On simple and semisimple quantales

TL;DR

This paper provides new characterizations of simple and semisimple quantales, avoiding the notion of discrete quantale, and explores their connections with quantale modules and earlier research.

Contribution

It introduces novel characterizations of simple and semisimple quantales based on separating and cyclic sets, expanding the theoretical understanding.

Findings

New characterization based on separating and cyclic sets

Links established between simple quantale modules and quantales

Discussion of applications to semisimple quantales

Abstract

In a recent paper, J. W. Pelletier and J. Rosicky published a characterization of *-simple *-quantales. Their results were adapted for the case of simple quantales by J. Paseka. In this paper we present similar characterizations which do not use a notion of discrete quantale. We also show a completely new characterization based on separating and cyclic sets. Further we explain a link to simple quantale modules. To apply these characterizations, we study (*-)semisimple (*-)quantales and discuss some other perspectives. Our approach has connections with several earlier works on the subject.