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 $V\left(D\right)=U_0$ and $$T_0(u,v,w)\text{if}\text{and}\text{only}\text{if}{d}_D(u,v)+d_D(v,w)=d_D(u,w)$$ 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 $𝐬$ of the language of the first-order logic such that a ternary structure is a directed geodetic structure if and only if it satisfies...

By a ternary structure we mean an ordered pair $(X_0,T_0)$, where $X_0$ is a finite nonempty set and $T_0$ is a ternary relation on $X_0$. By the underlying graph of a ternary structure $(X_0,T_0)$ we mean the (undirected) graph $G$ with the properties that $X_0$ is its vertex set and distinct vertices $u$ and $v$ of $G$ are adjacent if and only if $$\{x\in X_0{T}_0(u,x,v)\}\cup \{x\in X_0{T}_0(v,x,u)\}=\{u,v\}.$$ A ternary structure $(X_0,T_0)$ is said to be the B-structure of a connected graph $G$ if $X_0$ is the vertex set of $G$ and the following statement holds for all $u,x,y\in X_0$: $T_0(x,u,y)$ if and only if $u$ belongs to an induced $x-y$...