Similarity relations and fuzzy orderings

An extended version of Szpilrajn's theorem is proved and various properties of similarity relations and fuzzy orderings are investigated and, as an illustration, a fuzzy preordering is investigated which is reflexive and antisymmetric.

Abstract

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 μs(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, μs(x,x) = 1 (reflexivity), μs(x,y) = μs(y,x) (symmetry), and μs(x,z) ⩾ ∨ (μs(x,y) Å μs(y,z)) (transitivity), where ∀ and Å 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, (μP(x,y) > 0 and x ≠ y) ⇒ μP(y,x) = 0. A fuzzy linear ordering is a fuzzy partial ordering in which x ≠ y ⇒ μs(x,y) > 0 or μs(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 ⇒ μs(x,y) > 0 or μs(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.