Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines

Abstract

Abstract We construct a fuzzy multiplication for the case L a chain in the L-fuzzy real line R (L) which is jointly fuzzy continuous and which is the unique, piecewise consistent extension of ordinary multiplication to R (L) distributing over the fuzzy addition constructed by the author in [47]; we also specify the fuzzy integers and give an extensive analysis of fuzzy contractions and expansions. These results, the constructions of this paper, and the results of [47] are used to prove R (L) and the stratified R c(L) are nontopologically isomorphic complete fuzzy topological hyperfields if L is a chain with |L|⩾3. We can now completely answer for L a chain the following question of [27] in the affirmative: can a set of fuzzy numbers, its fuzzy topology, and its fuzzy addition and multiplication be defined such that these operations are jointly fuzzy continuous extensions of the usual addition and multiplication of R and can we specify the fuzzy integers? These results, in combination with recent papers of several authors, would seem to answer in the affirmative the following philosophical question (cf. [34]): does fuzzy topology have deep, specific, canonical examples?