Some Properties of Semi-Prime Ideals in Lattices

Abstract

Recently Yehuda Rav has given the concept of Semi-prime
ideals in a general lattice by generalizing the notion of 0-distributive
lattices. In this paper we have included several characterizations of Semiprime
ideals. We give a simpler proof of a prime Separation theorem in a
general lattice by using semi-prime ideals. We also study different
properties of minimal prime ideals containing a semi prime ideal in
proving some interesting results. By defining a p-algebra L relative to a
principal semi prime ideal J , we prove that when L is 1-distributive,
then L is a relative S-algebra if and only if every prime ideal containing
J contains a unique minimal prime ideal containing J , which is also
equivalent to the condition that for any x, y Î L , x Ù y Î J implies
Ú + x = 1 + y . Finally, we prove that every relative S-algebra is a
relative D- algebra if L is 1-distributive and modular with respect to J .