From ordered semigroups to ordered Gamma-hypersemigroups

In an attempt to show the way we pass from ordered semigroups to ordered P-hypersemigroups, we examine the results of Semigroup Forum (1992; 46: 341-346) for an ordered P-hypersemigroup. It has been shown that the concept of semisimple ordered P-hypersemigroup S is identical with the concept "the ideals of S are idempotent" and the ideals of S are idempotent if and only if for all ideals A, B of S , we have A boolean AND B = (APB]. The main results of the paper are the following: The ideals of an ordered P-hypersemigroup S are weakly prime if and only if they form a chain and S is semisimple. The ideals of S are prime if and only if they form a chain and S is intraregular. It should be finally mentioned that the concepts "prime ideal" and "both semiprime and weakly prime ideal" are the same; and that in commutative ordered P-hypersemigroups the prime and weakly prime ideals coincide. For an abstract formulation of the above statements we refer to Turk J Math (2016; 40: 310-316).