Weighted ordinal means Anna Kolesa ´rova ´ a , Gaspar Mayor b , Radko Mesiar c,d, * a Institute IAM, FCHFT, Slovak University of Technology, Radlinske ´ho 9, 812 7 Bratislava, Slovakia b Department of Mathematics and Computer Science, University of the Balearic Islands, Palma de Mallorca, Spain c Department of Mathematics and Descriptive Geometry, SvF, Slovak University of Technology, Radlinske ´ho 11, 813 68 Bratislava, Slovakia d U ´ TIA AV C ˇ R, 182 08 Prague, Czech Republic Received 24 November 2006; received in revised form 2 March 2007; accepted 10 March 2007 Abstract The concept of weighted ordinal arithmetic means and other related weighted ordinal means is studied. Based on the relevant results on the scale [0, 1], new types of weighted ordinal means are proposed. In some cases these ordinal means coincide with those proposed by Godo and Torra, but not in the case when ordinal means introduced by them are not idempotent. Based on divisible ordinal t-conorms and modifying the approach of Godo and Torra, we show how the pre- viously introduced weighted ordinal means can be obtained without exploiting the formal similarity of the structure of continuous t-conorms on [0, 1] and divisible ordinal t-conorms. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Ordinal aggregation function; Ordinal mean; Divisible t-conorm 1. Introduction Typical means on the cardinal scale [0, 1] are the arithmetic mean M and several of its generalizations, such as quasi-arithmetic means, weighted arithmetic means, weighted quasi-arithmetic means, OWA operators, weighted ordered quasi-arithmetic means. Recall that OWA operators can be viewed as weighted arithmetic means applied not directly to given input values, but to ordered ones. Weighted ordered quasi-arithmetic means are defined in a similar way. Moreover, the class of weighted quasi-arithmetic means contains M, and also quasi-arithmetic and weighted arithmetic means. Calvo and Mesiar [3] have introduced weighted (continuous) t-conorms and their class of aggregation func- tions also covers all weighted quasi-arithmetic means for which 0 is not an annihilator. Observe that by duality weighted (continuous) t-norms can be introduced, covering all weighted quasi-arithmetic means for which the element 1 is not an annihilator, and thus the union of both these classes contains all quasi-arithmetic means. 0020-0255/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2007.03.003 * Corresponding author. Address: Department of Mathematics and Descriptive Geometry, SvF, Slovak University of Technology, Radlinske ´ho 11, 813 68 Bratislava, Slovakia. Fax: +421 252967027. E-mail addresses: anna.kolesarova@stuba.sk (A. Kolesa ´rova ´), gmayor@uib.es (G. Mayor), mesiar@math.sk (R. Mesiar). Information Sciences 177 (2007) 3822–3830 www.elsevier.com/locate/ins