On groupoids defined by commutators
On groupoids defined by commutators
We study matrices <italic>R, L</italic> which count the numbers of solutions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i x equals j"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">ix = j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x i equals j"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> …