On a special class of left-continuous uninorms

Gang Li

Kybernetika (2018)

  • Volume: 54, Issue: 3, page 427-442
  • ISSN: 0023-5954

Abstract

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This paper is devoted to the study of a class of left-continuous uninorms locally internal in the region A ( e ) and the residual implications derived from them. It is shown that such uninorm can be represented as an ordinal sum of semigroups in the sense of Clifford. Moreover, the explicit expressions for the residual implication derived from this special class of uninorms are given. A set of axioms is presented that characterizes those binary functions I : [ 0 , 1 ] 2 → [ 0 , 1 ] for which a uninorm U of this special class exists in such a way that I is the residual implications derived from U .

How to cite

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Li, Gang. "On a special class of left-continuous uninorms." Kybernetika 54.3 (2018): 427-442. <http://eudml.org/doc/294184>.

@article{Li2018,
abstract = {This paper is devoted to the study of a class of left-continuous uninorms locally internal in the region $A(e)$ and the residual implications derived from them. It is shown that such uninorm can be represented as an ordinal sum of semigroups in the sense of Clifford. Moreover, the explicit expressions for the residual implication derived from this special class of uninorms are given. A set of axioms is presented that characterizes those binary functions $I: [0,1]^\{2\}\rightarrow [0,1]$ for which a uninorm $U$ of this special class exists in such a way that $I$ is the residual implications derived from $U$.},
author = {Li, Gang},
journal = {Kybernetika},
keywords = {uninorm; internal operator; ordinal sum; residual implication; triangular subnorm},
language = {eng},
number = {3},
pages = {427-442},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On a special class of left-continuous uninorms},
url = {http://eudml.org/doc/294184},
volume = {54},
year = {2018},
}

TY - JOUR
AU - Li, Gang
TI - On a special class of left-continuous uninorms
JO - Kybernetika
PY - 2018
PB - Institute of Information Theory and Automation AS CR
VL - 54
IS - 3
SP - 427
EP - 442
AB - This paper is devoted to the study of a class of left-continuous uninorms locally internal in the region $A(e)$ and the residual implications derived from them. It is shown that such uninorm can be represented as an ordinal sum of semigroups in the sense of Clifford. Moreover, the explicit expressions for the residual implication derived from this special class of uninorms are given. A set of axioms is presented that characterizes those binary functions $I: [0,1]^{2}\rightarrow [0,1]$ for which a uninorm $U$ of this special class exists in such a way that $I$ is the residual implications derived from $U$.
LA - eng
KW - uninorm; internal operator; ordinal sum; residual implication; triangular subnorm
UR - http://eudml.org/doc/294184
ER -

