Some Studies On A Class Of U-rpp Semigroups

In recent years, the research of some generalized regular semigroups has attracted a lot of attention of many researchers. As a generalization of abundant semigroups, the study of U-semiabundant semigroups and their subclasses has become an important topic in the study of semigroup theory.Let E(S) be the set of all idempotents of a semigroup S and let U C E(S) be a non-empty subset. A semigroup S is said to be U-semiabundant, if every LU-class and every RU-class of S contains some projections of S, denoted by (S,U). A U-semiabundant semigroup (S, U) is called U-abundant if (S, U) satisfies the congruence condition, that is, LU is a right congruence on (S, U)and RU is a left congruence on (S, U). In this paper, we mainly study two kinds of U-abundant semigroups:left U-rpp semigroups and PI-U-rpp semigroups.Firstly, we discuss a U-rpp semigroup which has left projection, that is left U-rpp semi-group. U-rpp semigroup is called left U-rpp semigroup if U forms a band and xey= exy for all x, y∈S1, y≠1. In this capter,we give the notion and properties of left U-rpp semigroups, constructure the algebra structures for this kind of semigroups and prove that a semigroup (S, U) is left U-rpp if and only if (S, U) is a semilattice of semigroups Sa and each Sa is a direct product of a U-left cancellative monoid and a right zero band. Also, such a semigroup can be described as a strong semilattice of semigroups Sa and each Sa is a direct product of a U-left cancellative monoid and a right zero band.Finally, We study the PI-U-rpp semigroup, that is, a U-rpp semigroup which satisfies permutation identities. A structure theorem of this kind of semigroups is established by introducing the L-admissible congruence on semigroups. It is proved that a semigroup (S, U) is PI-U-rpp if and only if (S, U) is isomorphic to the weakly spined product of a left normal U-rpp semigroup and a right normal band.