Travel groupoids

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 ∈ V:

$$\begin{gathered} (u * v) * u = u; \hfill \\ if (u * v) * v = u, then u = v \hfill \\ \end{gathered} $$

. Let (V, *) be a travel groupoid. It is easy to show that if x, y ∈ 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

$$E(G) = \{ \{ u,v\} : u, v \in V and u \ne u * v = v\} $$

. Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.