A class of aggregation functions encompassing two-dimensional OWA operators H. Bustince a, * , T. Calvo b , B. De Baets c , J. Fodor d , R. Mesiar e,f , J. Montero g , D. Paternain a , A. Pradera h a Departamento de Automática y Computación, Universidad Pública de Navarra, Campus Arrosadia s/n, P.O. Box 31006, Pamplona, Spain b Universidad de Alcalá, Spain c Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, 9000 Gent, Belgium d Institute of Intelligent Engineering Systems, Budapest Tech, Bécsi út 96/b, H-1034 Budapest, Hungary e Department of Mathematics and Descriptive Geometry, Slovak University of Technology, SK-813 68 Bratislava, Slovakia f Institute of Information Theory and Automation, Czech Academy of Sciences, CZ-182 08 Prague, Czech Republic g Facultad de Matemáticas, Universidad Complutense, 28040 Madrid, Spain h Departamento de Ciencias de la Computación, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain article info Article history: Received 3 August 2009 Received in revised form 20 November 2009 Accepted 21 January 2010 Keywords: OWA operators Interval-valued fuzzy sets Ka operators Generalized Ka operators Dispersion abstract In this paper we prove that, under suitable conditions, Atanassov’s K a operators, which act on intervals, provide the same numerical results as OWA operators of dimension two. On one hand, this allows us to recover OWA operators from K a operators. On the other hand, by analyzing the properties of Atanassov’s operators, we can generalize them. In this way, we introduce a class of aggregation functions – the generalized Atanassov operators – that, in particular, include two-dimensional OWA operators. We investigate under which condi- tions these generalized Atanassov operators satisfy some properties usually required for aggregation functions, such as bisymmetry, strictness, monotonicity, etc. We also show that if we apply these aggregation functions to interval-valued fuzzy sets, we obtain an ordered family of fuzzy sets. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction In 1983 Atanassov introduced a new operator [2] allowing to associate a fuzzy set with each Atanassov intuitionistic fuzzy set or interval-valued fuzzy set (IVFS) [17,20]. In fact, this operator, which we denote by K a , takes a value from the interval representing the membership to the IVFS and defines that value to be the membership degree to a fuzzy set [26,27]. In this way, it is possible, for instance, to recover all the usual fuzzy set theoretic results when dealing with IVFS. In 1988 Yager presented the definition of an OWA operator [22]. Comparison of the results of Atanassov and Yager reveals that in two dimensions the numerical results provided by Ata- nassov operators and OWA operators are the same. This numerical coincidence prompted us to introduce and define new operators by suitably modifying the domain for the definition of Atanassov’s operators. Analysis of the properties required for Atanassov’s operators has allowed us to consider a class of aggregation functions that are a generalization of Atanassov’s operators [6–8]. In particular, it would be interesting to determine whether some of the properties that are usually required for aggregation functions, such as bisymmetry, strictness, monotonicity, etc., also hold for this class of generalized Atanassov operators. 0020-0255/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2010.01.022 * Corresponding author. E-mail addresses: bustince@unavarra.es (H. Bustince), daniel.paternain@unavarra.es (D. Paternain). Information Sciences 180 (2010) 1977–1989 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins