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Travel groupoids

Authors:

Ladislav Nebeský

Affiliation:

(1) Univerzita Karlova v Praze, Filozofická fakulta, nám. J. Palacha 2, 116 38 Praha 1, Czech Republic

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.

Keywords:

travel groupoid graph path geodetic graph

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