Nuclei and applications to star, semistar, and semiprime operations

TL;DR

This paper explores the deep connections between quantale theory and various types of star operations, providing new representation theorems and generalizations that unify these algebraic structures.

Contribution

It introduces novel representation theorems for precoherent prequantales and extends the theory of semistar operations within the quantale framework.

Findings

Representation theorems for precoherent prequantales

Characterizations of simple prequantales

Generalization of semistar operation constructions

Abstract

We show that the theory of quantales and quantic nuclei motivate new results on star operations, semistar operations, semiprime operations, ideal systems, and module systems, and conversely the latter theories motivate new results on quantales and quantic nuclei. Results include representation theorems for precoherent prequantales and multiplicative semilattices; characterizations of the simple prequantales; and a generalization to the setting of precoherent quantales of the construction of the largest finite type semistar operation and the largest stable semistar operation smaller than a given semistar operation.