Similarity relations and fuzzy orderings

• Authors: L. A. Zadeh
• Venue: Information Sciences: an International Journal
• Year: 1971

The notion of ''similarity'' as defined in this paper is essentially a generalization of the notion of equivalence. In the same vein, a fuzzy ordering is a generalization of the concept of ordering. For example, the relation x @? y (x is much larger than y) is a fuzzy linear ordering in the set of real numbers. More concretely, a similarity relation, S, is a fuzzy relation which is reflexive, symmetric, and transitive. Thus, let x, y be elements of a set X and @m"s(x,y) denote the grade of membership of the ordered pair (x,y) in S. Then S is a similarity relation in X if and only if, for all x, y, z in X, @m"s(x,x) = 1 (reflexivity), @m"s(x,y) = @m"s(y,x) (symmetry), and @m"s(x,z) = @? (@m"s(x,y) A @m"s(y,z)) (transitivity), where @? and A denote max and min, respectively. ^y A fuzzy ordering is a fuzzy relation which is transitive. In particular, a fuzzy partial ordering, P, is a fuzzy ordering which is reflexive and antisymmetric, that is, (@m"P(x,y) 0 and x y) @? @m"P(y,x) = 0. A fuzzy linear ordering is a fuzzy partial ordering in which x y @? @m"s(x,y) 0 or @m"s(y,x) 0. A fuzzy preordering is a fuzzy ordering which is reflexive. A fuzzy weak ordering is a fuzzy preordering in which x y @? @m"s(x,y) 0 or @m"s(y,x) 0. Various properties of similarity relations and fuzzy orderings are investigated and, as an illustration, an extended version of Szpilrajn's theorem is proved.