JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (9): 36-42.doi: 10.6040/j.issn.1671-9352.0.2018.560

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Distributivity of fuzzy implications over additively generated overlap and grouping functions

ZHANG Ting-hai, QIN Feng   

  1. College of Mathematics and Informatics, Jiangxi Normal University, Nanchang 330022, Jiangxi, China
  • Online:2019-09-20 Published:2019-07-30

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Abstract: Among the researches of the fuzzy system, the distributivity between fuzzy implications and some special aggregation functions(e.g. t-norms, t-conorms, uninorms, t-operators, semi-uninorms and semi-t-operators)has been studied by many authors. Overlap functions and grouping functions have been followed with interest for their applications in image processing, classification problems and decision making based on fuzzy preference relations. In this paper, we give necessary and sufficient conditions for two distributivity equations I(O(x,y),z)=G(I(x,z),I(y,z)) and I(x,G1(y,z))=G2(I(x,y),I(x,z)), where O is an additively generated overlap function, G1 and G2 are additively generated grouping functions and I is a binary function satisfying boundary conditions.

Key words: fuzzy implication, overlap function, grouping function, Cauchy function equation, distributivity

CLC Number: 

  • O159
[1] BACZYNSKI M, JAYARAM B. Fuzzy implications[M]. Berlin: Springer, 2008.
[2] 杨丽,覃锋.构造模糊蕴涵的新序和方法[J].江西师范大学学报(自然科学版),2018,42(3):254-259. YANG Li, QIN Feng. The novel method of constructing fuzzy implications by ordinal sum[J]. Journal of Jiangxi Normal University(Natural Science), 2018, 42(3):254-259.
[3] BACZYNSKI M, JAYARAM B.(S,N)-and R-implications: a state-of-the-art survey[J]. Fuzzy Sets and Systems, 2008, 159(14):1836-1859.
[4] BACZYNSKI M, JAYARAM B. On the characterization of (S,N)-implications[J]. Fuzzy Sets and Systems, 2007, 158:1713-1727.
[5] FODOR J C. Contrapositive symmetry of fuzzy implications[J]. Fuzzy Sets and Systems, 1995, 69(2):141-156.
[6] JAYARAM B. On the law of importation (x∧y)→z≡x→(y→z) in fuzzy logic[J]. IEEE Transactions on Fuzzy Systems, 2008,16(1):130-144.
[7] MAS M, MONSERRAT M, TORRENS J. QL-implications versus D-implications[J]. Kybernetika, 2006, 42(3):351-366.
[8] DE BAETS B. Idempotent uninorms[J]. European Journal of Operational Research, 1999, 118(3):631-642.
[9] DIMURO G P, BEDREGAL B. On residual implications derived from overlap functions[J]. Information Sciences, 2015, 312:78-88.
[10] DIMURO G P, BEDREGAL B, SANTIAGO R H N. On(G,N)-implications derived from grouping functions[J]. Information Sciences, 2014, 379:1-17.
[11] BALASUBRAMANIAM J, RAO C J M. On the distributivity of implication operators over T and S norms[J]. IEEE Transactions on Fuzzy Systems, 2004, 12(2):194-198.
[12] RUIZ-AGUILERA D, TORRENS J. Distributivity of strong implications over conjunctive and disjunctive uninorms[J]. Kybernetika, 2006, 42(3):319-336.
[13] RUIZ-AGUILERa D, TORRENS J. Distributivity of residual implications over conjunctive and disjunctive uninorms[J]. Fuzzy Sets and Systems, 2007, 158(1):23-37.
[14] QIN F, YANG L. Distributive equations of implications based on nilpotent triangular norms[J]. International Journal of Approximate Reasoning, 2010, 51(8):984-992.
[15] QIN F, BACZYNSKI M, XIE A F. Distributive equations of implications based on continuous triangular norms(I)[J]. IEEE Transactions on Fuzzy Systems, 2012, 20(1):153-167.
[16] QIN F, YANG P C. On the distributive equations of implications based on a continuous t-norms and a continuous Archimedean t-conorm[C] // International Conference on Biomedical Engineering and Informatics. USA: IEEE, 2011: 2290-2294.
[17] QIN F, BACZYNSKI M. Distributive Equations of implications based on continuous triangular conorms(II)[J]. Fuzzy Sets and Systems, 2014, 240:86-102.
[18] QIAO J S, HU B Q. The distributive laws of fuzzy implications over overlap and grouping functions[J]. Information Sciences, 2018, 438:107-126.
[19] BUSTINCE H, PAGOLA M, MESIAR R, et al. Grouping, overlaps and generalized bientropic functions for fuzzy modeling of pairwise comparisons[J]. IEEE Transactions on Fuzzy Systems, 2012, 20:405-415.
[20] DIMURO G P, BEDREGAL B, BUSTINCE H, et al. On additive generators of overlap functions[J]. Fuzzy Sets and Systems, 2016, 287:76-96.
[21] DIMURO G P, BEDREGAL B, BUSTINCE H, et al. On additive generators of grouping functions[M] // LAURENT A, STRAUSS O, BOUCHON-MEUNIER B, et al. Information Processing and Management of Uncertainty in Knowledge-Based Information Science. Berlin: Springer, 2014, 444:252-261.
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