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Linearly ordered semigroups for fuzzy set theory. (English) Zbl 1125.03040
Summary: The fuzzy set theory initiated by Zadeh was based on the real unit interval \([0,1]\) for support of membership functions with the natural product for the intersection operation. This paper proposes to extend this definition by using the more general linearly ordered semigroup structure. As Gr. C. Moisil [Essais sur les logiques non chrysippiennes. Bucuresti: Éditions de l’Académie de la République Socialiste de Roumanie (1972; Zbl 0241.02006), p. 162] proposed to define Łukasiewicz logics on an abelian ordered group as a truth value set, we give a simple negative answer to the question on the possibility to build a many-valued logic on a finite abelian ordered group. In a constructive way characteristic properties are deduced step by step from the corresponding set theory to the semigroup order structure. Some results of A. H. Clifford on topological semigroups [Proc. Am. Math. Soc. 9, 682–687 (1958; Zbl 0100.02403); Trans. Am. Math. Soc. 88, 80–98 (1958; Zbl 0082.02401)], A. B. Paalman-de Miranda’s work on I-semigroups [Topological semigroups. Amsterdam: Mathematisch Centrum Amsterdam (1964; Zbl 0136.26904)] and B. Schweizer and A. Sklar’s results on t-norms [Publ. Math. Debrecen 10, 69–81 (1963); Pac. J. Math. 10, 313–334 (1960; Zbl 0091.29801); Publ. Math. Debrecen 8, 169–186 (1961)] are revisited in this framework. As a simple consequence of W. M. Faucett’s theorems [Proc. Am. Math. Soc. 6, 741–747 (1955; Zbl 0065.25204)], we prove how canonical properties from the fuzzy set theory point of view lead to the Zadeh choice thus giving another proof of the representation theorem of t-norms. This structural approach gives a new perspective to tackle the question of G. Moisil about the definition of discrete many-valued logics as approximation of fuzzy continuous ones.
MSC:
03E72 Theory of fuzzy sets, etc.
03B50 Many-valued logic
06F05 Ordered semigroups and monoids
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