On hypersemigroups.

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 ?= Ø." Although according to Mittas "it is very easy to prove that xy ?= Ø", 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ordered hypersemigroups are copies from corresponding results on ordered semigroups in which the multiplication "·" has been replaced by "?". [ABSTRACT FROM AUTHOR]