Fixed point results for finitely supported algebraic structures

doi:10.1016/j.fss.2019.09.014

Fuzzy Sets and Systems

Volume 397, 15 October 2020, Pages 1-27

https://doi.org/10.1016/j.fss.2019.09.014 Get rights and content

Abstract

We present a collection of fixed point theorems in the framework of finitely supported structures, preserving the validity of several classical Zermelo-Fraenkel fixed point theorems such as Tarski strong theorem, Bourbaki-Witt theorem, Scott theorem and Tarski-Kantorovitch theorem. We also prove several specific fixed point properties in the framework of finitely supported algebraic structures, results that are not reformulations of some corresponding Zermelo-Fraenkel results. Applications of the fixed point theorems are emphasized by presenting many examples of finitely supported ordered structures for which these theorems can be used. Particularly, the results provide properties of L-fuzzy sets defined in the world of finitely supported structures.

Section snippets

Finitely supported structures: an introduction

The theory of finitely supported structures is known in the literature both under the name of ‘nominal sets’ [11] when dealing with computer science applications, and ‘finitely supported mathematics’ [1] when dealing with the set theory foundations. Finitely supported structures provide an alternative framework for studying infinite structures hierarchically constructed by involving some basic elements (called atoms) by analyzing only finitely many entities that form their supports. As an

Finitely supported sets: preliminary results

A finite set is referred to a set of the form ${x_{1}, \dots, x_{n}}$ . Let A be a fixed infinite (non-finite) ZF-set. We do not involve any internal structure on the elements of A. Alternatively, the theory of invariant sets can be adequately reformulated if A is the set of atoms from ZFA set theory (see [1]). We refer to A simply as an ‘infinite ZF set’ without discussing its ZF cardinality. This is because one can prove that there does not exist a finitely supported bijection between A and a non-atomic ZF

Fixed points results for transformations of atoms

A finitely supported function $f : A \to A$ which is not injective, nor surjective, could have no fixed points. For example, let us consider two atoms $a, b \in A$ , and $f : A \to A$ defined by $f (x) = {\begin{matrix} a & for x \in A ∖ {a} \\ b & for x = a \end{matrix}$ . From Proposition 8 it follows that f is supported by $supp (a) \cup supp (b) = {a, b}$ . However, f does not have a fixed point. We prove below that a finitely supported function $f : A \to A$ which is injective or surjective has an infinite set of fixed points. More exactly, all but finitely many atoms are fixed by such

Fixed point results for invariant lattices

We introduce and study invariant lattices by equipping invariant sets with equivariant lattice orders. The main goal of this section is to prove that Tarski's fixed point theorem for ZF complete lattices remains valid in FSM. Such a result can be applied for the particular FSM complete lattices constructed in the next section. FSM Tarski's fixed point theorem can also be applied in an FSM theory of abstract interpretation to prove the existence of least fixed points for specific finitely

Powersets

Theorem 23

Let $(X, \cdot)$ be an invariant set. Then the set $(℘_{f s} (X), ⋆, \subseteq)$ is an invariant complete lattice.

Proof

According to the definition of the action ⋆ in the statement of Proposition 2(6), for any finitely supported subsets Y and Z of X with $Y \subseteq Z$ , we have $π ⋆ Y = {π \cdot y | y \in Y} \subseteq {π \cdot z | z \in Z} = π ⋆ Z$ . Thus, ⊆ is an equivariant order relation on $℘_{f s} (X)$ .

Let $F = {(X_{i})}_{i \in I}$ be a finitely supported family of finitely supported subsets of X. We know that $\cup F = \underset{i \in I}{\cup} X_{i}$ exists in X. We have to prove that $\underset{i \in I}{\cup} X_{i} \in ℘_{f s} (X)$ . We claim that $supp (F)$

Fixed point results for invariant inductive sets

Definition 37

1.
An invariant inductive set is an invariant partially ordered set $(X, \cdot, ⊑)$ with the property that every finitely supported totally ordered subset (chain) of X has an upper bound in X.
2.
A strong invariant inductive set is an invariant partially ordered set $(X, \cdot, ⊑)$ with the property that every finitely supported totally ordered subset (chain) of X has a least upper bound in X.
3.
Let $(X, \cdot, ⊑)$ be an invariant poset. A finitely supported countable chain in X is a sequence ${(x_{n})}_{n \in N} \subseteq X$ such that the mapping $n \mapsto x_{n}$

Constructions of invariant inductive sets

Starting from a certain FSM algebraic structure Y, we can construct FSM Y-fuzzy sets that have the same structure as Y. In this way, the results presented in this paper lead to valid properties of such FSM Y-fuzzy sets.

Theorem 45

1.
Let $(X, \cdot)$ be an invariant set and $(Y, ⋄, \leq)$ a strong invariant inductive set. The family of those finitely supported functions $f : X \to Y$ (i.e. the family of all FSM Y-fuzzy sets over X) is a strong invariant inductive set, with the order relation ⊑ defined by $f ⊑ g$ if and only if $f (x) \leq g (x$

Conclusion

Finitely Supported Mathematics (FSM) is a theory of finitely supported algebraic structures. FSM is related to the theory of nominal sets and to the theory of FM sets, representing an alternative study of the sets by dealing with infinite algebraic structures that are finitely supported modulo certain permutation actions. More exactly, the elements outside the support of an (infinite) FSM object are somehow “similar”, and so the objects are characterized by their supports. The motivation of

Acknowledgements

The authors are grateful to the anonymous referees for various comments and suggestions which improve the paper.

References (12)

A. Alexandru et al.
Fuzzy sets within finitely supported mathematics
Fuzzy Sets Syst.
(2018)
N. Ajmal et al.
The lattices of fuzzy subgroups and fuzzy normal subgroups
Inf. Sci.
(1994)
A. Alexandru et al.
Finitely Supported Mathematics: An Introduction
(2016)
A. Alexandru et al.
Abstract interpretations in the framework of invariant sets
Fundam. Inform.
(2016)
A. Alexandru et al.
On the foundations of finitely supported sets
J. Mult.-Valued Log. Soft Comput.
(2019)
A. Alexandru et al.
Properties of the atoms in finitely supported structures
Arch. Math. Log.
(2019)

There are more references available in the full text version of this article.

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Fixed point results for finitely supported algebraic structures

Abstract

Section snippets

Finitely supported structures: an introduction

Finitely supported sets: preliminary results

Fixed points results for transformations of atoms

Fixed point results for invariant lattices

Powersets

Fixed point results for invariant inductive sets

Constructions of invariant inductive sets

Conclusion

Acknowledgements

Fuzzy Sets Syst.

Inf. Sci.

Finitely Supported Mathematics: An Introduction

Abstract interpretations in the framework of invariant sets

Fundam. Inform.

On the foundations of finitely supported sets

J. Mult.-Valued Log. Soft Comput.

Properties of the atoms in finitely supported structures

Arch. Math. Log.