# zbMATH — the first resource for mathematics

$$\Gamma$$-semihypergroups and their properties. (English) Zbl 1197.20062
Summary: Algebraic hyperstructures are a suitable generalization of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. The concept of $$\Gamma$$-semihypergroups is a generalization of semigroups, a generalization of semihypergroups and a generalization of $$\Gamma$$-semigroups. In this paper, we define the notion of ideal, prime ideal, extension of an ideal in $$\Gamma$$-semihypergroups, then we prove some results in this respect and present many examples of $$\Gamma$$-semihypergroups. We also introduce the notion of quotient $$\Gamma$$-semihypergroup by using a congruence relation, and introduce the notion of right Noetherian $$\Gamma$$-semihypergroups. Finally, we study some properties of fundamental relations on a special kind of $$\Gamma$$-semihypergroups.

##### MSC:
 20N20 Hypergroups 20M99 Semigroups 20M12 Ideal theory for semigroups