Aggregating Fuzzy QL- and (S,N)-Subimplications: Conjugate and Dual Constructions

Renata Hax Sander Reiser, Íbero C. K. Benítez, Adenauer C. Yamin, Benjamín R. C. Bedregal

Abstract


Fuzzy (S,N)- and QL-subimplication classes can be obtained by a distributive n-ry aggregation  performed over the families of t-subnorms and t-subconorms along with a fuzzy negation. Since these classes of subimplications are explicitly represented by t-subconorms and t-subnorms verifying the generalized associativity, the corresponding (S,N)- and QL-subimplications,  referred as I(S,N) and I_(S,T,N), are  characterized as distributive n-ary aggregation together with related generalizations as the exchange and neutrality principles. Based on these results, the both subclasses I_(S,n) and I_QL of (S,N)- and QL-subimplications which are obtained by the median aggregation operation performed over the standard negation N_S together with  the families  of t-subnorms and t-subconorms S_P and T_P, respectively. In particular, the subclass T_P extends the product t-norm T_P as well as S_P extends the algebraic sum S_P. As the main results, the family of subimplications I_(S_P,N) and I_(S_P,T_P,N) extends the implication class by preserving the corresponding properties. We also present an extension from (S,N)- and QL-subimplications to (S,N)- and QL-implications and discuss dual and conjugate constructions.


Keywords


Median aggregation; t-sub(co)norms; Fuzzy (sub)implications; QL-implications; (S,N)-implications.

Full Text:

PDF

References


G. Mayor and J.Monreal, “Additive generators of discrete conjunctive aggregation op- erations”, IEEE Transactions on Fuzzy Systems, 15, No. 6 (2007) 1046–1052.

S.Rasheed, D.Stashuk, and M.S.Kamel, “Integrating heterogeneous classifier ensembles for EMG-signal decomposition based on classifier agreement”, IEEE Transactions on Fuzzy Systems, 14, No. 3 (2010) 866–882.

H. Izakian, W. Pedrycz and I. Jam, “Clustering spatio temporal data: an augmented fuzzy C-Means”, IEEE Transaction on Fuzzy Systems, 21, No.5 (2013), 855–868.

B. Geng, J. K. Mills and D. Sun, “Two-Stage charging strategy for plug-In electric vehicles at the residential transformer level”, IEEE Transaction on Smart Grid, 4, No.3 (2013), 1442–1452.

E. P. Klement, R. Mesiar and E. Pap, Triangular Norms, Dordrecht: Kluwer Academic Publisher, 2000.

T. Calvo, A. Kolesárová, M. Kormoníková, and R. Mesiar, “Aggregation operators: Properties, classes and construction methods”, in Aggregation Operators New Trends and Applications (T. Calvo, G. Mayor, and R. Mesiar, Eds.), Studies in Fuzziness and Soft Computing, Vol. 97, pp. 3–106, Springer, 2002.

V. Torra, “Aggregation operators and models”, Fuzzy Sets Systems, 156 No. 3 (2005) 407–410.

H. Bustince, T. Calvo, B. D. Baets, J. C. Fodor, R. Mesiar, J. Monteiro, D Paternain and A. Pradera, “A class of aggregation fucntions encopassing two-dimensional OWA operations”, Inf. Science, 180 No. 10 (2010) 1977–1989.

G. Beliakov, H. Bustince and J. Fernandez, “The median and its extensions”, Fuzzy Sets and Systems, 1175 (2011) 36–47.

J. Wang, K. Li and H. Zhang, “Multi-criteria decision-making method based on in- duced intuitionistic normal fuzzy related aggregation operators”, Intl. Journal of Un- certainty, Fuzziness and Knowledge-Based Systems, 20 No. 04 (2012) 559–578.

H. Bustince, M. Galar, B. Bedregal, A. K. a, and R. Mesiar, “A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications”, IEEE Transaction on Fuzzy Systems, 21 No. 6 (2013) 1150–1162.

G. Cornelis, G. Deschrijver and E. Kerre “Implications in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification and application”, Intl. Journal of Approximate Reasoning, 35 (2004) 55–95.

G. Deschrijver and E. Kerre, “Implicators based on binary aggregation operators in interval-valued fuzzy set theory”, Fuzzy Sets and Systems, 153 No. 2 (2005) 229–248.

K. T. Atanassov, “My personal view on intuitionistic fuzzy sets theory”, in Fuzzy Sets and Their Extensions: Representation, Aggregation and Models, (H. B. Sola, F. Her- rera, and J. Montero, Eds.) Ser. Studies in Fuzziness and Soft Computing, Vol. 220, pp. 23–43, Springer, 2008.

B. Bedregal, G. Dimuro, R. Santiago and R. Reiser, “On interval fuzzy S- implications”, Inf. Science, 180 (2010) 1373–1389.

