Aggregation operators on partially ordered sets and their categorical foundations

Mustafa Demirci

Kybernetika (2006)

  • Volume: 42, Issue: 3, page 261-277
  • ISSN: 0023-5954

Abstract

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In spite of increasing studies and investigations in the field of aggregation operators, there are two fundamental problems remaining unsolved: aggregation of L -fuzzy set-theoretic notions and their justification. In order to solve these problems, we will formulate aggregation operators and their special types on partially ordered sets with universal bounds, and introduce their categories. Furthermore, we will show that there exists a strong connection between the category of aggregation operators on partially ordered sets with universal bounds (Agop) and the category of partially ordered groupoids with universal bounds (Pogpu). Moreover, the subcategories of Agop consisting of associative aggregation operators, symmetric and associative aggregation operators and associative aggregation operators with neutral elements are, respectively, isomorphic to the subcategories of Pogpu formed by partially ordered semigroups, commutative partially ordered semigroups and partially ordered monoids in the sense of Birkhoff. As a justification of the present notions and results, some relevant examples for aggregations operators on partially ordered sets are given. Particularly, aggregation process in probabilistic metric spaces is also considered.

How to cite

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Demirci, Mustafa. "Aggregation operators on partially ordered sets and their categorical foundations." Kybernetika 42.3 (2006): 261-277. <http://eudml.org/doc/33804>.

@article{Demirci2006,
abstract = {In spite of increasing studies and investigations in the field of aggregation operators, there are two fundamental problems remaining unsolved: aggregation of $L$-fuzzy set-theoretic notions and their justification. In order to solve these problems, we will formulate aggregation operators and their special types on partially ordered sets with universal bounds, and introduce their categories. Furthermore, we will show that there exists a strong connection between the category of aggregation operators on partially ordered sets with universal bounds (Agop) and the category of partially ordered groupoids with universal bounds (Pogpu). Moreover, the subcategories of Agop consisting of associative aggregation operators, symmetric and associative aggregation operators and associative aggregation operators with neutral elements are, respectively, isomorphic to the subcategories of Pogpu formed by partially ordered semigroups, commutative partially ordered semigroups and partially ordered monoids in the sense of Birkhoff. As a justification of the present notions and results, some relevant examples for aggregations operators on partially ordered sets are given. Particularly, aggregation process in probabilistic metric spaces is also considered.},
author = {Demirci, Mustafa},
journal = {Kybernetika},
keywords = {category theory; aggregation operator; associative aggregation operator; partially ordered groupoid; partially ordered semigroup; partially ordered monoid; category theory; associative aggregation operator; partially ordered groupoid; partially ordered semigroup; partially ordered monoid},
language = {eng},
number = {3},
pages = {261-277},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Aggregation operators on partially ordered sets and their categorical foundations},
url = {http://eudml.org/doc/33804},
volume = {42},
year = {2006},
}

TY - JOUR
AU - Demirci, Mustafa
TI - Aggregation operators on partially ordered sets and their categorical foundations
JO - Kybernetika
PY - 2006
PB - Institute of Information Theory and Automation AS CR
VL - 42
IS - 3
SP - 261
EP - 277
AB - In spite of increasing studies and investigations in the field of aggregation operators, there are two fundamental problems remaining unsolved: aggregation of $L$-fuzzy set-theoretic notions and their justification. In order to solve these problems, we will formulate aggregation operators and their special types on partially ordered sets with universal bounds, and introduce their categories. Furthermore, we will show that there exists a strong connection between the category of aggregation operators on partially ordered sets with universal bounds (Agop) and the category of partially ordered groupoids with universal bounds (Pogpu). Moreover, the subcategories of Agop consisting of associative aggregation operators, symmetric and associative aggregation operators and associative aggregation operators with neutral elements are, respectively, isomorphic to the subcategories of Pogpu formed by partially ordered semigroups, commutative partially ordered semigroups and partially ordered monoids in the sense of Birkhoff. As a justification of the present notions and results, some relevant examples for aggregations operators on partially ordered sets are given. Particularly, aggregation process in probabilistic metric spaces is also considered.
LA - eng
KW - category theory; aggregation operator; associative aggregation operator; partially ordered groupoid; partially ordered semigroup; partially ordered monoid; category theory; associative aggregation operator; partially ordered groupoid; partially ordered semigroup; partially ordered monoid
UR - http://eudml.org/doc/33804
ER -

