Triangular norms. Position paper III: continuous t-norms

This third and last part of a series of position papers on triangular norms (for Parts I and II see (E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Position paper I: basic analytical and algebraic properties, Fuzzy Sets and Systems, in press; E.P. Klement, R. Mesiar, E. Pap, Triangular norms. Position paper II: general constructions and parameterized families, submitted for publication) presents the representation of continuous Archimedean t-norms by means of additive generators, and the representation of continuous t-norms by means of ordinal sums with Archimedean summands, both with full proofs. Finally some consequences of these representation theorems in the context of comparison and convergence of continuous t-norms, and of the determination of continuous t-norms by their diagonal sections are mentioned.

[1]  R. Yager.  On a general class of fuzzy connectives , 1980 .

[2]  J. Aczél,et al.  Sur les opérations définies pour nombres réels , 1948 .

[3]  Liu Dsosu,et al.  Fuzzy random measure and its extension theorem , 1983 .

[4]  Fritz Klein-Barmen,et al.  Über gewisse Halbverbände und kommutative Semigruppen , 1942 .

[5]  J. Dombi.  A general class of fuzzy operators, the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators , 1982 .

[6]  J. Aczél,et al.  Lectures on Functional Equations and Their Applications , 1968 .

[7]  R. Moynihan,et al.  On τT semigroups of probability distribution functions II , 1977 .

[8]  Petr Hájek,et al.  Observations on the monoidal t-norm logic , 2002, Fuzzy Sets Syst..

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

[10]  Radko Mesiar,et al.  Triangular Norms , 2000, Trends in Logic.

[11]  Lotfi A. Zadeh,et al.  Fuzzy sets and systems , 1990 .

[12]  S. Weber.  A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms , 1983 .

[13]  S. E. Rodabaugh,et al.  Topological and algebraic structures in fuzzy sets : a handbook of recent developments in the mathematics of fuzzy sets , 2003 .

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

[15]  Barbara Baccheli.  Representation of continuous associative functions. , 1986 .

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

[17]  M. J. Frank.  On the simultaneous associativity of F(x, y) and x+y-F(x, y). (Short Communication). , 1978 .

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

[19]  Radko Mesiar,et al.  Diagonals of continuous triangular norms , 1999, Fuzzy Sets Syst..

[20]  Karl H. Hofmann,et al.  Linearly ordered semigroups: Historical origins and A.H. Clifford's influence , 1995 .

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

[22]  Radko Mesiar,et al.  Triangular norms. Position paper II: general constructions and parameterized families , 2004, Fuzzy Sets Syst..

[23]  Radko Mesiar,et al.  Triangular norms. Position paper I: basic analytical and algebraic properties , 2004, Fuzzy Sets Syst..

[24]  Radko Mesiar,et al.  A characterization of the ordering of continuous t-norms , 1997, Fuzzy Sets Syst..

[25]  C. Alsina.  On a method of Pi-Calleja for describing additive generators of associative functions , 1992 .

[26]  Zsolt Páles,et al.  The associativity equation revisited , 1989 .

[27]  J. Aczél,et al.  Characterizations of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgements , 1982 .

[28]  A. Clifford,et al.  The algebraic theory of semigroups , 1964 .

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

[30]  János C. Fodor,et al.  On continuous triangular norms , 1998, Fuzzy Sets Syst..

[31]  Sándor Jenei,et al.  On Archimedean triangular norms , 1998, Fuzzy Sets Syst..

[32]  Vladik Kreinovich,et al.  Strict Archimedean t-norms and t-conorms as universal approximators , 1998, Int. J. Approx. Reason..

[33]  F. Klein-Barmen.  Über gewisse Halbverbände und kommutative Semigruppen , 1942 .

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