Commutative Quantale and Localization

TL;DR

This paper introduces a localization construction for quantales, extending algebraic and geometric concepts, and proves related theorems including a novel algebraic version of the Baire Category Theorem.

Contribution

It develops the theory of localization for quantales using multiplicative filters and establishes foundational theorems analogous to sheaf properties and the Baire Category Theorem.

Findings

Constructed localization of quantales at multiplicative filters

Proved theorems similar to '$ ext{Spec } R$ is a sheaf'

Presented a new algebraic version of the Baire Category Theorem

Abstract

In this paper we introduce the localization construction for quantales. A quantale is a complete semilattice combined with a multiplication. We mimic the notion of filter in a lattice to define multiplicative filters in a quantale, and construct the localization of the quantale at a multiplicative filter. We prove theorems with similar structure as "$\Spec R$ is a sheaf" and use them to obtain several results in algebra and geometry, including the Baire Category Theorem and some of its generalizations. We also present an algebraic version of Baire Category Theorem, which we believe has not appeared in literature.