Commutativity theorems for cancellative semigroups

Moghaddam, G.; Padmanabhan, R.
December 2017
Semigroup Forum;Dec2017, Vol. 95 Issue 3, p448
Academic Journal
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.


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