}. HerepT ₁ andpt _l are the upper and lower ends of the interval which makes up the _i -class of , _i being the left trace relation onC(S). The remaining symbols have the analogous meaning relative to the kernel and the right trace relations. We also consider the lattice generated by {T _l, K, T_r, t_l, _k, t_r} where and are the equality and the universal relations onS, respectively. In both cases, we find lattices freest relative to these lattices and represent them as distributive lattices with generators and relations.With 3 Figures     

On finite completely simple semigroups having no finite basis of identities

The purpose of this note is to provide a correct proof of the fact that there exist finite completely simple semigroups having no finite basis of identities.     

Varieties generated by free completely simple semigroups

Communicated by N. R. Reilly