A study of the interrelation between fuzzy topological systems and logic

TL;DR

This thesis explores the relationship between fuzzy topological systems and logic, introducing new concepts like L topological systems and fuzzy geometric logic, and establishing categorical relationships and dualities.

Contribution

It introduces L topological systems, fuzzy geometric logic, and new categorical relationships, advancing the theoretical framework connecting fuzzy topology and logic.

Findings

Categorical relationships between fuzzy topological systems and frames established.

Introduction of L topological systems and variable basis fuzzy topologies.

Proof of a generalized Stone duality for fuzzy Boolean systems.

Abstract

The major part of this thesis deals with fuzzy geometric logic and fuzzy geometric logic with graded consequence. The first chapter mainly contains the concept of topological system introduced by S. Vickers in 1989. In Chapter 2 the notion of fuzzy topological system is introduced and categorical relationship with fuzzy topology and frame is discussed in detail. Also this chapter contains some methodology to make new fuzzy topological systems from the old one.Chapter 3 provides a generalization of fuzzy topological system which shall be called L topological system and categorical relationships with appropriate topological space and frame. Furthermore, two ways of constructing subspaces and subsystems of an L topological space and an L topological system are respectively provided.Chapter 4 deals with the concept of variable basis fuzzy topological space on fuzzy sets and contains a new notion of variable basis fuzzy topological systems whose underlying sets are fuzzy sets. In this chapter categorical relationship between space and system is established. Chapter 5 contains a different proof of one kind of generalized stone duality, which was done directly by Maruyama, introducing a notion of n fuzzy Boolean system. The last two chapters, Chapter 6 and Chapter 7 deal the ultimate objective. Chapter 6 deals with the question- From which logic fuzzy topology can be studied?. To answer this, the notion of fuzzy geometric logic is invented.