Travel groupoids

Nebeský, Ladislav. "Travel groupoids." Czechoslovak Mathematical Journal 56.2 (2006): 659-675. <http://eudml.org/doc/31057>.

@article{Nebeský2006,
abstract = {In this paper, by a travel groupoid is meant an ordered pair $(V, *)$ such that $V$ is a nonempty set and $*$ is a binary operation on $V$ satisfying the following two conditions for all $u, v \in V$: \[ (u * v) * u = u; \text\{ if \}(u * v ) * v = u, \text\{ then \} u = v. \] Let $(V, *)$ be a travel groupoid. It is easy to show that if $x, y \in V$, then $x * y = y$ if and only if $y * x = x$. We say that $(V, *)$ is on a (finite or infinite) graph $G$ if $V(G) = V$ and \[ E(G) = \lbrace \lbrace u, v\rbrace \: u, v \in V \text\{ and \} u \ne u * v = v\rbrace . \] Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.},
author = {Nebeský, Ladislav},
journal = {Czechoslovak Mathematical Journal},
keywords = {travel groupoid; graph; path; geodetic graph; travel groupoids; paths; geodetic graphs},
language = {eng},
number = {2},
pages = {659-675},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Travel groupoids},
url = {http://eudml.org/doc/31057},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Nebeský, Ladislav
TI - Travel groupoids
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 2
SP - 659
EP - 675
AB - In this paper, by a travel groupoid is meant an ordered pair $(V, *)$ such that $V$ is a nonempty set and $*$ is a binary operation on $V$ satisfying the following two conditions for all $u, v \in V$: \[ (u * v) * u = u; \text{ if }(u * v ) * v = u, \text{ then } u = v. \] Let $(V, *)$ be a travel groupoid. It is easy to show that if $x, y \in V$, then $x * y = y$ if and only if $y * x = x$. We say that $(V, *)$ is on a (finite or infinite) graph $G$ if $V(G) = V$ and \[ E(G) = \lbrace \lbrace u, v\rbrace \: u, v \in V \text{ and } u \ne u * v = v\rbrace . \] Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.
LA - eng
KW - travel groupoid; graph; path; geodetic graph; travel groupoids; paths; geodetic graphs
UR - http://eudml.org/doc/31057
ER -