References

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  1. Adámek J., Herrlich, H., Strecker G. E., Abstract and Concrete Categories, Wiley, New York 1990 Zbl1113.18001MR1051419
  2. Atanassov K. T., 10.1016/S0165-0114(86)80034-3, Fuzzy Sets and Systems 20 (1986), 87–96 (1986) Zbl0631.03040MR0852871DOI10.1016/S0165-0114(86)80034-3
  3. Birkhoff G., Lattice Theory, Third edition. Amer. Math. Soc. Colloquium Publications, Amer. Math. Soc., Providence, RI 1967 Zbl0537.06001MR0227053
  4. Calvo T., Mayor, G., (eds.) R. Mesiar, Aggregation Operators, New Trends and Applications. (Studies in Fuzziness and Soft Computing, Vol. 97.) Physica-Verlag, Heidelberg 2002 Zbl0983.00020MR1936383
  5. Deschrijver G., Kerre E. E., On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems 133 (2003), 227–235 Zbl1013.03065MR1949024
  6. Deschrijver G., Kerre E. E., Implicators based on binary aggregation operators in interval-valued fuzzy set theory, Fuzzy Sets and Systems 153 (2005), 229–248 Zbl1090.03024MR2150282
  7. Deschrijver G., Kerre E. E., Triangular norms and related operators in L * -fuzzy set theory, In: Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms (E. P. Klement and R. Mesiar, eds.), Elsevier, Amsterdam 2005, pp. 231–259 Zbl1079.03043MR2165237
  8. Dubois D., Prade H., Fuzzy Sets and Systems, Theory and Applications. Academic Press, New York 1980 Zbl0444.94049MR0589341
  9. Fodor J. C., Yager R. R., Rybalov A., 10.1142/S0218488597000312, Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 5 (1997), 411–427 (1997) Zbl1232.03015MR1471619DOI10.1142/S0218488597000312
  10. Höhle U., (eds.) S. E. Rodabaugh, Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer Academic Publishers, Boston 1999 Zbl0942.00008MR1788899
  11. Klement E. P., Mesiar, R., Pap E., Triangular Norms, Vol. 8 of Trends in Logic. Kluwer Academic Publishers, Dordrecht 2000 Zbl1087.20041MR1790096
  12. Lázaro J., Calvo T., XAO Operators – The interval universe, In: Proc. 4th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2005), pp. 198–203 
  13. Mizumoto M., Tanaka K., 10.1016/S0019-9958(76)80011-3, Inform. and Control 31 (1976), 312–340 (1976) Zbl0331.02042MR0449947DOI10.1016/S0019-9958(76)80011-3
  14. Sambuc R., Fonctions Φ -floues, Application à l’aide au diagnostic en pathologie thyroidienne, Ph. D. Thesis, Université de Marseille 1975 
  15. Saminger S., Mesiar, R., Bodenhofer U., 10.1142/S0218488502001806, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 10 (2002), 11–35 Zbl1053.03514MR1962666DOI10.1142/S0218488502001806
  16. Schweizer B., Sklar A., Probabilistic Metric Spaces, North–Holland, New York 1983 Zbl0546.60010MR0790314
  17. Zadeh L. A., 10.1016/S0019-9958(65)90241-X, Inform. and Control 8 (1965), 338–353 (1965) Zbl0139.24606MR0219427DOI10.1016/S0019-9958(65)90241-X

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