Fuzzy Lie Groups

TL;DR

This paper explores the integration of fuzzy set theory with Lie groups, defining fuzzy submanifolds and related concepts to facilitate fuzzy differential invariants.

Contribution

It introduces the concept of ${ m C}^1$ fuzzy submanifolds and extends Lie group theory to fuzzy contexts, providing foundational structures for fuzzy differential invariants.

Findings

Defined ${ m C}^1$ fuzzy submanifolds

Introduced fuzzy transformation groups and fuzzy G-invariance

Established conditions for fuzzy differential invariants

Abstract

In this paper, we have tried to apply the concepts of fuzzy sets to Lie groups and its relative concepts. First, we define a ${\cal C}^1$ fuzzy submanifold after reviewing ${\cal C}^1-$fuzzy manifold definition. In main section, we defined the Lie group and some its relative concepts such as fuzzy transformation group, fuzzy $G$-invariant. Our aim is to prepare the suitable conditions for defining fuzzy differential invariant by constructing suitable objects.