Abstract
Fuzzy implication functions are logical connectives commonly used to model fuzzy conditional and consequently they are essential in fuzzy logic and approximate reasoning. From the theoretical point of view, the study of how to construct new implication functions from old ones is one of the most important topics in this field. In this paper new ordinal sum construction methods of implication functions based on fuzzy negations N are presented. Some general properties are analysed and particular cases when the considered fuzzy negation is the classical one or any strong negation are highlighted.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aguiló, I., Suñer, J., Torrens, J.: New types of contrapositivisation of fuzzy implications with respect to fuzzy negations. Inf. Sci. 322, 223–226 (2015)
Baczyński, M., Beliakov, G., Bustince Sola, H., Pradera, A. (eds.): Advances in Fuzzy Implication Functions. Studies in Fuzziness and Soft Computing, vol. 300. Springer, Heidelberg (2013)
Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing, vol. 231. Springer, Heidelberg (2008)
Baczyński, M., Jayaram, B., Massanet, S., Torrens, J.: Fuzzy implications: past, present, and future. In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pp. 183–202. Springer, Heidelberg (2015)
Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Studies in Fuzziness and Soft Computing, vol. 221. Springer, Heidelberg (2007)
Clifford, A.H.: Naturally totally ordered commutative semigroups. Am. J. Math. 76(3), 631–646 (1954)
De Baets, B., Mesiar, R.: Residual implicators of continuous t-norms. In: Proceedings of EUFIT 1996, pp. 37–41 (1996)
Fodor, J.C., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)
Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation Functions. Encyclopedia of Mathematics and Its Applications, vol. 127. Cambridge University Press, New York (2009)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)
Klement, E.P., Mesiar, R., Pap, E.: Triangular norms as ordinal sums of semigroups in the sense of A.H. Clifford. Semigroup Forum 65, 71–82 (2002)
Hlinená, D., Kalina, M., Kral, P.: Implications functions generated using functions of one variable. In: [2], pp. 125–153 (2013)
Mas, M., Monserrat, M., Torrens, J., Trillas, E.: A survey on fuzzy implication functions. IEEE Trans. Fuzzy Syst. 15(6), 1107–1121 (2007)
Massanet, S., Torrens, J.: On a new class of fuzzy implications: h-Implications and generalizations. Inf. Sci. 181, 2111–2127 (2011)
Massanet, S., Torrens, J.: Threshold generation method of construction of a new implication from two given ones. Fuzzy Sets Syst. 205, 50–75 (2012)
Massanet, S., Torrens, J.: On the vertical threshold generation method of fuzzy implication and its properties. Fuzzy Sets Syst. 226, 232–252 (2013)
Massanet, S., Torrens, J.: An overview of construction methods of fuzzy implications. In: [2], pp. 1–30 (2013)
Mesiar, R., Mesiarová, A.: Residual implications and left-continuous t-norms which are ordinal sums of semigroups. Fuzzy Sets Syst. 143, 47–57 (2004)
Mesiarová, A.: Continuous triangular subnorms. Fuzzy Sets Syst. 142, 75–83 (2004)
Shi, Y., Van Hasse, B., Ruan, D., Kerre, E.: Fuzzy implications: classification and a new class. In: [2], pp. 31–53 (2013)
Su, Y., Xie, A., Liu, H.: On ordinal sum implications. Inf. Sci. 293, 251–262 (2015)
Trillas, E., Mas, M., Monserrat, M., Torrens, J.: On the representation of fuzzy rules. Int. J. Approximate Reason. 48, 583–597 (2008)
Vemuri, N.R., Jayaram, B.: Representations through a monoid on the set of fuzzy implications. Fuzzy Sets Syst. 247, 51–67 (2014)
Vemuri, N.R., Jayaram, B.: The composition of fuzzy implications: closures with respect to properties, powers and families. Fuzzy Sets Syst. 275, 58–87 (2015)
Acknowledgement
This paper has been partially supported by the Spanish Grant TIN2013-42795-P.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Massanet, S., Riera, J.V., Torrens, J. (2016). A New Look on the Ordinal Sum of Fuzzy Implication Functions. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 610. Springer, Cham. https://doi.org/10.1007/978-3-319-40596-4_34
Download citation
DOI: https://doi.org/10.1007/978-3-319-40596-4_34
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-40595-7
Online ISBN: 978-3-319-40596-4
eBook Packages: Computer ScienceComputer Science (R0)