Divisorial Elements in Lattice-Ordered Monoids

Abstract

Let S be a commutative lattice-ordered monoid that is conditionally complete and admits residuals. Imitating the definition of divisorial ideals in commutative ring theory, we study divisorial elements in S. The archimedean divisorial elements behave especially nicely. We establish a Galois correspondence of the divisorial elements in a finite interval. Assuming the maximum condition on integral divisorial elements, it is shown that their Krull associated primes are divisorial and the integral divisorial elements admit irredundant representations as intersections of finitely many p-components that are p-primal divisorial elements.