On the structure of continuous uninorms

Paweł Drygaś

Kybernetika (2007)

  • Volume: 43, Issue: 2, page 183-196
  • ISSN: 0023-5954

Abstract

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Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation U in the unit interval with the neutral element e ∈ [ 0 , 1 ] . If operation U is continuous, then e = 0 or e = 1 . So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element e ∈ ( 0 , 1 ) , which is continuous in the open unit square may be given in [ 0 , 1 ) 2 or ( 0 , 1 ] 2 as an ordinal sum of a semigroup and a group. This group is isomorphic to the positive real numbers with multiplication. As a corollary we obtain the results of Hu, Li [7].

How to cite

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Drygaś, Paweł. "On the structure of continuous uninorms." Kybernetika 43.2 (2007): 183-196. <http://eudml.org/doc/33850>.

@article{Drygaś2007,
abstract = {Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation $U$ in the unit interval with the neutral element $e\in [0,1]$. If operation $U$ is continuous, then $e=0$ or $e=1$. So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element $e\in (0,1)$, which is continuous in the open unit square may be given in $[0,1)^2$ or $(0,1]^2$ as an ordinal sum of a semigroup and a group. This group is isomorphic to the positive real numbers with multiplication. As a corollary we obtain the results of Hu, Li [7].},
author = {Drygaś, Paweł},
journal = {Kybernetika},
keywords = {uninorms; continuity; $t$-norms; $t$-conorms; ordinal sum of semigroups; uninorms; continuity; ordinal sum of semigroups},
language = {eng},
number = {2},
pages = {183-196},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On the structure of continuous uninorms},
url = {http://eudml.org/doc/33850},
volume = {43},
year = {2007},
}

TY - JOUR
AU - Drygaś, Paweł
TI - On the structure of continuous uninorms
JO - Kybernetika
PY - 2007
PB - Institute of Information Theory and Automation AS CR
VL - 43
IS - 2
SP - 183
EP - 196
AB - Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation $U$ in the unit interval with the neutral element $e\in [0,1]$. If operation $U$ is continuous, then $e=0$ or $e=1$. So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element $e\in (0,1)$, which is continuous in the open unit square may be given in $[0,1)^2$ or $(0,1]^2$ as an ordinal sum of a semigroup and a group. This group is isomorphic to the positive real numbers with multiplication. As a corollary we obtain the results of Hu, Li [7].
LA - eng
KW - uninorms; continuity; $t$-norms; $t$-conorms; ordinal sum of semigroups; uninorms; continuity; ordinal sum of semigroups
UR - http://eudml.org/doc/33850
ER -

References

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  8. Jenei S., A note on the ordinal sum theorem and its consequence for the construction of triangular norm, Fuzzy Sets and Systems 126 (2002), 199–205 MR1884686
  9. Klement E. P., Mesiar, R., Pap E., Triangular Norms, Kluwer Academic Publishers, Dordrecht 2000 Zbl1087.20041MR1790096
  10. Li Y.-M., Shi Z.-K., Remarks on uninorm aggregation operators, Fuzzy Sets and Systems 114 (2000), 377–380 Zbl0962.03052MR1775275
  11. Mas M., Monserrat, M., Torrens J., On left and right uninorms, Internat. J. Uncertain. Fuzziness Knowledge–Based Systems 9 (2001), 491–507 Zbl1045.03029MR1852342
  12. Sander W., Associative aggregation operators, In: Aggregation Operators (T. Calvo, G. Mayor, and R. Mesiar, eds), Physica–Verlag, Heidelberg 2002, pp. 124–158 Zbl1025.03054MR1936386
  13. Yager R., Rybalov A., Uninorm aggregation operators, Fuzzy Sets and Systems 80 (1996), 111–120 (1996) Zbl0871.04007MR1389951

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