In this paper, we introduce six basic types of composition of ternary relations, four of which are associative. These compositions are based on two types of composition of a ternary relation with a binary relation recently introduced by Zedam et al. We study the properties of these compositions, in particular the link with the usual composition of binary relations through the use of the operations of projection and cylindrical extension.
By a ternary structure we mean an ordered pair $(U_0, T_0)$, where $U_0$ is a finite nonempty set and $T_0$ is a ternary relation on $U_0$. A ternary structure $(U_0, T_0)$ is called here a directed geodetic structure if there exists a strong digraph $D$ with the properties that $V(D) = U_0$ and \[ T_0(u, v, w)\quad \text{if} \text{and} \text{only} \text{if}\quad d_D(u, v) + d_D(v, w) = d_D(u, w) \] for all $u, v, w \in U_0$, where $d_D$ denotes the (directed) distance function in $D$. It is proved in this paper that there exists no sentence ${\mathbf s}$ of the language of the first-order logic such that a ternary structure is a directed geodetic structure if and only if it satisfies ${\mathbf s}$.
By a ternary structure we mean an ordered pair (X0, T0), where X0 is a finite nonempty set and T0 is a ternary relation on X0. By the underlying graph of a ternary structure (X0, T0) we mean the (undirected) graph G with the properties that X0 is its vertex set and distinct vertices u and v of G are adjacent if and only if {x ∈ X0 ; T0(u, x, v)}∪{x ∈ X0 ; T0(v,x,u)} = {u, v}. A ternary structure (X0, T0) is said to be the B-structure of a connected graph G if X0 is the vertex set of G and the following statement holds for all u, x,y ∈ X0: T0(x, u, y) if and only if u belongs to an induced x − y path in G. It is clear that if a ternary structure (X0, T0) is the B-structure of a connected graph G, then G is the underlying graph of (X0, T0). We will prove that there exists no sentence σ of the first-order logic such that a ternary structure (X0, T0) with a connected underlying graph G is the B-structure of G if and only if (X0, T0) satisfies σ.