Action Theory of Fuzzy Bornological Groups

This paper explores the action theory of fuzzy bornological groups. We investigate the scenario where a fuzzy bornological group acts on a fuzzy bornological set such that the action map is fuzzy bounded. Initially, we establish that both left and right translation functions are fuzzy bounded maps. This motivates us to consider a new action called a fuzzy bounded action. The primary goal of studying this action is to examine the effects of the fuzzy bornological groups on the fuzzy bornological sets by partitioning the fuzzy bornological sets into orbits. Furthermore, we define an invariant subset of under the fuzzy bounded action of . Furthermore, the intersection and the union for a collection of invariant fuzzy bounded sets are also invariant fuzzy bounded sets. Finally, we characterize the orbital structure and prove a fundamental result a fuzzy bornological group action effectively partitions the fuzzy bornological set into orbits through fuzzy bornological isomorphism.