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Generalizing Leximin to t-norms and t-conorms

By Ronald R. Yager, Carol L. Walker and Elbert A. Walker

Abstract

of [0; 1] n. It is based on the min t-norm. It …rst compares two tuples with respect to their smallest value. If these values are not equal, it indicates the tuple producing the larger of these values as being preferred. If they are equal, then it looks to the second largest argument value, and repeats the process until one is preferred or it runs out of terms to compare, in which case it considers the tuples as equivalent. Thus the Leximin ordering of [0; 1] n is to put the entries into increasing order and use lexicographic ordering. A standard notation for elements in [0; 1] n with coordinates in increasing order is [0; 1] [n]. So the Leximin ordering boils down to lexicographic ordering of [0; 1] [n]. A natural generalization to any t-norm would be this. For a,b 2 [0; 1] [n], if T (a)> T (b) we say that a is preferred over b. In case T (a) = T (b), compare T (a2;:::; an) and T (b2;:::; bn); and if these are equal, compare T (a3;:::; an) and T (b3;:::; bn); and so on. Unfortunately, this method of comparing does not always distinguish between distinct elements of [0; 1] [n], as easy examples with the ×ukasiewicz t-norm show. The LexiT ordering is this. Let a = (a1; a2;:::; an) 2 [0; 1] [n] and let T be an

Topics: Leximin, triangular norms In [1, Dubois and Prade introduced the Leximin ordering>Leximin of elements
Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.4309
Provided by: CiteSeerX
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