Equations of tournaments are not finitely based (Q1969795)

Multiplication on the nodes of a tournament \(T\) (with loops) can be defined by setting \(ab= ba=a\) whenever \(a\to b\); so a tournament can be regarded as a commutative groupoid satisfying \(ab\in\{a,b\}\) for all \(a\) and \(b\). Then all tournaments satisfy equations such as \(xx= x\), \(xy= yx\), \(x(xy)= xy\), and \(x((xy)(xz))= (xy)(xz)\), for example. The authors show that there is no finite list of equations such that any equation satisfied by all tournaments can be derived from equations in the list. They also obtain a generalization for directed graphs.