Fuzzy Sets and Systems46 (1992) 189-192 189 North-Holland Fuzzy sets as named sets Mark Burgin and Vladimir Kuznetsov Institute of Pedagogics of the Ukrainian SSR and Institute of Philosophy of the Academy of Sciences of the Ukrainian SSR, 252001, Kiev-1, USSR Received June 1989 Revised September 1990 Abstract: The paper contains a short summary of named set theory, some definitions and theorems from it and relations between fuzzy sets and named sets. The aim of the paper is to show that various approaches in fuzzy set theory are specifications of named set theory. Keywords: Fuzzy sets; named sets; algebra; category; morphism; methodologyof science. A principal feature of the development of mathematics is an elaboration of more and more general mathematical ideas and constructions. On one hand, they allow us to unite previously disconnected mathematical concepts and theories. On the other hand, on the basis of new general constructions it is possible to gain, as a rule, much more applicability in comparison with particular cases that existed before. The concept of a fuzzy set is a good illustration of this feature of the development of mathematics. Firstly, an 'ordinary' set is a particular case of a fuzzy set and so the theory of fuzzy sets generalizes in a way set theory. Secondly, the appearance of fuzzy sets theory extended essentially the area of practical implementation of set-theoretical ideas in physics, technology, management, huma- nities, etc. Nevertheless there are no grounds to think that the concept of a fuzzy set is the most general and abstract one. The appearance of the concept of a multiset, the alternative theory of sets and other new set-theoretical constructions exhibit this very vividly. In addition to this it is necesary to take into account the existence of different kinds of fuzzy sets: L-fuzzy sets, sub-determinated fuzzy sets, rough, completely fuzzy sets, etc. As a result it is necessary to put forward those general set-theoretical construc- tions which allow us to treat all above-mentioned concepts of sets as particular cases. The concept of a named set [1] is one of the most important of these constructions. In order to give an exact definition of the concept of a named set we fix three collections ~, See/, qCd of sets (or classes) and their morphisms. Totalities of all objects (i.e. sets and (or) classes) from these collections are denoted 6~, 6gfee¢, Ug~d, respectively. Totality of all morphisms (i.e. maps of sets (classes) or relations between them) from these collections are denoted ~toy~,~, ~5~d, ~c¢~u', respec- tively. The following conditions are valid: (1) ~?/,~J, U/~¢~ ~?//£d, (2) ~,,J, ~t.,,Se~t~_ Jt/~,~,/; (3) the collection ~rc~o/ is closed with respect to the product of morphisms, i.e. if maps a~:A-+B, fl:B---~C (relations c~_AxB, tic_ B ×C), belong to ~/£o¢', where A,B,C• ~?gc¢o/, then their product a~fl belongs to ~/o/¢M. Let us fix in J//o~q¢~/ some subclass M___ ~qCcu'. Using different conditions in M it is possible to define constructions necessary in each specific case. Definition 1. A named (with respect to M) set is a triple Y = (X, a, I) where X • 6g~J, ! • 695e~¢, a,•M, a~: X---~ I. The concept of a named set by means of appropriate specifications unites different exist- ing set-theoretic constructions (standard as well as nonstandard ones). Defmition 2. Let 3E = (X, tr, I) be a named set. (i) The set X is called the support of .~ and is denoted S(~). (ii) The set i is called the set of names of 3~ and is denoted N(3E). (iii) The map (relation) o: is called the naming map (relation) of ~ and is denoted n(~). Let us consider from the viewpoint of the 0165-0114/92/$05.00© 1992--Elsevier Science Publishers B.V. All rights reserved