On maximal cycles and prime ideals in trellises

Abstract

In this paper we obatin some characterizations of a partially ordered set 〈S*; ⊴*〉, where S* is the set of all maximal cycles of a pseudo-ordered set 〈S; ⊴〉. It is proved that the poset 〈S*; ⊴*〉 forms a lattice if and only if the pseudo-ordered set 〈S; ⊴〉 is pseudo-perfect. Also we proved that 〈S*; ⊴*〉 satisfies the minimum condition if and only if the psoset 〈S; ⊴〉 satisfies the minimum condition. Introduced the concept of prime ideals in trellises and generalized Prime ideal theorem for lattices to trellises.