Left and right semi-uninorms on a complete lattice

Yong Su; Zhudeng Wang; Keming Tang

Kybernetika (2013)

  • Volume: 49, Issue: 6, page 948-961
  • ISSN: 0023-5954

Abstract

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Uninorms are important generalizations of triangular norms and conorms, with a neutral element lying anywhere in the unit interval, and left (right) semi-uninorms are non-commutative and non-associative extensions of uninorms. In this paper, we firstly introduce the concepts of left and right semi-uninorms on a complete lattice and illustrate these notions by means of some examples. Then, we lay bare the formulas for calculating the upper and lower approximation left (right) semi-uninorms of a binary operation. Finally, we discuss the relations between the upper approximation left (right) semi-uninorms of a given binary operation and the lower approximation left (right) semi-uninorms of its dual operation.

How to cite

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Su, Yong, Wang, Zhudeng, and Tang, Keming. "Left and right semi-uninorms on a complete lattice." Kybernetika 49.6 (2013): 948-961. <http://eudml.org/doc/260821>.

@article{Su2013,
abstract = {Uninorms are important generalizations of triangular norms and conorms, with a neutral element lying anywhere in the unit interval, and left (right) semi-uninorms are non-commutative and non-associative extensions of uninorms. In this paper, we firstly introduce the concepts of left and right semi-uninorms on a complete lattice and illustrate these notions by means of some examples. Then, we lay bare the formulas for calculating the upper and lower approximation left (right) semi-uninorms of a binary operation. Finally, we discuss the relations between the upper approximation left (right) semi-uninorms of a given binary operation and the lower approximation left (right) semi-uninorms of its dual operation.},
author = {Su, Yong, Wang, Zhudeng, Tang, Keming},
journal = {Kybernetika},
keywords = {fuzzy connective; uninorm; left (right) semi-uninorm; upper (lower) approximation; fuzzy connective; uninorm; left (right) semi-uninorm; upper (lower) approximation},
language = {eng},
number = {6},
pages = {948-961},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Left and right semi-uninorms on a complete lattice},
url = {http://eudml.org/doc/260821},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Su, Yong
AU - Wang, Zhudeng
AU - Tang, Keming
TI - Left and right semi-uninorms on a complete lattice
JO - Kybernetika
PY - 2013
PB - Institute of Information Theory and Automation AS CR
VL - 49
IS - 6
SP - 948
EP - 961
AB - Uninorms are important generalizations of triangular norms and conorms, with a neutral element lying anywhere in the unit interval, and left (right) semi-uninorms are non-commutative and non-associative extensions of uninorms. In this paper, we firstly introduce the concepts of left and right semi-uninorms on a complete lattice and illustrate these notions by means of some examples. Then, we lay bare the formulas for calculating the upper and lower approximation left (right) semi-uninorms of a binary operation. Finally, we discuss the relations between the upper approximation left (right) semi-uninorms of a given binary operation and the lower approximation left (right) semi-uninorms of its dual operation.
LA - eng
KW - fuzzy connective; uninorm; left (right) semi-uninorm; upper (lower) approximation; fuzzy connective; uninorm; left (right) semi-uninorm; upper (lower) approximation
UR - http://eudml.org/doc/260821
ER -