References

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  1. Aguiló, I., Suñer, J., Torrens, J., 10.1016/j.ins.2010.06.023, Inform. Sci. 180 (2010), 3992-4005. MR2671753DOI10.1016/j.ins.2010.06.023
  2. Alsina, C., Frank, M. J., Schweizer, B., 10.1142/9789812774200, World Scientific, New Jersey 2006. MR2222258DOI10.1142/9789812774200
  3. Baczyński, M., Jayaram, B., Fuzzy Implications., Springer, Berlin, Herdelberg 2008. Zbl1293.03012
  4. Baets, B. De, 10.1007/978-1-4615-2357-4_3, In: Fuzzy Set Theory and Advanced Mathemtical Applications (D. Ruan, ed.), Kluwer, Dordrecht 1995, pp. 67-87. DOI10.1007/978-1-4615-2357-4_3
  5. Baets, B. De, Fodor, J., 10.1007/s005000050057, Soft Comput. 3 (1999), 89-100. MR2391544DOI10.1007/s005000050057
  6. Baets, B. De, Fodor, J., 10.1016/s0165-0114(98)00265-6, Fuzzy Sets Systems 104 (1999), 133-136. Zbl0928.03060MR1685816DOI10.1016/s0165-0114(98)00265-6
  7. Baets, B. De, 10.1016/s0377-2217(98)00325-7, Eur. J. Oper. Res. 118 (1998), 631-642. Zbl1178.03070DOI10.1016/s0377-2217(98)00325-7
  8. Baets, B. De, Kwasnikowska, N., Kerre, E., Fuzzy morphology based on uninorms., In: Seventh IFSA World Congress, Prague, 220 (1997), 215-220. 
  9. Baets, B. De, Fodor, J., Ruiz-Aguilera, D., Torrens, J., 10.1142/s021848850900570x, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 17 (2009), 1-14. Zbl1178.03070MR2514519DOI10.1142/s021848850900570x
  10. Clifford, A. H., 10.2307/2372706, Amer. J. Math. 76 (1954), 631-646. MR0062118DOI10.2307/2372706
  11. Csiszár, O., Fodor, J., On uninorms with fixed values along their border., Ann. Univ. Sci. Bundapest., Sect. Com. 42 (2014), 93-108. MR3275665
  12. Czogała, E., Drewniak, J., 10.1016/0165-0114(84)90072-1, Fuzzy Sets Systems 12 (1984), 249-269. MR0740097DOI10.1016/0165-0114(84)90072-1
  13. Drygaś, P., 10.1016/j.fss.2015.05.018, Kybernetika 41 (2005), 213-226. Zbl1249.03093MR2138769DOI10.1016/j.fss.2015.05.018
  14. Drygaś, P., On the structure of continuous uninorms., Kybernetika 43 (2007), 183-196. Zbl1132.03349MR2343394
  15. Drygaś, P., 10.1016/j.fss.2009.09.017, Fuzzy Sets Systems 161 (2010), 149-157. Zbl1191.03039MR2566236DOI10.1016/j.fss.2009.09.017
  16. Drygaś, P., Ruiz-Aguilera, D., Torrens, J., 10.1016/j.fss.2015.07.015, Fuzzy Sets Systems 287 (2016), 137-153. MR3447023DOI10.1016/j.fss.2015.07.015
  17. Esteva, F., Godo, L., 10.1016/s0165-0114(01)00098-7, Fuzzy Sets Systems 124 (2001), 271-288. MR1860848DOI10.1016/s0165-0114(01)00098-7
  18. Fodor, J., Yager, R. R., Rybalov, A., 10.1142/s0218488597000312, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 5 (1997), 411-427. Zbl1232.03015MR1471619DOI10.1142/s0218488597000312
  19. Fodor, J., Baets, B. De, 10.1016/j.fss.2011.12.001, Fuzzy Sets Systems 202 (2012), 89-99. Zbl1268.03027MR2934788DOI10.1016/j.fss.2011.12.001
  20. Hu, S., Li, Z., 10.1016/s0165-0114(00)00044-0, Fuzzy Sets Systems 124 (2001), 43-52. MR1859776DOI10.1016/s0165-0114(00)00044-0
  21. Jenei, S., 10.1016/s0165-0114(01)00040-9, Fuzzy Sets Systems 126 (2002), 199-205. MR1884686DOI10.1016/s0165-0114(01)00040-9
  22. Klement, E. P., Mesiar, R., Pap, E., Triangular Norms., Kluwer Academic Publishers, Dordrecht 2000. Zbl1087.20041MR1790096
  23. Klement, E. P., Mesiar, R., Pap, E., 10.1142/s0218488500000514, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 8 (2000), 707-717. MR1803475DOI10.1142/s0218488500000514
  24. Li, G., Liu, H-W., Fodor, J., 10.1142/s0218488514500299, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 22 (2014), 591-604. MR3252143DOI10.1142/s0218488514500299
