Several Kind Of Abundant Semigroups And Its Structure,

The relations L* and R* are generalizations of Green's relations ?and R : elements a and b of a semigroup S are related by L*(R*) in S if and only if they are related by L(R) in some oversemigroup of S. A semigroup is left(right) abundant if each R*(L*)- class contains an idempotent. A semigroup is abundant if it is both left and right abundant. An abundant semigroup is quasi-adequate if its set of idempoents is a band. When the idempotents commute in an abundant semigroup, then it is called an adequate semigroup. The objective of this thesis is to investigate abundant semigroups and their constructions. The thesis is divided into four sections.In Section 1, we introduce some fundamental results on abundan semigroups and give some preliminaries.In Section 2, we investigate abundant semigroup whose idempotents form a left regular band. We define its adequate transversal. Then by making use of an adequate transversal S0 and a semilattice I of left zero semigroups, we define a semidirect product I x S0 and show that I x S0 is a left abundant semigroup whose idempotents form a left regular band and has an adequate transversal which isomorphic to S0.In Section 3, the structure of proper cover for left type-A is established. We introduce the concept of proper covers for left type-A semigroups, and prove a construction theorem.In Section 4, we introduce the concept of F-abundant semigroups. After obtaining some properties of F-abundant semigroups, the good homomorphic images of F-abundant semigroups is considered.