Strongly transitive fuzzy relations: An alternative way to describe similarity

The notion of a transitive closure of a fuzzy relation is very useful for clustering in pattern recognition, for fuzzy databases, etc. It is based on translating the standard definition of transitivity and transitive closure into fuzzy terms. This definition works fine, but to some extent it does not fully capture our understanding of transitivity. The reason is that this definition is based on fuzzifying only the positive side of transitivity: if R(a, b) and R(b, c), then R(a, c); but transitivity also includes a negative side: if R(a, b) and not R(a, c), then not R(b, c). In classical logic, this negative statement follows from the standard “positive” definition of transitivity. In fuzzy logic, this negative part of the transitivity has to be formulated as an additional demand. In the present paper, we define a strongly transitive fuzzy relation as the one that satisfies both the positive and the negative parts of the transitivity demands, prove the existence of strong transitive closure, and find the relationship between strongly transitive similarity and clustering.