G. Dimuro, B. Bedregal, R. Santiago and R. Reiser, “Interval additive generators of interval t-norms and interval t-conorms”, Inf. Sciences, 181 No. 18 (2011) 3898–3916.

G. Beliakov, H. Bustince, S. James, T. Calvo and J. Fernandez, “Aggregation for Atanassov’s intuitionistic and interval valued fuzzy sets: the median operator”, IEEE Transactions on Fuzzy Systems, 20, pp. 487–498, 2012.

R. Reiser and B. Bedregal, “Interval-valued intuitionistic fuzzy implications – con- struction, properties and representability”, Inf. Sciences, 248 no. 1 (2013) 68–88.

R.Reiser,B.Bedregal,andG.A.A.dosReis,“Interval-valuedfuzzycoimplications”, Journal of Computer and System Sciences, 80, No. 2 (2014) 10–425.

D. Li, L. Wang, and G. Chen, “Group decision making methodology based on the Atanassov’s intuitionistic fuzzy set generalized OWA operator”, Intl. Journal of Un- certainty, Fuzziness and Knowledge-Based Systems, 18 no. 6 (2010) 801–817.

C. da Costa, B. Bedregal, and A. D. Neto, “Relating De Morgan triples with Atanassov’s intuitionistic De Morgan triples via automorphisms”, Intl. J. Approx. Rea- soning, 52 no. 4 (2011) 473–487.

J. Lin and Q. Zhang, “Some continuous aggregation operators with interval-valued intuitionistic fuzzy Inf. and their application to decision making”, Intl. Journal of Un- certainty, Fuzziness and Knowledge-Based Systems, 20 no. 2 (2012) 185–209.

H. Bustince, E. Barrenechea, and M. Pagola, “Generation of interval-valued fuzzy and Atanassov’s intuitionistic fuzzy connectives from fuzzy connectives and from kα operators: Law of conjuntions and disjuntions, amplitute”, Intl. Journal of Intelligent Systems, 23 (2008) 680–714.

L. Visintin, R. Reiser, and B. Bedregal, “Interval-valued intuitionistic fuzzy impli- cations”, in IEEE Pos-Proceedings of the Workshop-School on Theoretical Computer Sciences, Pelotas, 2011, pp. 46–52, DOI:10.1109/WEIT.2011.22.

M. Xia and Z. Xu, “Hesitant fuzzy Inf. aggregation in decision making”, Intl. Journal of Approximate Reasoning, 52 no. 3 (2011) 395–407.

M. Xia, Z. Xu, and N. Chen, “Some hesitant fuzzy aggregation operators with their application in group decision making,” Group Decision and Negotiation, 22 no. 2 (2013) 259–279.

B. Bedregal, R. Reiser, H. Bustince, C. Lopez-Molina, and V. Torra, “Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms”, Inf. Sciences, 255 no. 1 (2014) 82–99.

R. Reiser, B. Bedregal and M. Baczyn ́ski, “Aggregating fuzzy implications”, Inf. Sci- ence, 253 (2013) 126–146.

I. Benitez, R. Reiser, A. Yamin and B. Bedregal, “Aggregating Fuzzy QL- Implications”, IEEE Xplore Digital Library Pos-Proceedings of 3rd Workshop-School on Theoretical Computer Science, pp. 121 – 128, DOI:10.1109/WEIT.2013.11.

Y. Shi, B. V. Gasse, D. Ruan and E. E. Kerre, “On the first place antitonicity in QL- implications”, Fuzzy Sets and Systems, 159 no. 22 (2008) 2988–3013.

E. Klement, R. Mesiar and E. Pap, “Triangular norms. position paper I: basic analyti- cal and algebraic properties”, Fuzzy Sets and Systems, 143 no. 1 (2004) 5–26.

M.Baczyn ́ski and B. Jayaram, “QL-implications: Some properties and intersections”, Fuzzy Sets and Systems, 161 (2010) 158–188.

L. Kitainik, Fuzzy Decision Procedures with Binary Relations.

Dordrecht: Kluwer Academic Publisher, 1993.

H. Bustince, P. Burillo and F. Soria, “Automorphisms, negations and implication operators”, Fuzzy Sets Systems, 134 no. 2 (2003) 209– 229.

M.Baczyn ́ski and B. Jayaram, “(S,N)- and R-implications: A state-of-the-art survey”, Fuzzy Sets and Systems, 159 no. 14 (2008) 1836–1859.

M. Mas, M. Monserrat and J. Torrens, “On interval fuzzy negations”, Fuzzy Sets and Systems, 158 (2007) 2612–2626.

G. Maksa, “Quasisums and generalized associativity”, Aequationes Mathematicae, 69 no. 1-2 (2005) 6–27.




DOI: https://doi.org/10.5540/tema.2015.016.03.0229

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Refbacks

  • There are currently no refbacks.



Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

 

Indexed in:

                       

 

Desenvolvido por:

Logomarca da Lepidus Tecnologia