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Γ-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 Γ-semihypergroups is a generalization of semigroups, a generalization of semihypergroups and a generalization of Γ-semigroups. In this paper, we define the notion of ideal, prime ideal, extension of an ideal in Γ-semihypergroups, then we prove some results in this respect and present many examples of Γ-semihypergroups. We also introduce the notion of quotient Γ-semihypergroup by using a congruence relation, and introduce the notion of right Noetherian Γ-semihypergroups. Finally, we study some properties of fundamental relations on a special kind of Γ-semihypergroups.
MSC:
20N20Hypergroups (group theory)
20M99Semigroups
20M12Ideal theory of semigroups