The Sandwich Set And Its Application On Γ-semigroup

In this thesis, we generalize the concept of sandwich set on semigroupto Γ-semigroup S. The aim of the thesis is to study the properties and itsapplication of sandwich set on Γ-semigroup S. The thesis has three chapters.In chapter 1, the basic definitions and some properties of Γ-semigroup S aregiven. For example, the Lemma of generalized associativity, the properties oflinked semigroup and some properties parallel to those obtained in semigrouptheory.As the particularity of Γ-semigroup S, one side Γ-semigroup havesome properties analogous with them on semigroup . For example, the Green'sequivalence and the Lemma of Green's theorem (they are di-erent with themon semigroup in form too ). on the other hand , there are some properties can'tbe obtained parallel to those in semigroup theory.For example,the H classes ofΓ-semigroup can contain not only one idempotent.As we know ,the sandwichset is introduced to study regular bi-order set by Indian mathematician K.S.S.,Nambooripad . It has important value to study regular semigroup. It isused widely. Scholars had already generalize the concept of sandwich set onidempotent of Γ-semigroup. In chapter 2, the concept of sandwich set on Γ-semigroup are generalized to arbitrary elements of Γ-semigroup .We give theproperties and the structure of sandwich set on Γ-semigroup .The relationbetween sandwich set and the Green's equivalent classes of Γ-semigroup arediscussed. We use the sandwich set to study the relations of the inverse ofa product b with the inverse of a product the inverse of b too.For arbitraryη∈Γ,Sη(a,b) is a rectangular band ,but we will use example to prove S(a,b)not a rectangular band . In chapter 3, as the use of sandwich, we obtain anecessary and su-cient condition for the Green's equivalence H of regular Γ-semigroup S to be a congruence. In section 2 of chapter 3, we generalize theconcept of natural partial order on regular semigroup to regular Γ-semigroupS. Finally, we will prove the su-cient and necessary condition of natural par-tial order compatible with the multiplication is Γ-semigroup S is locally inverse.