A Study On Γ - Super Semigroups

In the algebraic theory of semigroups, ideals play a important role in studying the elemen-tary properties of semigroups. As a generalization of semigroups, we introduce the concepts of (m, n) quasi-hyperideals, (m, n) hyperideals and so on, using them to study the properties of Γ-hypersemigroups. Meanwhile, we present the notions of regular Γ-hyper-odered-semigroups and quasi-hyperideals of Γ-hyper-odered-semigroups, applying them to give the characteriza-tions of regular Γ-hyper-odered-semigroups.In chapter 3, we introduce the concepts of (m, n) quasi-hyperideals, m-left hyperideal-s and n-right hyperideals of Γ-semihypergroups. We obtain the representation theorems of (m, n) quasi-hyperideals by any nonempty subset of a Γ-semihypergroup, and we also prove that any (m, n) quasi-hyperideal of Γ-semihypergroups can be expressed as the intersection of a m-left hyperideal and a n-right hyperideal. Moreover, any (m, n) quasi-hyperideal of Γ-semihypergroups is minimal if and only if it can be expressed as the intersection of a minimal m-left hyperideal and a minimal n-right hyperideal of Γ-semihypergroups. At last, we get the characterizations of (m, n) quasi simple Γ-semihypergroups.In chapter 4, firstly, we introduce the concept of (m, n) hyperideals of Γ-hypersemigroups, the characterizations of (m, n) hyperideals of Γ-hypersemigroups and the generalized theorem of (m, n) hyperideals obtained by every nonempty set of Γ-hypersemigroups. Secondly, the characterizations of (m, n) simple Γ-hypersemigroups, and (m, n) regular Γ-hypersemigroups are given by (m, n) hyperideals. Finally, as an application of results of this chapter, some characterizations of regular semigroups are obtained in terms of their (m, n) ideals.In chapter 5, we introduce the concept of quasi-hyperideals of ordered Γ-hypersemigroups, which is a generalization of quasi-ideals of ordered semigroups, and describe some related prop-erties of them. Furthermore, we characterize regular ordered Γ-hypersemigroups in term of their ordered quasi-hyperideals, and obtain the fact that for any regular ordered Γ-hypersemigroups S, the sets Qs of all quasi-hyperideals of S is a regular ordered Γ-semigroup, with the multi-plications "oΓ"; for ∨ X, Y ∈ Qs there has X oΓ Y= (XYY]. In addition, we consider some special classes of regular ordered Γ-semigroup, when the regular ordered Γ-semigroups (Qs, o) are bands, left regular bands and semilattices.