Narrow Results

A U-abundant semigroup whose subset U satisfies a permutation identity is said to be U^σ-abundant. In this paper, we consider the minimum Ehresmann congruence δ on a U^σ-abundant semigroup and explore the relationship between the category of U^σ-abundant semigroups (S,U) and the category of Ehresmann semigroups (S,U)/δ. We also establish a structure theorem of U^σ-abundant semigroups by using the concept of quasi-spined product of semigroups. This generalizes a result of Yamada for regular semigroups in 1967 and a result of Guo for abundant semigroups in 1997.

A class of weakly U-abundant semigroups satisfying the congruence condition (C) is investigated, which contains both the class of regular semigroups and the class of abundant semigroups as its subclasses. In particular, the class of weakly U-abundant semigroups satisfying the congruence condition (C) with Ehresmann transversals is studied. After giving some properties of U-abundant semigroups with Ehresmann transversals, we establish some characterization theorems for such semigroups. Our results may be regarded as an extension of the work of Blyth, McAlister and McFadden in regular semigroups and that of El-Qallali, Guo and Chen in abundant semigroups.

In this survey article, we will briefly introduce the recent development of rpp semigroups, its generalizations and some of its special subclasses. Some methods of constructions for such semigroups are introduced. Several structure theorems of such semigroups are described.

As generalizations of groups, inverse semigroups were first studied by V. V. Vagner (1952) and independently by G. B. Preston (1954). The theory of inverse semigroups playa major rule in development of the algebraic theory of semigroups. The main purpose of this paper is to give a brief survey of inverse semigroups and their generalizations in the class of regular semigroups, in the class of abundant semi groups and within the class of U-abundant semigroups. In particular, we exhibit some basic properties and structures of inverse semigroups and their generalizations, for example, left inverse semigroups, quasiinverse semigroups, orthodox semigroups, adequate semigroups, type A semigroups, -inverse semigroups, -inverse semigroups, type W semigroups, Ehresmann semigroups and U-orthodox semigroups and others. We establish a corresponding hierarchy of some important semigroups which generalize the inverse semigroups for regular semigroups, abundant semigroups and weakly abundant semigroups, respectively.