Additive decomposition of fuzzy pre-orders

Fuzzy pre-orders (reflexive and min-transitive fuzzy relations) constitute an important class of fuzzy relations. By means of an indifference generator, a fuzzy pre-order can be decomposed additively into two parts: an indifference relation and a strict preference relation. When using a Frank t-norm as indifference generator, we fully characterize the transitivity of these parts. Only in case the minimum operator is used as generator, both parts are min-transitive. The transitivity of the indifference relation is determined by the Frank t-norms, while the transitivity of the strict preference relation is determined by transforms of the nilpotent minimum t-norm.