Multiplicative lattices: Maximal implies prime and related questions

Abstract

The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counterpart of multiplicative closed sets, that of [Formula: see text]-systems. The notion of [Formula: see text]-system is also studied from the topological point of view.