This paper is devoted to a class of fuzzy orderings which play a fundamental role in decision analysis and fuzzy control|strongly linear fuzzy (weak) orderings. First, we see that any relation of that kind can be decomposed into a crisp linear ordering and a fuzzy equivalence relation. As a consequence, a general representation theorem follows. Finally, a method for constructing strongly linear fuzzy orderings from pseudo-metrics is presented.

This contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings. Different existing concepts are evaluated with respect to three fundamental correspondences from the classical case—linearizability of partial orderings, intersection representation, and one-to-one correspondence between linearity and maximality. As a main result, we obtain that it is virtually impossible to simultaneously preserve all these three properties in the fuzzy case. If we do not require a one-to-one correspondence between linearity and maximality, however, we obtain that an implication-based definition appears to constitute a sound compromise, in particular, if Łukasiewicz-type logics are considered. Key words: completeness, fuzzy ordering, fuzzy preference modeling, fuzzy relation,