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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.

20N20 Hypergroups
20M99 Semigroups
20M12 Ideal theory for semigroups