Some Studies On Several Classes Of Weakly Adequate Semigroups

With the development of semigroup theory, special attention has been concentratedon the generalized regular semigroups. In particular, the study of generalized regularsemigroups in which the set of idempotents satisfying some properties has become animportant topic in the study of semigroup theory. And some of its special subclasseshave been extensively studied by many authors. In this paper, we concentrate on theclass of r-wide semigroups whose idempotents commute.Firstly, by using the generalized Green’s relation, that is,(*,~)-Green’s relation,the definitions of weakly adequate semigroup and weakly type A semigroup areintroduced and their elementary properties are discussed. Where，weakly adequatesemigroups and weakly type A semigroups are generalizations of adequate semigroupsand type A semigroups in the class of r-wide semigroups.Secondly, the largest congruence μ contained in H~*,~on a weakly adequatesemigroup is defined and the necessary and sufficient conditions which leads to theS/μ be isomorphic to the semilattice of idempotents E (S)of S is found, that is,E (S)is central in S. The result extends a theorem of adequate semigroups establishedby Fountain in1979. Based on the largest congruence μ, a concept of fundamentalweakly adequate semigroup is introduced. Furthermore, by applying some properties offundamental weakly adequate semigroup, we show that S/μ is fundamental if S/μis a weakly adequate semigroup.Finally, the concept of proper weakly type A monoid is introduced. In this paper,we consider a class of McAlister monoids, that is, M|--monoid. We will investigate thestructure of proper weakly type A monoid and show that every proper weakly type Amonoids is isomorphic to a M|--monoid. Our result not only generalizes the knownMcAlister theorem of proper inverse semigroups, but also extends the structure theorem about proper type A monoids.