Isomorphisms of Fuzzy Sets and Cut Systems

Abstract

Any fuzzy set X in a classical set A with values in a complete (residuated) lattice Ω can be identified with a system of α-cuts X _α , α ∈ Ω. Analogical results were proved for sets with similarity relations with values in Ω (e.g. Ω-sets) which are objects of two special categories K = Set(Ω) or SetR(Ω) of Ω-sets and for fuzzy sets defined as morphisms from Ω-set into a special Ω-set (Ω, ↔ ). These fuzzy sets can be defined equivalently as special cut systems (C _α ) _α , called f-cuts. That equivalence then represents a natural isomorphism between covariant functor of fuzzy sets ${\cal F}_{\bf K}$ and covariant functor of f-cuts ${\cal C}_{\bf K}$. In the paper we are interested in relationships between sets of fuzzy sets and sets of f-cuts in an Ω-set (A,δ) in corresponding categories Set(Ω) and SetR(Ω), which are endowed with binary operations extended either from binary operations in the lattice Ω, or from binary operations defined in a set A by the generalized Zadeh’s extension principle. We prove that the final binary structures are (under some conditions) isomorphic.

This work was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070).