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${\Gamma }$-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 (group theory) 20M99 Semigroups 20M12 Ideal theory of semigroups