TY - JOUR
T1 - A formal study of linearity axioms for fuzzy orderings
AU - Bodenhofer, Ulrich
AU - Klawonn, Frank
N1 - Funding Information:
Ulrich Bodenhofer gratefully acknowledges support of the Kplus Competence Center Program which is funded by the Austrian Government, the Province of Upper Austria, and the Chamber of Commerce of Upper Austria.
PY - 2004/8/1
Y1 - 2004/8/1
N2 - 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.
AB - 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.
KW - Completeness
KW - Fuzzy ordering
KW - Fuzzy preference modeling
KW - Fuzzy relation
KW - Linearity
KW - Szpilrajn theorem
UR - http://www.scopus.com/inward/record.url?scp=2942644489&partnerID=8YFLogxK
U2 - 10.1016/S0165-0114(03)00128-3
DO - 10.1016/S0165-0114(03)00128-3
M3 - Article
AN - SCOPUS:2942644489
SN - 0165-0114
VL - 145
SP - 323
EP - 354
JO - Fuzzy Sets and Systems
JF - Fuzzy Sets and Systems
IS - 3
ER -