Abstract
The rearrangement inequality states that the sum of products of permutations of 2 sequences of real numbers are maximized when the terms are similarly ordered and minimized when the terms are ordered in opposite order. We show that similar inequalities exist in algebras of multi-valued logic when the multiplication and addition operations are replaced with various T-norms and T-conorms respectively. For instance, we show that the rearrangement inequality holds when the T-norms and T-conorms are derived from Archimedean copulas.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1952)
Oppenheim, A.: Inequalities connected with definite Hermitian forms, II. Am. Math. Mon. 61, 463–466 (1954)
Minc, H.: Rearrangements. Trans. Am. Math. Soc. 159, 497–504 (1971)
Wu, C.W.: On rearrangement inequalities for multiple sequences. Math. Inequal. Appl. 25(2), 511–534 (2022)
Gottwald, S.: Many-valued logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Stanford (2020)
Fodor, J.: Left-continuous t-norms in fuzzy logic: an overview. Acta Polytech. Hung. 1(2) (2004)
Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. position paper I: basic analytical and algebraic properties. Fuzzy Sets Syst. 143(1), 5–26 (2004). (Advances in Fuzzy Logic)
Jenei, S.: How to construct left-continuous triangular norms-state of the art. Fuzzy Sets Syst. 143(1), 27–45 (2004). (Advances in Fuzzy Logic)
Trillas, E.: Sobre functiones de negación en la teoría de conjuntas difusos. Stochastica 3, 47–60 (1979)
Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications, vol. 144. Academic Press, Cambridge (1980)
Mayor, G., Torrens, J.: On a family of t-norms. Fuzzy Sets Syst. 41(2), 161–166 (1991)
Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper II: general constructions and parameterized families. Fuzzy Sets Syst. 145(3), 411–438 (2004)
Nelsen, R.B.: An Introduction to Copulas. Springer, Berlin (2006)
Clifford, A.H.: Naturally totally ordered commutative semigroups. Am. J. Math. 76, 631–646 (1954)
Jenei, S.: A note on the ordinal sum theorem and its consequence for the construction of triangular norms. Fuzzy Sets Syst. 126(2), 199–205 (2002)
Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets Syst. 80(1), 111–120 (1996)
Fodor, J.C., Yager, R.R., Rybalov, A.: Structure of uninorms. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 5(4), 411–427 (1997)
Janssens, S., de Baets, B., de Meyer, H.: Bell-type inequalities for parametric families of triangular norms. Kybernetika 40(1), 89–106 (2004)
Moynihan, R.: On \(\tau _{T}\) semigroups of probability distribution functions II. Aequationes Math. 17, 19–40 (1978)
Bacigál, T., Najjari, V., Mesiar, R., Bal, H.: Additive generators of copulas. Fuzzy Sets Syst. 264, 42–50 (2015). (Special issue on Aggregation functions at AGOP2013 and EUSFLAT 2013)
Mesiar, R., Sempi, C.: Ordinal sums and idempotents of copulas. Aequationes Math. 79(1–2), 39–52 (2010)
Klement, E.P., Mesiar, R., Pap, E.: Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms. Fuzzy Sets Syst. 104(1), 3–13 (1999)
Kauers, M., Pillwein, V., Saminger-Platz, S.: Dominance in the family of Sugeno–Weber t-norms. Fuzzy Sets Syst. 181, 07 (2010)
Klement, E., Mesiar, R., Spizzichino, F., Stupnanová, A.: Universal integrals based on copulas. Fuzzy Optim. Decis. Mak. 13(3), 273–286 (2014)
Yu, H.: Circular rearrangement inequality. J. Math. Inequal. 12(3), 635–643 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Wu, C.W. On Rearrangement Inequalities for Triangular Norms and Co-norms in Multi-valued Logic. Log. Univers. 17, 331–346 (2023). https://doi.org/10.1007/s11787-023-00332-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11787-023-00332-0