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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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##### References:
 [1] Clifford, A.H.: Ordered commutative semigroups of the second kind. Proc. Amer. Math. Soc. 9, 682–687 (1958) · Zbl 0100.02403 [2] Clifford, A.H.: Connected ordered topological semigroups with idempotent endpoints. I. Trans. Amer. Math. Soc. 88, 80–98 (1958) · Zbl 0082.02401 [3] Faucett, W.M.: Compact semigroups irreducibly connected between two idempotents. Proc. Amer. Math. Soc. 6, 741–747 (1955) · Zbl 0065.25204 [4] Fuchs, L.: Partially Ordered Algebraic Structures. Pergamon, Oxford (1963) · Zbl 0137.02001 [5] Georgescu, G., Iorgulescu, A., Rudeanu, S.: Grigore C. Moisil (1906–1973) and his School in Algebraic Logic. IJCCC 1(1), 81–99 (2006) (on the web) [6] Hájek, P.: Basic Fuzzy Logic and BL-algebras. TR.V736, ICS, Ac. of Sciences of the Czech Republic (1996) · Zbl 1018.03021 [7] Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer (1998) · Zbl 0937.03030 [8] Hájek, P., et al.: Complexity of t-tautologies. Ann. Pure Appl. Logic 113, 3–11 (2002) · Zbl 1006.03022 [9] Hofmann, K.H., Lawson, J.D.: Linearly ordered semigroups: historic origins and A.H. Clifford’s Influence. In: Hofmann, K.H., Mislove, M.W. (eds.) Semigroup theory and its applications. London Math. Soc. Lecture Notes 231, 15–39 (1996) · Zbl 0901.06012 [10] Moisil, Gr.C.: Essais sur les Logiques non Chrysippiennes. Académie des Sciences de Roumanie, Bucarest (1972) · Zbl 0241.02006 [11] Paalman de Miranda, A.B.: Topological Semigroups. Mathematical Centre Tracts, Amsterdam. ( http://www.math.buffalo.edu/mad/PEEPS/paalman_aida.html ) (1964) · Zbl 0136.26904 [12] Pin, J.E.: Finite Semigroups as categories, ordered semigroups or compact semigroups. Semigroup Forum, Clifford Conference Tulane (1994) [13] Ponasse, D.: Séminaire de Mathématiques Floues. Polycopiés, Vol. I et II , Université de Lyon I, (1978, 1979) [14] Ribenboïm, P.: Théorie des Groupes Ordonnés. University Press, Bahia Blanca, Argentina (1959) [15] Ruspini, E., Bonissone, P., Pedrycz, W. (eds.) Handbook of fuzzy computation. Institute of Physics Publ., London (1998) · Zbl 0902.68068 [16] Rutherford, D.E.: Introduction to Lattice Theory. Oliver & Boyd Publ. (1965) · Zbl 0127.24904 [17] Schweizer, B., Sklar, A.: Associative functions and abstract semigroups. Publ. Math. Debrecen 10, 69–81 (1963) · Zbl 0119.14001 [18] Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313–334 (1960) · Zbl 0091.29801 [19] Schweizer, B., Sklar, A.: Associative functions and statistical triangle inequalities. Publ. Math. Debrecen 8, 169–186 (1961) · Zbl 0107.12203 [20] Turunen, E.: Mathematics Behind Fuzzy Logic. Advances in Soft Computing, Physica Verlag, Springer (1999) · Zbl 0940.03029 [21] Wallace, A.D.: The structure of topological semigroups. Bul. AMS 61, 95–112 (1955) · Zbl 0065.00802 [22] Zadeh, L.A.: Fuzzy sets. Information Control 8, 338–353 (1965) · Zbl 0139.24606
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