TY - JOUR
TI - On ordered hypersemigroups with idempotent ideals, prime and weakly prime ideals
PY - 2018/01/30
Y2 - 2024/07/21
JF - European Journal of Pure and Applied Mathematics
JA - Eur. J. Pure Appl. Math.
VL - 11
IS - 1
LA - en
DO - 10.29020/nybg.ejpam.v11i1.3085
UR - https://doi.org/10.29020/nybg.ejpam.v11i1.3085
SP - 10-22
AB - 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.
ER -