Homomorphisms of Fuzzy Logic Models Based on Sets With Similarities

We investigate interpretations ‖ψ‖D of formulas ψ in a first order fuzzy logic in models D which are based on Ω-sets, i.e. sets with similarity relations with values in a complete MV-algebra Ω. These interpretations are fuzzy sets in some Ωsets. We show that if φ : D1 → D2 is a strong homomorphism between two models then there are also strong relationships between interpretations ‖ψ‖D1 and ‖ψ‖D2 . Finally, for any model D based on an Ω-set (A, δ) we construct another model S(D) based on a set S(A, δ) of all ”continuous” maps from (A, δ) to (Ω,↔) and such that a singleton map {−} is a strong homomorphism D → S(D).