Quantale Modules and their Operators, with Applications

TL;DR

This paper explores the categorical structure of modules over unital quantales, introduces Q-module transforms, and demonstrates their applications in fuzzy image compression and mathematical morphology.

Contribution

It establishes categorical properties of quantale modules and defines Q-module transforms as homomorphisms, applying them to image processing tasks.

Findings

Q-module transforms are precisely the homomorphisms between free modules.

Applications in fuzzy image compression improve efficiency.

Mathematical morphology benefits from the introduced operators.

Abstract

The central topic of this work is the categories of modules over unital quantales. The main categorical properties are established and a special class of operators, called Q-module transforms, is defined. Such operators - that turn out to be precisely the homomorphisms between free objects in those categories - find concrete applications in two different branches of image processing, namely fuzzy image compression and mathematical morphology.