Fuzzy sets and cut systems in a category of sets with similarity relations

Abstract

Let X be a complete residuated lattice. Let SetRðXÞ be the category of sets with similarity relations with values in X (called X-sets), which is an analogy of the category of classical sets with relations as morphisms. A fuzzy set in an X-set in the category SetRðXÞ is a morphism from X-set to a special X-set ðX; $Þ; where $ is the biresiduation operation in X: In the paper, we prove that fuzzy sets in X-sets in the category SetRðXÞ can be expressed equivalently as special cut systems ðC a Þ a2X

α-Cuts and models of fuzzy logic

“…For details of rather technical proofs see Močkoř (2012). The principal result of this section is the following theorem which states that fuzzy sets (with respect to both categories Set(V) and SetR(V)) can be equivalently defined by f-cuts.…”

“…To define an action of C SetR(V) on morphisms we need the following two technical lemmas, details of proofs can be found in Močkoř (2012). Now we will continue with the definition of the functor C SetR(V) .…”

“…V with some special but natural properties. For more details about such fuzzy sets see Novák et al (1999) and Močkoř (2009Močkoř ( , 2012.…”

“…For more details about V-sets and categories Set(V) and SetR(V) see e.g. Höhle (1992), Novák, Perfilijeva and Močkoř (1999) and Močkoř (2006Močkoř ( , 2009Močkoř ( , 2010bMočkoř ( , 2012.…”

Spaces with fuzzy partitions and fuzzy sets

“…These fuzzy objects generalize classical fuzzy sets A → Q and in facts, a fuzzy objects (A, δ ) → (Q, ↔) is nothing else than extensional map in a Q-set (see, e.g. [6,5,7]). Another generalization of universes used in fuzzy theory was done by I. Perfilieva [12,13], who introduce the notion of F-transform.…”