The concept of  -semihypergroups is a generalization of semigroups, a generalization of semihypergroups and a generalization of  -semigroups. In this paper, we study the concept of semiprime ideals in a  -semihypergroup and prove some results. Also, we introduce the notion of  -hypergroups and closed  -subhypergroups. Finally, we study the concept of  -semihypergroups associated to binary relations and give necessary and sufficient conditions on a set of binary relations  on a non-empty set S such that S becomes a  -semihypergroup or a  -hypergroup.

-hypergroups and -semihypergroups associated to binary relations

In this paper we show that the left operator and right operator semi groups of a Γ-semigroup with unities are Morita equivalent monoids. It is also deduced that if L and R are two Morita equivalent monoids then a Γ-semigroup with unities can be constructed such that its left and right operator monoids are isomorphic to L and R respectively. This result is used to obtain some properties of monoids which remain invariant under Morita equivalence.

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Gamma-semigroups with Unities and Morita Equivalence for Monoids

The concept of Γ-semigroups is a generalization of semigroups. In this paper, we associate two transformation semigroups to a Γ-semigroup and we call them the left and right transformation semigroups. We prove some relationships between the ideals of a Γ-semigroup and the ideals of its left and right transformation semigroups. Finally, we study some relationships between Green’s equivalence relations of a Γ-semigroup and its left (right) transformation semigroup.

http://www.discuss.wmie.uz.zgora.pl/al/33_2/a_gaa33heam.pdf

Transformation semigroups associated to Γ-semigroups

The concepts of ⁄-relation of a (i-)semigroup and "-adequate transversal of a (i-)abundant semigroup are deflned in this note. Then we develop a matrix type theory for abundant semigroups. We give some equivalent conditions of a Rees matrix semigroup being abundant and some equivalent conditions of an abundant Rees matrix semigroup having an adequate transversal. Then we obtain some Rees matrix representa- tions for abundant semigroups with adequate transversals by the above theories.

On rees matrix representations of abundant semigroups with adequate transversals

The concept of Γ-semigroups is a generalization of semigroups In this paper, we consider Γ-groups and prove that every Γ-group is derived from a group then, we give the number of Γ-groups of small order

Enumeration of Γ-groups of finite order

In this paper we introduce the notion of prime one-sided ideal in S where is a and show that there is a one-to-one correspondence between the set of all prime right ideals of S and the set of all prime right ideals of M where M is the left operator semigroup of the . We also introduce the notions of right Noetherian on S where is a and Noetherian and study it. Lastly we show that a commutative with unities is Noetherian if and only if every prime ideal of S is finitely generated, thus extending a result of Cohen's Theorem [1] for semi group to .

T.K. Dutta

Papers

On Noetherian ${\Gamma}-semigroup$