A Note On Topological D-Posets Of Fuzzy Sets

Abstract

. Kopka and Chovanec in [KCH] defined the difference poset (D-poset) as a partially ordered set with a partial difference operation. We show in this paper that every difference operation on a dense subset of h0; 1i is continuous with respect to the usual topology of the real line. We prove also some consequences for the continuity of the difference operation on D-posets of fuzzy sets. Difference posets were defined by Kopka and Chovanec in [KCH] and they are investigated in many recent papers (see for example [DR], [NP], [P] and [RB]). Definition 1. Difference poset (briefly D-poset) is a couple (D; \Psi), where D is a partially ordered set with the largest element 1 and the difference \Psi is the partial operation, which defines for every a; b 2 D, a b, an element b \Psi a in such a way that the following conditions are satisfied: i) b \Psi a b ii) b \Psi (b \Psi a) = a iii) if a b c, then c \Psi b c \Psi a and (c \Psi a) \Psi (c \Psi b) = b \Psi a. Special cases of D-posets..