In an attempt to show the way we pass from ordered semigroups to ordered $\Gamma$-hypersemigroups, we examine the results of Semigroup Forum (1992; 46: 341-346) for an ordered $\Gamma$-hypersemigroup. It has been shown that the concept of semisimple ordered $\Gamma$-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 $A\cap B=(A\Gamma B]$. The main results of the paper are the following: The ideals of an ordered $\Gamma$-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 $\Gamma$-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).
Ordered $\Gamma$-hypersemigroup, prime, weakly prime, semisimple, intraregular
"From ordered semigroups to ordered $\Gamma$-hypersemigroups,"
Turkish Journal of Mathematics: Vol. 46:
8, Article 15.
Available at: https://journals.tubitak.gov.tr/math/vol46/iss8/15