Morphisms in Categories of Sets with Similarity Relations

Morphisms of some categories of sets with similarity relations (Ω-sets) are investigated, where Ω is a complete residuated lattice. Namely a category SetF(Ω) with morphisms (A, δ) → (B, γ) defined as special maps A → B and a category SetR(Ω) with morphisms defined as a special relations A × B → Ω. It is proved that arbitrary maps A → Ω and A × B → Ω can be extended onto morphisms (A, δ) → (Ω,↔) in SetF(Ω) and morphisms (A, δ) → (B, γ) in SetR(Ω), respectively and that these extension processes are from categorical point of view special reflections. Moreover, if Ω is a complete Heyting algebra we also investigate morphisms in another category Set(Ω) which consists of classical fuzzy sets f : A → Ω. Keywords— residuated lattice, Heyting algebra, similarity relations, fuzzy sets.