Isomorphisms of fuzzy sets and cut systems

• Authors: Jiří Močkoř
• Venue: IWANN'13 Proceedings of the 12th international conference on Artificial Neural Networks: advances in computational intelligence - Volume Part I
• Year: 2013

Any fuzzy set X in a classical set A with values in a complete (residuated) lattice Ω can be identified with a system of º-cuts Xº, º∈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 (Cº)º, 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.