Abstract. In this paper, we define $z$-ideals in bounded lattices. A separation theorem for the existence of prime $z$-ideals is proved in distributive lattices. As a consequence, we prove that every $z$-ideal is the intersection of some prime $z$-ideals. Lastly, we prove a characterization of dually semi-complemented lattices.

DOI: 10.14232/actasm-016-012-2

AMS Subject Classification (1991): 06B10, 06D75

Keyword(s): $z$-ideals, Baer ideal, $0$-ideal, closed ideal, minimal prime ideal, maximal ideal, dense ideal, dually semi-complemented lattice

Received February 22, 2016 and in final form September 3, 2018. (Registered under 12/2016.)