On hypersemigroups

This is from the paper "Hypergroupes canoniques values et hypervalues" by J. Mittas in Mathematica Balkanica 1971: "The concept of hypergroup introduced by Fr. MARTY in 1934 [Actes du Congres 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 is an element of H, the following hold: (1) xy subset of 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 is an element of H, we have xy not equal empty set" Although according to Mittas "it is very easy to prove that xy not equal empty set", 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 boolean OR(u is an element of yz) xu = boolean OR(u is an element of 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 " has been replaced by "o".