Fixed Point Results for Infinite Fuzzy Sets with Atoms
Andrei Alexandru and Gabriel Ciobanu
The theory of finitely supported structures, which is original from Fraenkel and Mostowski, allows a discrete representation of infinite structures by analyzing their finite supports.We present fixed point theorems (of Bourbaki-Witt type and Knaster-Tarski type) for finitely supported order-preserving functions defined on finitely supported sets that are chain complete, results which generalize the classical fixed point theorems in Zermelo-Fraenkel framework. We provide various examples of finitely supported structures in which the related theorems can be applied, particularly sets which are potentially large but they do not contain infinite uniformly supported subsets. Then we focus on finitely supported fuzzy sets called 𝑃-fuzzy sets, where 𝑃 is either a finitely supported chain complete set or a finitely supported monoid.
Keywords: Finitely supported structures, finitely supported chain complete sets, fixed point results, 𝑇-fuzzy sets, 𝑀-fuzzy sets