T-quasigroups (Part II.)
In this paper, by a travel groupoid is meant an ordered pair such that is a nonempty set and is a binary operation on satisfying the following two conditions for all : Let be a travel groupoid. It is easy to show that if , then if and only if . We say that is on a (finite or infinite) graph if and Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.
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 and a binary operation on satisfying two axioms. We can associate a graph with a travel groupoid. We say that a graph has a travel groupoid if the graph associated with the travel groupoid is equal to . Nebeský gave a characterization of finite graphs having a travel groupoid. In this paper, we study travel groupoids on infinite graphs....
A characterization of all classes of idempotent groupoids having no more than two essentially binary term operations with respect to small finite models is given.
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...