Geometric properties of ternary fuzzy relations

Ternary fuzzy relations, and fuzzy betweenness relations in particular, are witnessing increasing attention in recent years. A key reason is that axiomatic properties of ternary fuzzy relations seem to be ideally suited to capture geometric characteristics of the abstract notion of betweenness. In this paper, we introduce several new properties of ternary fuzzy relations, including the Peano property, the Pasch property and the sand-glass property, that can be qualified as geometric properties. We investigate their interrelationships as well as their connections with various types of fuzzy betweenness relations. Additionally, in the context of our study of the Pasch property and the sand-glass property, we introduce the convexity property of ternary fuzzy relations by taking inspiration from the solid theoretical basis of the theory of fuzzy convex structures.