On a categorical analysis of Zadeh generalized subsets of sets, II

Abstract

The hypothesis of right regularity of non-zero truth-values of a category C with fuzzy subsets is proved to be equivalent to the fact that surjections in C are exactly C-epimorphisms. C has associative images if and only if the ∧-semilattice L of truth-values is a sup-complete lattice which satisfies left distributivity law. As a consequence, L is a sup-distributive lattice when the canonical multiplication on L is commutative. It is proved that any topos of j-sheaves for a Grothendieck topology j on the monoid L ∗ of non-zero truth-values is a good toposophical approximation of C when L ∗ is a j-sheaf. A consequence is that the topos of j-sheaves, where j is the canonical topology is the best toposophical approximation of C. When the monoid L is commutative, the corresponding ∧-semilattice is a sup-distributive lattice and L ∗ satisfies a strong properties of density then the sup-operation on left ideals of L ∗ determines a Grothendieck topology ( −) on L ∗ such that the category of ( −)-separated objects is a good completion of C.