Similarity relations and fuzzy orderings

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 p&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, &A = 1 (reflexivity), pdx.~) = PS(Y.X) (symmetry), and &,z) P V (&x,y) A &y,z)) (transitivity), where V 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, &(x, y) > 0 and x # y) =z~~(y,x) = 0. A fuzzy linear ordering is a fuzzy partial ordering in which x # y ps(x, y) > 0 or pS(y,x) > 0. A fuzzypreordering is a fuzzy ordering which is reflexive. A fuzzy weak ordering is a fuzzy preordering in which x # y * &x, y) > 0 or ps( 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.