Some Studies For U-semiabundant Semigroups

In the recent years,the research of some generalized regular semigroups which has attracted a lot of attention of many researchers forms an important research domain of semigroup theory.As an extensive generalization of abundant semigroups,the study of U-semiabundant semigroups and their subclasses has also become a hot topic in the study of semigroup theory.We say that a semigroup S is U-semiabundant,if both every(?)~U-class and every (?)~U-class of S contain idempotents of U.A U-semiabtmdant semigroup S is called Uabundant, if S satisfies the congruence condition,that is,(?)~U is a right congruence and (?)~U is a left congruence on S.In this paper,We mainly study two kinds of U-abundant semigroups:U~σ-abundant semigroups and U-cyber semigroups.The main aim of this thesis is to study the structure of U~σ-abundant semigroups. So-called U~σ-abundant semigroups are U-abundant semigroups whose subset U satisfies permutation identities.We first give a construction of U~σ-abundant semigroups by using the quasi-spined product of semigroups.Secondly,a structure theorem of U~σ-abundant semigroups is established through introducing the concept of comprehensive congruences on U-abundant semigroups.It is proved that a semigroup S is U~σ-abundant if and only if S is isomorphic to the quasi-spined product QS(L,T(Y),R;Y)of a left normal band L, an Ehresmann semigroup T(Y)and a right normal band R.Finally,we introduce the concept of U-cyber semigroups by giving the definition of idempotent-linked(IL)U-semiabundant semigroups.So-called U-cyber semigroups are idempotent-linked(IL)U-abundant semigroups whose subset U forms a subsemigroup. Furthermore,we study the admissible relationσdefined on a U-semiabundant semigroup and it is proved that the admissible relationσon a U-cyber semigroup is a comprehensive congruence.