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Travel groupoids

Ladislav Nebeský (2006)

Czechoslovak Mathematical Journal

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 : ( u * v ) * u = u ; if ( u * v ) * v = u , then u = v . 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 V and u u * v = v } . Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.

Travel groupoids on infinite graphs

Jung Rae Cho, Jeongmi Park, Yoshio Sano (2014)

Czechoslovak Mathematical Journal

The notion of travel groupoids was introduced by L. Nebeský in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set V and a binary operation * on V satisfying two axioms. We can associate a graph with a travel groupoid. We say that a graph G has a travel groupoid if the graph associated with the travel groupoid is equal to G . Nebeský gave a characterization of finite graphs having a travel groupoid. In this paper, we study travel groupoids on infinite graphs....

Unions of subquasigroups

Tomáš Kepka, Pavel Příhoda, Jan Šťovíček (2004)

Acta Universitatis Carolinae. Mathematica et Physica

Units in quasigroups with classical Bol--Moufang type identities

Natalia Didurik, Viktor Alekseevich Shcherbakov (2020)

Commentationes Mathematicae Universitatis Carolinae

We proceed with Kunen's research about existence of units (left, right, two-sided) in quasigroups with classical Bol--Moufang type identities, listed in paper Extra loops II, by F. Fenyves (1969). We consider those Bol--Moufang identities where it has not been decided yet whether a quasigroup fulfilling this identity has to possess a left or right identity. We also provide a table of all Moufang--Bol identities, indicating at each whether it describes the variety of groups, and whether it forces...

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