Reticulation of a quantale, pure elements and new transfer properties

TL;DR

This paper explores the relationship between the reticulation of a coherent quantale and its pure elements, establishing transfer properties and new characterizations for various classes of quantales.

Contribution

It introduces new transfer properties between the reticulation's ideals and pure elements, leading to novel characterizations of several quantale classes.

Findings

Pure elements are characterized via reticulation properties.

Transfer of properties from sigma-ideals to pure elements is established.

New theorems provide characterizations for normal, mp, PF, purified, and PP quantales.

Abstract

We know from a previous paper that the reticulation of a coherent quantale $A$ is a bounded distributive lattice $L(A)$ whose prime spectrum is homeomorphic to $m$ - prime spectrum of $A$. In this paper we shall prove several results on the pure elements of the quantale $A$ by means of the reticulation $L(A)$. We shall investigate how the properties of $\sigma$ - ideals of $L(A)$ can be transferred to pure elements of $A$. Then the pure elements of $A$ are used to obtain new properties and characterization theorems for some important classes of quantales: normal quantales, $mp$ - quantales, $PF$ - quantales, purified quantales and $PP$ - quantales.