Generalized Semigroups And R - Semigroups

In this paper, we mainly study several congruences on generalized eventually regular semigroups and properties of R-semigroups. It can be divided into four chapters.In chapter 1, we introduce concepts of a few class of semigroups and indicate the relationship between them. Moreover, the outline of this paper is given.In chapter 2, we research properties of generalized eventually regular semigroups and their congruences. It contains three sections. In section 1, we prove that every regular congruence on regular-surjective semigroups is uniquely determined by its P-partial kernel normal system. Hence, we generalize the result of theorem 5 in [14]. In section 2, we discuss structurally eventually regular semigroups. We introduce the concepts of eventually regular-surjective semigroups and weak ER-semigroups, and prove that structurally eventually regular semigroups are idempotent-surjective eventually regular-surjective weak ER-semigroups. Thus, we generalize the relative results in [18]. In section 3,we prove that every strongly regular congruence on E-inversive semigroups are uniquely determined by its kernel normal system in different method from [20]. Moreover, we introduce the concept of ES-surjective semigroups and prove that every left inverse congruence on ES-surjective regular-surjective semigroups can be uniquely determined by its kernel and hyper-trace. Therefore, we generalize the corresponding results in [19].In chapter 3, we characterize the least Clifford congruence on idempotent-surjective weak ER-semigroups. In addition, a characterization of all clifford congruences on such semigroups is given. Therefore, we generalize the relative results in [5].In chapter 4, we study R-semigroups. It contains three sections. In section 1, several properties of R-semigroups are given. In section 2, we discuss sandwich sets S(e, f),one sided sandwich sets eS(e, f) and S(e, f)f of R-semigroups. Thus, we generalize the corresponding results for regular semigroups. In section 3, some properties of EIsemigroup are researched.