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.