Fuzzy Topological Systems

TL;DR

This paper extends fuzzy Petri nets to fuzzy topological systems using Dialectica categories with the unit interval [0,1] as a dualizing object, generalizing Vickers's topological systems.

Contribution

It introduces a novel categorical framework for fuzzy topological systems based on Dialectica categories and the unit interval, expanding the modeling capabilities of fuzzy logic.

Findings

Develops a categorical model for fuzzy topological systems.

Generalizes Vickers's notion of topological systems.

Provides a foundation for further research in fuzzy logic and topology.

Abstract

Dialectica categories are a very versatile categorical model of linear logic. These have been used to model many seemingly different things (e.g., Petri nets and Lambek's calculus). In this note, we expand our previous work on fuzzy petri nets to deal with fuzzy topological systems. One basic idea is to use as the dualizing object in the Dialectica categories construction, the unit real interval [0,1], which has all the properties of a {\em lineale}. The second basic idea is to generalize Vickers's notion of a topological system.