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 tran sitive closure into fuzzy terms. This definition works fine, but to so me extent it does not fully capture our understanding of transitivity. The reason is that this definition is based on fuzzifying only the po sitive 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 follo ws 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 article, we define a strongly tr ansitive fuzzy relation as the one that satisfies both the positive an d the negative parts of the transitivity demands, prove the existence of strong transitive closure, and find the relationship between strong ly transitive similarity and clustering. (C) 1995 John Wiley & Sons, I nc.