Some Properties of Prime and Z-Semi-Ideals in Posets

Document Type : Original Article


Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore - 641 114, India.


We define the notion of z-semi-ideals in a poset $P$ and we show that if a z-semi-ideal $J$ satisfies $(\ast )$-property, then every minimal prime semi-ideal containing $J$ is also a z-semi-ideal of $P.$ We also show that every prime semi-ideal is a  z-semi-ideal or the maximal z-semi-ideals contained in it are prime z-semi-ideals. Further, we characterize some properties of union of prime semi-ideals of $P$ provided the prime semi-ideals are contained in the unique maximal semi-ideal of $P.$