Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines
We construct a fuzzy multiplication for the case L a chain in the L-fuzzy real line \({\mathbb{R}}(L)\) which is jointly fuzzy continuous and which is the unique, piecewise consistent extension of ordinary multiplication to \({\mathbb{R}}(L)\) distributing over the fuzzy addition constructed by the author in ibid. 8, 39-52 (1982; Zbl 0508.54002); 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 the paper cited above are used to prove \({\mathbb{R}}(L)\) and the stratified \({\mathbb{R}}^ c(L)\) are nontopologically isomorphic complete fuzzy topological hyperfields if L is a chain with \(| L| \geq 3\). We can now completely answer for L a chain the following question [\textit{E. P. Klement} (ed.), Open questions, Proc. 2nd. Int. Semin. on Fuzzy Set Theory, Johannes Kepler Universität Linz (1981; Zbl 0456.00008)] in the affirmative: can a set of fuzzy numbers, its fuzzy topology, and its fuzzy addition and multiplication be defined such that these operation are jointly fuzzy continuous extensions of the usual addition and multiplication of \({\mathbb{R}}\), 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. \textit{R. Lowen}, Fuzzy Sets Syst. 9, 278-311 (1983; Zbl 0546.54010)]: does fuzzy topology have deep, specific, canonical examples?
scientific article; zbMATH DE number 3913222