On complete fuzzy preorders and their characterizations

Abstract

In the context of crisp or classical relations, one may find several alternative characterizations of the concept of a total preorder. In this contribution, we first discuss the way of translating those characterizations to the framework of fuzzy relations. Those new properties depend on t-norms. We focus on two important families of t-norms, namely those that do not admit zero divisors and those that are rotation invariant. For these families, we study whether or not the properties shown for fuzzy relations lead to characterizations of complete fuzzy preorders. Special attention is paid to the minimum operator, which shows the best behaviour in preserving most of the characterizations known for crisp relations.

Conditional extensions of fuzzy preorders

“…Montes et al [34] is a good example of the relevance of the t-norm chosen. The T -transitivity of a fuzzy relation Q is usually denoted by Q • T Q ⊆ Q.…”

Decomposition and Arrow-Like Aggregation of Fuzzy Preferences

“…Furthermore, the fact of existence of different non-equivalent kinds of transitivity definitions and connectedness as well as completeness (see also [27][28][29][30]), tells us that the consideration in the fuzzy context of some kind of fuzzy total preorder is not unique. (Other non-equivalent definitions of transitivity have been introduced in this literature, see e.g., [27,28,31,32]).…”

Numerical representability of fuzzy total preorders

“…However, it is indeed true that many equivalent definitions that appear in the crisp setting (e.g. : a total preorder, an interval order or a semiorder) fail to be equivalent when extended (in a natural way) to the fuzzy setting 1,2,3,4 .…”