caa a22 4500 463187237 CHVBK 20180406164857.0 cr unu---uuuuu 170326e20070401xx s 000 0 eng 10.1007/s10472-007-9060-7 doi (NATIONALLICENCE)springer-10.1007/s10472-007-9060-7 Singer Daniel LITA, UFR MIM, Université Paul Verlaine de Metz, Ile de Saulcy, 57045, Metz Cedex 1, France aut Linearly ordered semigroups for fuzzy set theory [Elektronische Daten] [Daniel Singer] The fuzzy set theory initiated by Zadeh (Information Control 8:338-353, 1965) was based on the real unit interval [0,1] for support of membership functions with the natural product for intersection operation. This paper proposes to extend this definition by using the more general linearly ordered semigroup structure. As Moisil (Essais sur les Logiques non Chrysippiennes. Académie des Sciences de Roumanie, Bucarest, 1972, p. 162) proposed to define Lukasiewicz logics on an abelian ordered group for truth values 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 step by step deduced from the corresponding set theory to the semigroup order structure. Some results of Clifford on topological semigroups (Clifford, A.H., Proc. Amer. Math. Soc. 9:682-687, 1958; Clifford, A.H., Trans. Amer. Math. Soc. 88:80-98, 1958), Paalman de Miranda work on I-semigroups (Paalman de Miranda, A.B., Topological Semigroups. Mathematical Centre Tracts, Amsterdam, 1964) and Schweitzer, Sklar on T-norms (Schweizer, B., Sklar, A., Publ. Math. Debrecen 10:69-81, 1963; Schweizer, B., Sklar, A., Pacific J. Math. 10:313-334, 1960; Schweizer, B., Sklar, A., Publ. Math. Debrecen 8:169-186, 1961) are revisited in this framework. As a simple consequence of Faucett theorems (Proc. Amer. Math. Soc. 6:741-747, 1955), 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 shall give a new perspective to tackle the question of G. Moisil about the definition of discrete Many-valued logics as approximation of fuzzy continuous ones. Springer Science+Business Media B.V., 2007 Clifford nationallicence Fuzzy sets nationallicence Many-valued logic nationallicence Moisil nationallicence Ordered semigroups nationallicence T-norms nationallicence Topological semigroups nationallicence Annals of Mathematics and Artificial Intelligence Springer Netherlands 49/1-4(2007-04-01), 207-220 1012-2443 49:1-4<207 2007 49 10472 https://doi.org/10.1007/s10472-007-9060-7 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10472-007-9060-7 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Singer Daniel LITA, UFR MIM, Université Paul Verlaine de Metz, Ile de Saulcy, 57045, Metz Cedex 1, France aut NATIONALLICENCE 773 0- Annals of Mathematics and Artificial Intelligence Springer Netherlands 49/1-4(2007-04-01), 207-220 1012-2443 49:1-4<207 2007 49 10472 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer