(untitled) (European Journal of Pure and Applied Mathematics, 2018)
Some well known results on ordered semigroups are examined in case of ordered hypersemigroups. Following the paper in Semigroup Forum 44 (1992), 341–346, we prove the following: The ideals of an ordered hypergroupoidH are idempotent if and only if for any two ideals A and B of H, we have AB=(A*B]. Let now H be an ordered hypersemigroup. Then, the ideals of H are idempotent if and only if H is semisimple. The ideals of H are weakly prime if and only if they are idempotent and they form a chain. The ideals of H are prime if and only if they form a chain and H is intra-regular. The paper serves as an example to show how we pass from ordered semigroups to ordered hypersemigroups.