Tolerances and congruences on tree groupoids (Q2759859)

The concept of a tree groupoid was introduced by L. Nebeský for an algebraization of trees in graph theory. Let \(T\) be a tree. The tree groupoid \(G\) corresponding to \(T\) has as its support the vertex set of \(T\) and for \(a,b\) of \(G\) the product \(a\cdot b=a\) if \(a=b\) and \(a\cdot b\) is the vertex of \(T\) adjacent to \(a\) and lies on the path connecting \(a\) and \(b\) in the case \(a\neq b\).NEWLINENEWLINENEWLINEIt is shown that every tree groupoid is simple and several elementary properties of tolerances are listed. Among other results, the join of two tolerances on \(G\) is equal to their set-theoretical union and hence the lattice of tolerances on \(G\) is distributive.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00014].