Order closed ideals in pre-Riesz spaces and their relationship to bands

TL;DR

This paper explores the relationships between different types of ideals and bands in Archimedean pre-Riesz spaces, extending classical vector lattice concepts and introducing the notion of supremum closed ideals.

Contribution

It generalizes the concept of bands and ideals from vector lattices to pre-Riesz spaces, establishing conditions under which these notions coincide.

Findings

In pervasive pre-Riesz spaces, the three notions of bands coincide for directed ideals.

Every directed band in pervasive pre-Riesz spaces is supremum closed.

Counterexamples show the notions differ in general Archimedean pre-Riesz spaces.

Abstract

In Archimedean vector lattices bands can be introduced via three different coinciding notions. First, they are order closed ideals. Second, they are precisely those ideals which equal their double disjoint complements. The third concept is that of an ideal which contains the supremum of any of its bounded subsets, provided the supremum exists in the vector lattice. We investigate these three notions and their relationships in the more general setting of Archimedean pre-Riesz spaces. We introduce the notion of a supremum closed ideal, which is related to the third aforementioned notion in vector lattices. We show that for a directed ideal $I$ in a pervasive pre-Riesz space with the Riesz decomposition property these three concepts coincide, provided the double disjoint complement of $I$ is directed. In pervasive pre-Riesz spaces every directed band is supremum closed and every supremum closed directed ideal $I$ equals its double disjoint complement, provided the double disjoint complement of $I$ is directed. In general, in Archimedean pre-Riesz spaces the three notions differ. For this we provide appropriate counterexamples.