Fixed point results for finitely supported algebraic 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 . 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 which is not injective, nor surjective, could have no fixed points. For example, let us consider two atoms , and defined by . From Proposition 8 it follows that f is supported by . However, f does not have a fixed point. We prove below that a finitely supported function 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 be an invariant set. Then the set 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 , we have . Thus, ⊆ is an equivariant order relation on . Let be a finitely supported family of finitely supported subsets of X. We know that exists in X. We have to prove that . We claim that
Fixed point results for invariant inductive sets
Definition 37 An invariant inductive set is an invariant partially ordered set with the property that every finitely supported totally ordered subset (chain) of X has an upper bound in X. A strong invariant inductive set is an invariant partially ordered set with the property that every finitely supported totally ordered subset (chain) of X has a least upper bound in X. Let be an invariant poset. A finitely supported countable chain in X is a sequence such that the mapping
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 Let be an invariant set and a strong invariant inductive set. The family of those finitely supported functions (i.e. the family of all FSM Y-fuzzy sets over X) is a strong invariant inductive set, with the order relation ⊑ defined by if and only if
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)
- et al.
Fuzzy sets within finitely supported mathematics
Fuzzy Sets Syst.
(2018) - et al.
The lattices of fuzzy subgroups and fuzzy normal subgroups
Inf. Sci.
(1994) - et al.
Finitely Supported Mathematics: An Introduction
(2016) - et al.
Abstract interpretations in the framework of invariant sets
Fundam. Inform.
(2016) - et al.
On the foundations of finitely supported sets
J. Mult.-Valued Log. Soft Comput.
(2019) - et al.
Properties of the atoms in finitely supported structures
Arch. Math. Log.
(2019)
Cited by (8)
THE CATEGORY REL(NOM)
2023, arXivFixed Point Results for Infinite Fuzzy Sets with Atoms
2023, Journal of Multiple-Valued Logic and Soft ComputingFuzzy results for finitely supported structures
2021, MathematicsProperties of finitely supported self - mappings on the finite powerset of atoms
2021, Computer Science Journal of MoldovaFoundations of finitely supported structures: A set theoretical viewpoint
2020, Foundations of Finitely Supported Structures: A Set Theoretical ViewpointUniformly supported sets and fixed points properties
2020, Carpathian Journal of Mathematics