A clone-based representation of the fuzzy tolerance or equivalence relations a strict order relation is compatible with

Abstract

We show that although there exists no non-trivial (fuzzy) tolerance relation a partial order relation is compatible with (in the sense of Bělohlávek), the situation is quite different when considering its strict part. More specifically, we provide a representation of all fuzzy tolerance (and, in particular, all fuzzy equivalence) relations a strict order relation is compatible with. To that end, we introduce the notion of clone relation associated with a partially ordered set and discuss its basic properties. The mentioned representation is intimately connected with this clone relation.