Similarity-Based Fuzzy Orderings: Recent Advances

Fuzzy orderings with reference to an underlying context of similarity were first defined by Höhle and Blanchard [9]. Since then, several authors have contributed to the development of this class of fuzzy relations, both from the theoretical and the practical side (e.g. [1–4, 7, 8]). Two problems remained open throughout several years: how to define strict fuzzy orderings and how to define lexicographic compositions of fuzzy orderings. It is clear, of course, that these two open problems are strongly related, since lexicographic composition is only possible if a meaningful way to extract the strict part of a fuzzy ordering is available. This contribution aims at presenting an overview of recent advances in this direction. Firstly, we summarize the content of a recent paper devoted to strict fuzzy orderings in the similaritybased framework [5]. We introduce and justify a similarity-based concept of strict fuzzy orderings and provide constructions how fuzzy orderings can be transformed into strict fuzzy orderings and vice versa. We demonstrate that there is a meaningful correspondence between fuzzy orderings and strict fuzzy orderings. Unlike the classical case, however, we do not obtain a general one-to-one correspondence. We observe that the strongest results are achieved if the underlying t-norm induces a strong negation, which, in particular, includes nilpotent t-norms and the nilpotent minimum. Secondly, we highlight the results of recent research on lexicographic composition of similaritybased fuzzy orderings. Starting from the approach to strict fuzzy orderings presented in [5], we provide a construction principle for lexicographic composition of fuzzy orderings. Note, however, that an approach to lexicographic composition is only meaningful if the lexicographic composition of linear orderings is linear again. Therefore, we approach the question if (or under which conditions) this fundamental property holds, assuming that T -linearity [6, 9] is the appropriate concept to express linearity of fuzzy orderings. It turns out that, once again, we only obtain meaningful results if the underlying t-norm induces a strong negation.