The directed geodetic structure of a strong digraph

Nebeský, Ladislav. "The directed geodetic structure of a strong digraph." Czechoslovak Mathematical Journal 54.1 (2004): 1-8. <http://eudml.org/doc/30834>.

@article{Nebeský2004,
abstract = {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 $\{\mathbf \{s\}\}$.},
author = {Nebeský, Ladislav},
journal = {Czechoslovak Mathematical Journal},
keywords = {strong digraph; directed distance; ternary relation; finite structure; strong digraph; directed distance; ternary relation; finite structure},
language = {eng},
number = {1},
pages = {1-8},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The directed geodetic structure of a strong digraph},
url = {http://eudml.org/doc/30834},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Nebeský, Ladislav
TI - The directed geodetic structure of a strong digraph
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 1
EP - 8
AB - 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 ${\mathbf {s}}$.
LA - eng
KW - strong digraph; directed distance; ternary relation; finite structure; strong digraph; directed distance; ternary relation; finite structure
UR - http://eudml.org/doc/30834
ER -