On hypersemigroups

Abstract

This is from the paper “Hypergroupes canoniques valués et hypervalués” by J. Mittas in Mathematica Balkanica 1971: “The concept of hypergroup introduced by Fr. MARTY in 1934 [Actes du Congrès des Math. Scand. Stocholm 1935, p. 45] is as follows: “A hypergroup is a nonempty set H endowed with a multiplication xy such that, for every x, y, z ∈ H, the following hold: (1) xy ⊆ H ; (2) x(yz) = (xy)z and (3) xH = Hx = H . The first condition expresses that the multiplication is an hyperoperation on H , in other words, the composition of two elements x, y of H is a subset of H . It is very easy to prove that for any x, y ∈ H , we have xy ̸= ∅.” Although according to Mittas “it is very easy to prove that xy ̸= ∅”, this is not possible. The notation x(yz) has a meaning of course if we identify the x by {x} and define an operation between sets. The authors working on hypersemigroups added in the definition by Mittas, the following: x(yz) = (xy)z means that ∪ u∈yz xu = ∪ v∈xy vz . But we never use this last equality in the papers on hypersemigroups in which we always use the x(yz) = (xy)z . As a result, most of the results of ordered hypersemigroups are copies from corresponding results on ordered semigroups in which the multiplication “·” has been replaced by “◦”.