Compatibility of fuzzy relations - KHENICHE, Azzedine (Université de M'sila, 2016)
: The first aim of the present thesis is to strip the compatibility property of its originating context of fuzzy equivalence relations, since only the notion of compatibility of a fuzzy relation with respect to a fuzzy equality (equivalence) was introduced in the early years. Indeed, we have generalized this compatibility property to arbitrary fuzzy relations; this property is dissected here into left and right compatibility and studied in detail in the context of arbitrary fuzzy relations. Relying on the notions of left and right traces of a fuzzy relation, the main findings of the first part of the thesis is that the left and right compatibility can be expressed in terms of inclusions, which facilitates their understanding. On the other hand, we have shown that although there exists no non-trivial (fuzzy) tolerance relation a partial order relation is compatible with, the situation is quite different when considering its strict part. Indeed, We provide a characterization of all fuzzy tolerance (and, in particular, fuzzy equivalence) relations a strict order relation is left compatible, right compatible and compatible with, by using the notion of the clone relation associated with a partially ordered set introduced here and discuss its basic propertiesDate
Keywords: compatibility, equivalence relations, tolerance relations, Clone relations, left and right compatibility, residuated lattices