Fuzzy preference structures without incomparability

In this paper, we establish important relationships between the basic properties of the components of a fuzzy preference structure without incomparability. This study is carried out for the fuzzy preference structures introduced recently by De Baets, Van de Walle and Kerre. A set of remarkable theorems gives detailed insight in the relationships between the sup-T transitivity of the fuzzy preference and indifference relations and the sup-T transitivity of the fuzzy large preference relation. Several paths of thought, involving t-norms with or without zero-divisors, are explored and, where required, illustrative counterexamples confirm the falsity of certain implications. Finally, we introduce the (T, N)-Ferrers property of a binary fuzzy relation and show that the fuzzy preference and fuzzy large preference relations share certain types of this Ferrers property.