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. © 2003 Elsevier B.V. All rights reserved.
Bodenhofer, U., & Klawonn, F. (2004). A formal study of linearity axioms for fuzzy orderings. Fuzzy Sets and Systems, 145(3), 323–354. https://doi.org/10.1016/S0165-0114(03)00128-3