A basis of identities Σ for a semigroup S is a set of identities satisfied by S from which all other identities of S can be derived. The basis Σ is said to be irredundant (or irreducible) if no proper subset of Σ is a basis for the identities of S. If the basis Σ is finite, then it is always possible to extract an irredundant basis, however if Σ is infinite then it is conceivable that no irredu...

Epigroups are semigroups equipped with an additional unary operation called pseudoinversion. Each finite semigroup can be considered as an epigroup. We prove the following theorem announced by Zhil’tsov in 2000: the equational theory of the class of all epigroups coincides with the equational theory of the class of all finite epigroups and is decidable. We show that the theory is not finitely b...

The Hemavend, a Kurdish tribe, were exiled from their original settlements of Kirkuk, Aleppo and Mosul to the Eyalet Archipelago, especially Chios, Kos Rhodes Island. This population, whose total number was close 200, did not see this province, which place exile, as stay due insecurity disorder, weather. people who made situation worse by escaping lived separately families, lives worse. financi...

This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell's aperiodic pointlike theorem, using a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided s...

We consider the Krohn-Rhodes complexity of certain semigroups of upper triangular matrices over finite fields. We show that for any n > 1 and finite field k, the semigroups of all n × n upper triangular matrices over k and of all n × n unitriangular matrices over k have complexity n− 1. A consequence is that the complexity c > 1 of a finite semigroup places a lower bound of c+1 on the dimension...

In this paper we discuss various aspects of the problem of determining the minimal dimension of an injective linear representation of a finite semigroup over a field. We outline some general techniques and results, and apply them to numerous ex-