Commutativity theorems for cancellative semigroups

Abstract

Psomopoulos has proved that $[x^n, y] = [x, y^{n+1}]$ for a positive integer n implies commutativity in groups. Here we show that cancellative semigroups admitting commutators and satisfying the identity $[x^n, y] = [x, y^{n+k}]$ implies that the element $y^k$ is central. The special case of $k=1$ yields the above mentioned commutativity theorem. To accommodate negative exponents, we consider the functional equation $[f(x), y] = [x, g(y)f(y)]$ where f and g are unary functions satisfying certain formal syntactic rules and prove that in cancellative semigroups admitting commutators, the functional equation $[f(x), y] = [x, g(y)f(y)]$ implies that the element g(y) is central i.e. $xg(y) = g(y)x$ for all x and y. By the way, these results are new even in group theory.