Triangular norms as ordinal sums of semigroups in the sense of A. H. Clifford

Ordinal sums of semigroups in the sense of [3] leading to triangular norms [29] are studied, generalizing the ordinal sum of triangular norms [30]. The summands of these general ordinal sums are fully characterized, and the ordinal irreducibility of triangular norms is investigated, inducing a natural partition of the class of all triangular norms.

[1]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[2]  S. Gottwald.  A Treatise on Many-Valued Logics , 2001 .

[3]  Karl H. Hofmann,et al.  Semigroup theory and its applications , 1996 .

[4]  A. H. Clifford,et al.  Naturally Totally Ordered Commutative Semigroups , 1954 .

[5]  菅野 道夫,et al.  Theory of fuzzy integrals and its applications , 1975 .

[6]  D. Butnariu,et al.  Triangular Norm-Based Measures and Games with Fuzzy Coalitions , 1993 .

[7]  W. M. Faucett.  Topological semigroups and continua with cut points , 1955 .

[8]  Siegfried Weber,et al.  Generalized measures , 1991 .

[9]  W. M. Faucett.  Compact semigroups irreducibly connected between two idempotents , 1955 .

[10]  R. Nelsen.  An Introduction to Copulas , 1998 .

[11]  E. Pap.  Null-Additive Set Functions , 1995 .

[12]  Hung T. Nguyen,et al.  A First Course in Fuzzy Logic , 1996 .

[13]  P. Mostert,et al.  On the Structure of Semigroups on a Compact Manifold With Boundary , 1957 .

[14]  M. J. Frank.  On the simultaneous associativity ofF(x,y) andx +y -F(x,y) , 1979 .

[15]  Sándor Jenei,et al.  Structure of left-continuous triangular norms with strong induced negations (I) Rotation construction , 2000, J. Appl. Non Class. Logics.

[16]  G. Preston,et al.  A.H. Clifford: An appreciation of his work on the occasion of his sixty-fifth birthday , 1974 .

[17]  B. Schweizer,et al.  Statistical metric spaces. , 1960 .

[18]  R. J. Koch,et al.  The theory of topological semigroups , 1986 .

[19]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[20]  Francesc Esteva,et al.  Review of Triangular norms by E. P. Klement, R. Mesiar and E. Pap. Kluwer Academic Publishers , 2003 .

[21]  Hung T. Nguyen,et al.  Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference , 1994 .

[22]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[23]  K. Menger.  Statistical Metrics. , 1942, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Sándor Jenei,et al.  A note on the ordinal sum theorem and its consequence for the construction of triangular norms , 2002, Fuzzy Sets Syst..

[25]  A. H. Clifford,et al.  Connected ordered topological semigroups with idempotent endpoints. II. , 1958 .

[26]  Ronald R. Yager,et al.  Uninorm aggregation operators , 1996, Fuzzy Sets Syst..

[27]  J. Fodor.  Contrapositive symmetry of fuzzy implications , 1995 .

[28]  M. Tokizawa,et al.  On Topological Semigroups , 1982 .

[29]  S. Weber.  ⊥-Decomposable measures and integrals for Archimedean t-conorms ⊥ , 1984 .