  25. Li, G., Liu, H-W., Fodor, J., 10.14736/kyb-2015-4-0699, Kybernetika 51(4) (2015), 699-711. MR3423195DOI10.14736/kyb-2015-4-0699
  26. Li, G., Liu, H-W., 10.1016/j.fss.2015.01.019, Fuzzy Sets Systems 287 (2016), 154-171. MR3447024DOI10.1016/j.fss.2015.01.019
  27. Li, G., Liu, H-W., 10.1007/978-3-319-46206-6_25, In: Fan TH., Chen SL., Wang SM., Li YM. (eds) Quantitative Logic and Soft Computing 2016. Advances in Intelligent Systems and Computing, vol. 510. Springer, 2017, pp. 251-259. DOI10.1007/978-3-319-46206-6_25
  28. Li, G., Liu, H-W., 10.1016/j.fss.2017.07.014, Fuzzy Sets Systems 332 (2017), 116-128. MR3732254DOI10.1016/j.fss.2017.07.014
  29. Martin, J., Mayor, G., Torrens, J., 10.1016/s0165-0114(02)00430-x, Fuzzy Sets Systems 137(1) (2003), 27-42. Zbl1022.03038MR1992696DOI10.1016/s0165-0114(02)00430-x
  30. Massanet, S., Torrens, J., 10.1016/j.fss.2010.12.012, Fuzzy Sets Systems 168 (2011), 47-69. MR2772620DOI10.1016/j.fss.2010.12.012
  31. Mas, M., Massanet, S., Ruiz-Aguilera, D., Torrens, J., 10.3233/ifs-151728, J. Intell. Fuzzy Systems 29(3) (2015), 1021-1037. MR3414365DOI10.3233/ifs-151728
  32. Mesiarová, A., 10.1016/j.ins.2014.12.060, Inform. Sci. 301 (2015), 227-240. MR3311790DOI10.1016/j.ins.2014.12.060
  33. Mesiarová, A., Characterization of uninorms with continuous underlying t-norm and t-conorm by their set of discontinuity points., IEEE Trans. Fuzzy Systems PP (2017), in press. MR3614252
  34. Mesiarová, A., 10.1016/j.ijar.2017.01.007, Int. J. Approx. Reason. 87 (2017), 176-192. MR3614252DOI10.1016/j.ijar.2017.01.007
  35. Noguera, C., Esteva, F., Godo, L., 10.1016/j.ins.2009.12.011, Inform. Sci. 180 (2010), 1354-1372. MR2587910DOI10.1016/j.ins.2009.12.011
  36. Petrík, M., Mesiar, R., 10.1016/j.fss.2013.09.013, Fuzzy Sets Systems 240 (2014), 22-38. Zbl1315.03099MR3167510DOI10.1016/j.fss.2013.09.013
  37. Pouzet, M., Rosenberg, I. G., Stone, M. G., 10.1007/bf01234102, Algebra Univers. 36(2) (1996), 159-184. MR1402510DOI10.1007/bf01234102
  38. Qin, F., Zhao, B., 10.1016/j.fss.2005.04.010, Fuzzy Sets Systems 155 (2005), 446-458. Zbl1077.03514MR2181001DOI10.1016/j.fss.2005.04.010
  39. Ruiz, D., Torrens, J., Residual implications and co-implications from idempotent uninorms., Kybernetika 40 (2004), 21-38. Zbl1249.94095MR2068596
  40. Ruiz, D., Torrens, J., 10.1109/tfuzz.2005.864087, IEEE Trans. Fuzzy Systems 14 (2006), 2, 180-190. DOI10.1109/tfuzz.2005.864087
  41. Ruiz-Aguilera, D., Torrens, J., 10.1016/j.fss.2008.05.015, Fuzzy Sets Systems 160 (2009), 832-852. MR2493278DOI10.1016/j.fss.2008.05.015
  42. Ruiz-Aguilera, D., Torrens, J., Baets, B. De, Fodor, J., 10.1007/978-3-642-14049-5_44, In: IPMU 2010, LNAI 6178, Eds. E.Hüllermeier, R.Kruse and F.Hoffmann, Springer-Verlag Berlin Heidelberg 2010, pp. 425-434. DOI10.1007/978-3-642-14049-5_44
  43. Ruiz-Aguilera, D., Torrens, J., 10.1016/j.fss.2014.10.020, Fuzzy Sets Syst. 268 (2015), 44-58. MR3320246DOI10.1016/j.fss.2014.10.020
  44. Takács, M., Uninorm-based models for FLC systems., J. Intell. Fuzzy Systems 19 (2008), 65-73. 
  45. Yager, R., Rybalov, A., 10.1016/0165-0114(95)00133-6, Fuzzy Sets Systems 80 (1996), 111-120. Zbl0871.04007MR1389951DOI10.1016/0165-0114(95)00133-6
  46. Yager, R., Rybalov, A., 10.1007/s10700-010-9096-8, Fuzzy Optim. Decis. Making 10 (2011), 59-70. MR2799503DOI10.1007/s10700-010-9096-8
  47. Yager, R., 10.1016/s0165-0114(00)00027-0, Fuzzy Sets Systems 122 (2001), 167-175. Zbl0978.93007MR1839955DOI10.1016/s0165-0114(00)00027-0

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