Some Studies On Locally U-ample Semigroups

The research of regular semigroups is a main subject in the algebraic theory of semi-groups. As the development of semigroup theory, many authors generalized regular semi-groups to generalized regular semigroups. In recent years, the study of generalized regular semigroups and their subclasses has become an important topic in the study of semigroup theory.A semigroup S is called a U-semiabundant semigroup, if both every LU-class and every RU-class of S contain idempotents of U, called the set of projections. A U-semiabundant semigroup is called a U-abundant semigroup if LU is a congruence and RU is a right congru-ence on S. A U-abundant semigroup in which every HU-class contains an element of U is called U-superabundant semigroup.U-superabundant semigroups is a natural generalization of completely regular semigroups and superabundants for the class of U- abundant semi-groups. In this paper, we mainly study a class of U-superabundant semigroups, studying the characterizations of these semigroups and their algebraic structures.In the first chapter of the thesis, we firstly introduce basic definitions and basic results for semigroups. Recall the concept of regular semigroups and abundant semigroups. In particular, we introduce basic concepts and basic lemmas of U-abundant semigroups.In the second chapter of the thesis, we discuss some of U- ortho-abundant semigroups, U-ortho-abundant semigroups which are U-superabundant semigroups in which sets of projec-tions form subsemigroups are discussed. Furthermore, by using the definition of U-rectangular monoid which are the product of a monoid, a left zero band and a right zero band, we es-tablish a structure of U-ortho-abundant semigroups. It is proved that if semigroup S is a U-ortho-abundant semigroup, then S is a semilattice of U- rectangular monoids. In the third chapter of the thesis, we study some of U-ample semigroups. Particuly, we define locally U-ample semigroups. Furthermore, we prove that a U-superabundant semi-group S is locally a U-ample semigroup if and only if S satisfies the following conditions:(â…°) S is DU-majorization; (â…±) S satisfies PC condition; (â…²) U is an order ideal of the set E(S) of idempotents.