Similarity relations, fuzzy linear orders, and fuzzy partial orders
We show that for a fuzzy partial order R on a finite universe Omega, there is a finite family of fuzzy linear orders {L-i: 1 less than or equal to i less than or equal to k} such that mu(R)(x, y) = min{mu(Li)(x, y): 1 less than or equal to i less than or equal to k} for all x and y. This generalizes a well-known result on crisp partial orders, which states that each partial order on a finite set Omega is a finite intersection of linear orders. As a special case, a similarity relation S, where mu(S)(x, y) < 1 for x not equal y, is shown to be the intersection mu(R)(x, y) boolean AND mu(R)(y, x) for a fuzzy linear ordering R. (C) 2000 Elsevier Science B.V. All rights reserved.
similarity relation fuzzy linear order fuzzy partial order