On strict fuzzy betweenness relations

Abstract

Fuzzy betweenness relations have gained increasing attention since Zhang et al. recently established their extensive connections with fuzzy order relations. In this manuscript, we delve into the exploration of strict fuzzy betweenness relations from two aspects. First of all, in light of one of the most important examples of strict betweenness relations from strict order relations, we extend this paradigm to the fuzzy case. Specially, with respect to a t-norm ⁎ and a fuzzy equivalence relation E, we construct a strict ⁎-E-betweenness relation from a strict ⁎-E-order relation. Secondly, by establishing a one-to-one correspondence between strict betweenness relations and betweenness relations, we investigate whether this elegant result persists within the many-valued setting. Toward this end, we present the construction of a strict fuzzy betweenness relation from a fuzzy betweenness relation and vice versa, scrutinizing the conditions under which these constructions are inverse to each other. The conclusion turns out that the correspondence in the fuzzy context exhibits distinct differences from the crisp one, primarily due to the inherent complexity of fuzzy logic.