Some Studies On Eventually Regular Semigroups With A System Of Subsemigroups

This doctoral thesis is mainly devoted to do some researches on eventually regular semigroups with a system of their subsemigroups. Some meaningful results are obtained.The thesis consists of five chapters.Chapter 1 provides a brief introduction. The background and develop-ment of the theory concerning semigroups with their subsemigroup systems are comprehensively presented. Moreover, the main results of the thesis are systematically introduced.In Chapter 2, we introduce some fundamental definitions, notations and terminologies that will be used in the sequel. In addition, several necessary preliminary materials on the lattice theory are covered as well.Chapter 3 is devoted to study the relationships between epigroups and their subepigroup lattices. The main aim of the chapter is to describe the structure of epigroups for which the subepigroup lattices satisfy some lattice-theoretic conditions. Among various of lattice-theoretic conditions, we investigate dis-tributive lattice, meet semidistributive lattice, 0-semidistributive lattice,0-modular lattice and chain, respectively.In Chapter 4, we explore the connections between eventually regular semi-groups and their full subsemigroup lattices. The sufficient and necessary con-ditions for eventually regular semigroups whose full subsemigroup lattices are 0-semidistributive lattices, distributive lattices and chains, respectively, are presented. Moreover, the particularly transparent structural descriptions are given in the case of strongly eventually inverse semigroups and of orthoepi-groups.The aim of Chapter 5 is to discuss various inter-relationships between even-tually regular semigroups and their particular subsemigroups (subsets). We mainly study the mutual reactions between the properties owned by eventually regular semigroups and those owned by their particular subsemigroups (sub-sets), including the set of idempotents, ideals and subsemigroups generated by the set of idempotents. Furthermore, semilattice decompositions of semi-groups are considered and some new results on characterizations of semilattices of nil-extensions of left groups are given.