The Directed Geodetic Structure of a Strong Digraph

By a ternary structure we mean an ordered pair (U ₀, T ₀), where U ₀is a finitenonempty set and T ₀is 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 Dwith the properties that V(D) = U ₀and T ₀ (u,v, w)if and only if d _D (u,v)+ d _D (v,w)= d _D (u, w) for all u, v, w U ₀, where d _Ddenotes the (directed) distance function in D. It is proved in this paper that there exists no sentence sof the language of the first-order logic such that a ternary structure is a directed geodetic structure if and only if it satisfies s.