Fuzzy logic models in a category of fuzzy relations

Abstract

We investigate interpretations \({\|\psi\|_{\mathcal E}}\) of formulas ψ in a first order fuzzy logic in models \({\mathcal {E}}\) which are based on objects of a category SetR(Ω) which consists of Ω-sets, i.e. sets with similarity relations with values in a complete MV-algebra Ω and with morphisms defined as special fuzzy relations between Ω-sets. The interpretations \({\|\psi\|_\mathcal {E}}\) are then morphisms in a category SetR(Ω) from some Ω-set to the object \({(\Omega,\leftrightarrow)}\). We define homomorphisms between models in a category SetR(Ω) and we prove that if \({\varphi : \mathcal {E}_1\rightarrow \mathcal {E}_2}\) is a (special) homomorphism of models in a category SetR(Ω) then there is a relation between interpretations \({\|\psi\|_{\mathcal {E}_i}}\) of a formula ψ in models \({\mathcal {E}_i}\).