On groupoids defined by commutators

Abstract:

We study matrices R, L which count the numbers of solutions of $ix = j$ and $xi = j$. For slight generalizations of R, L, the relation $RL = LR$ characterizes associativity of a groupoid. For groupoids defined by group commutators $xy{x^{ - 1}}{y^{ - 1}}$ the relation $RL = LR$ is valid. In addition one can study analogues of Green’s relations. Any $\mathcal {I}$-class contains at most four $\mathcal {H}$-classes in a commutator groupoid.