@article{On ordered hypersemigroups with idempotent ideals, prime and weakly prime ideals_2018, place={Maryland, USA}, volume={11}, url={https://ejpam.com/index.php/ejpam/article/view/3085}, DOI={10.29020/nybg.ejpam.v11i1.3085}, abstractNote={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 hypergroupoid$H$ are idempotent if and only if for any two ideals $A$ and $B$ of $H$, we have $A\cap B=(A*B]$. 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.}, number={1}, journal={European Journal of Pure and Applied Mathematics}, year={2018}, month={Jan.}, pages={10–22} }