A formal study of linearity axioms for fuzzy orderings

U. Bodenhofer, F. Klawonn. A formal study of linearity axioms for fuzzy orderings. Fuzzy Sets and Systems, volume 145, number 3, pages 323-354, 2004.

  • Ulrich Bodenhofer
  • Frank Klawonn
JournalFuzzy Sets and Systems
Abstract 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 betweenlinearity and maximality. As a main result, we obtain that it is virtually impossible to simultaneously preserve allthese three properties in the fuzzy case. If we do not require a one-to-one correspondence between linearity andmaximality, however, we obtain that an implication-based definition appears to constitute a sound compromise, inparticular, if Lukasiewicz-type logics are considered.