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 -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 -

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