Congruences On Some Irregular Semigroups

In this dissertation, we describe the semilattice of groups congruences and the least group congruences on some semigroups that are not regular. There are three chapters.In the first chapter, we deal with the GV-inverse semigroup S. Firstly, we give the definition of r-semiprime congruence pair (N, T) in this chapter. It is a pair composed of a normal subsemigroup of S and a semilattice congruence r on (E(S)). N and T satisfy the following conditions:(I.)r(a6)(r(b))-1(r(a))-1 N;(2.)ae N and (r(a))re a N;(3.)atb N and (r(ab)) T(r(t)) ab N. And conversely, ab N atb N, where a, b S, t NIn terms of its r-semiprime congruence pair (N,r), we describe the r-semiprime congruence on it this wayMoreover, we show that (kerp,p|{E(S))) is a r-semiprime congruence pair of S and that for all r-semiprime congruence p on it. So we have this result that there is a bijection between the set of all r-semiprime congruence pairs of S and the set of all r-semiprime congruences on S, given by the correspondence (N, T) P(N,r). During the proof of the above result, we obtained that (E(S)) is a GV-inverse subsemigroup.In the second chapter,we give the description of the least group congruence on a π-regular semigroup S.In the third chapter, we describe the group congruences on a semigroup S and construct the semilattice of groups congruence on it. In the description of group congruence, we give the generalization of the results of D.R.Latorre[1]. We give the equivalent definitions of group congruence on S as the following:(1) apb;(2) There exists b∈6 D(b) and x ∈ T (or for all b∈ D(b), there exists x ∈ T)such that axb ∈ T;(3) There exists a ∈ D(a) and x ∈ T(or for all a ∈ D(a), there exists x ∈ T) such that axb ∈ T;(4) There exists a ∈ D(a) and x∈ E T (or for all a ∈ D(a), there exists x ∈ T )such that bxa∈ T;(5) There exists b ∈ D(b) and x∈ T (or for all b ∈D(b), there exists x ∈ T)such that bxa ∈ T;(6) There exists x, y ∈ T such that ax = yb;(7) There exists x, y ∈ T such that xa = by;where p = pT and T is a seminormal subsemigroup. We also describe the group congruence class by the subset TU, and we show that TVp = ropoT, where p is a congruence and r is a group congruence of S. We describe some properties of a unitary and dense E-semigroup. Finally, we show that is a semilattice of groups congru-ence if and only if (Na)U is a seminormal subsemigroup on 5, where pNa is a groupcongruence on the semilattice congruence class Sa of 5.