Fuzzy Results for Finitely Supported Structures

Andrei Alexandru and Gabriel Ciobanu
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Andrei Alexandru: Institute of Computer Science, Romanian Academy, 700505 Iaşi, Romania
Gabriel Ciobanu: Faculty of Computer Science, Alexandru Ioan Cuza University, 700506 Iaşi, Romania

Mathematics, 2021, vol. 9, issue 14, 1-23

Abstract: We present a survey of some results published recently by the authors regarding the fuzzy aspects of finitely supported structures. Considering the notion of finite support, we introduce a new degree of membership association between a crisp set and a finitely supported function modelling a degree of membership for each element in the crisp set. We define and study the notions of invariant set, invariant complete lattices, invariant monoids and invariant strong inductive sets. The finitely supported (fuzzy) subgroups of an invariant group, as well as the L -fuzzy sets on an invariant set (with L being an invariant complete lattice) form invariant complete lattices. We present some fixed point results (particularly some extensions of the classical Tarski theorem, Bourbaki–Witt theorem or Tarski–Kantorovitch theorem) for finitely supported self-functions defined on invariant complete lattices and on invariant strong inductive sets; these results also provide new finiteness properties of infinite fuzzy sets. We show that apparently, large sets do not contain uniformly supported, infinite subsets, and so they are invariant strong inductive sets satisfying finiteness and fixed-point properties.

Keywords: invariant set; L -fuzzy set; T -fuzzy set; invariant complete lattice; invariant strong inductive set; fixed points; S -finite support principle; uniformly supported set (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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