References

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  1. Baets, B. De, Coimplicators, the forgotten connectives., Tatra Mountains Math. Publ. 12 (1997), 229-240. Zbl0954.03029MR1607142
  2. Baets, B. De, 10.1016/S0377-2217(98)00325-7, European J. Oper. Res. 118 (1999), 631-642. Zbl1178.03070DOI10.1016/S0377-2217(98)00325-7
  3. Baets, B. De, Fodor, J., Van Melle's combining function in MYCIN is a representable uninorm: an alternative proof., Fuzzy Sets and Systems 104 (1999), 133-136. Zbl0928.03060MR1685816
  4. Bassan, B., Spizzichino, F., 10.1016/j.jmva.2004.04.002, J. Multivariate Anal. 93 (2005), 313-339. Zbl1070.60015MR2162641DOI10.1016/j.jmva.2004.04.002
  5. Birkhoff, G., Lattice Theory., American Mathematical Society Colloquium Publishers, Providence 1967. Zbl0537.06001MR0227053
  6. Burris, S., Sankappanavar, H. P., A Course in Universal Algebra., Springer-Verlag, New York 1981. Zbl0478.08001MR0648287
  7. Cooman, G. De, Kerre, E. E., Order norms on bounded partially ordered sets., J. Fuzzy Math. 2 (1994), 281-310. Zbl0814.04005MR1280148
  8. Durante, F., Klement, E. P., al., R. Mesiar et, 10.1007/s00009-007-0122-1, Mediterranean J. Math. 4 (2007), 343-356. MR2349892DOI10.1007/s00009-007-0122-1
  9. Fodor, J., Yager, R. R., Rybalov, A., 10.1142/S0218488597000312, Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems 5 (1997), 411-427. Zbl1232.03015MR1471619DOI10.1142/S0218488597000312
  10. Gabbay, D., Metcalfe, G., 10.1007/s00153-007-0047-1, Arch. Math. Logic 46 (2007), 425-449. Zbl1128.03015MR2321585DOI10.1007/s00153-007-0047-1
  11. Gottwald, S., A Treatise on Many-Valued Logics., Studies in Logic and Computation Vol. 9, Research Studies Press, Baldock 2001. Zbl1048.03002MR1856623
  12. Jenei, S., A characterization theorem on the rotation construction for triangular norms., Fuzzy Sets and Systems 136 (2003), 283-289. Zbl1020.03021MR1984578
  13. Jenei, S., 10.1016/j.fss.2003.06.006, Fuzzy Sets and Systems 143 (2004), 27-45. Zbl1040.03021MR2060271DOI10.1016/j.fss.2003.06.006
  14. Jenei, S., Montagna, F., A general method for constructing left-continuous t -norms., Fuzzy Sets and Systems 136 (2003), 263-282. Zbl1020.03020MR1984577
  15. Liu, H. W., Semi-uninorm and implications on a complete lattice., Fuzzy Sets and Systems 191 (2012), 72-82. MR2874824
  16. Ma, Z., Wu, W. M., 10.1016/0020-0255(91)90007-H, Inform. Sci. 55 (1991), 77-97. Zbl0741.03010MR1080449DOI10.1016/0020-0255(91)90007-H
  17. Mas, M., Monserrat, M., Torrens, J., 10.1142/S0218488501000909, Internat. J. Uncertainly, Fuzziness and Knowledge-Based Systems 9 (2001), 491-507. Zbl1045.03029MR1852342DOI10.1142/S0218488501000909
  18. Mas, M., Monserrat, M., Torrens, J., On left and right uninorms on a finite chain., Fuzzy Sets and Systems 146 (2004), 3-17. Zbl1045.03029MR2074199
  19. Mas, M., Monserrat, M., Torrens, J., Two types of implications derived from uninorms., Fuzzy Sets and Systems 158 (2007), 2612-2626. Zbl1125.03018MR2363783
  20. Ruiz, D., Torrens, J., Residual implications and co-implications from idempotent uninorms., Kybernetika 40 (2004), 21-38. Zbl1249.94095MR2068596
  21. García, F. Suárez, Álvarez, P. Gil, 10.1016/0165-0114(86)90028-X, Fuzzy Sets and Systems 18 (1986), 67-81. MR0825620DOI10.1016/0165-0114(86)90028-X
  22. Tsadiras, A. K., Margaritis, K. G., the MYCIN certainty factor handling function as uninorm operator and its use as a threshold function in artificial neurons., Fuzzy Sets and Systems 93 (1998), 263-274. MR1605312
  23. Wang, Z. D., Yu, Y. D., Pseudo- t -norms and implication operators on a complete Brouwerian lattice., Fuzzy Sets and Systems 132 (2002), 113-124. Zbl1013.03020MR1936220
  24. Wang, Z. D., 10.1016/j.fss.2005.05.047, Fuzzy Sets and Systems 157 (2006), 398-410. Zbl1085.03020MR2186235DOI10.1016/j.fss.2005.05.047
  25. Wang, Z. D., Fang, J. X., Residual operators of left and right uninorms on a complete lattice., Fuzzy Sets and Systems 160 (2009), 22-31. MR2469427
  26. Wang, Z. D., Fang, J. X., Residual coimplicators of left and right uninorms on a complete lattice., Fuzzy Sets and Systems 160 (2009), 2086-2096. Zbl1183.03027MR2555022
  27. Yager, R. R., 10.1016/S0165-0114(00)00027-0, Fuzzy Sets and Systems 122 (2001), 167-175. MR1839955DOI10.1016/S0165-0114(00)00027-0
  28. Yager, R. R., 10.1016/S0377-2217(01)00267-3, European J. Oper. Res. 141 (2002), 217-232. Zbl0998.90046MR1925395DOI10.1016/S0377-2217(01)00267-3
  29. Yager, R. R., Kreinovich, V., Universal approximation theorem for uninorm-based fuzzy systems modeling., Fuzzy Sets and Systems 140 (2003), 331-339. Zbl1040.93043MR2021449
  30. Yager, R. R., Rybalov, A., 10.1016/0165-0114(95)00133-6, Fuzzy Sets and Systems 80 (1996), 111-120. Zbl0871.04007MR1389951DOI10.1016/0165-0114(95)00133-6

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