Transitivity of fuzzy relations under discretization

Fuzzy transitivity is a key property for many fuzzy relational structures, such as fuzzy preorders and equivalences. Theoretical models for practical problems which are based on fuzzy relations make use of continuous scales, mostly the unit interval [0, 1]. Practical implementation of these models though, involves their discretization into finite scales, which generally results in some loss of transitivity. In this paper we study if there are any transitivity preserving discretization strategies. Also, we evaluate the loss of transitivity in some commonly used discretization approaches.