On the compositional characterization of complete fuzzy pre-orders

Complete pre-orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations. In this paper, we investigate this characterization in a fuzzy setting. We consider two types of completeness (weak completeness and strong completeness) and decompose a fuzzy pre-order by means of an indifference generator, in particular a Frank t-norm. In the weakly complete case, we identify the strongest type of transitivity of the indifference and strict preference relations in function of the generator used for constructing them. In the strongly complete case, we lay bare a stronger type of transitivity of the strict preference relation. We conclude the paper with a rather negative result: there is no hope to obtain a compositional characterization of weakly complete fuzzy pre-orders, and hence also not of fuzzy pre-orders in general.