Classes of fuzzy sets with the same material conditional

Abstract

By the representation's theorem it is known that a fuzzy relation R is a preorder if and only if R = inf_μϵLI_μ^T, where $\text{L}$ is the set of logical states of R. In this paper it is shown how, in almost all cases, both for the t-norm Min and archimedean t-norms T, it is possible to consider that the fuzzy sets μ are normalized. Furthermore a partial order between logical states, defined from the elemental preorders I_μ^T, is studied.