Do the idempotents in an inverse semigroup commute?

Question

I have been looking at this for hours now. Why is it true that idempotents of an inverse semigroup commute? It seems like this should be straightforward but I just can't get it.

Any help is greatly appreciated.

Answer 1

Assume that $S$ is an inverse semigroup, and let $e, f\in S$ be arbitrary two idempotents. Then $$(ef)(f(ef)^{-1}e)(ef)=ef^2(ef)^{-1}e^2f=ef(ef)^{-1}ef=ef$$ $$(f(ef)^{-1}e)(ef)(f(ef)^{-1}e)=f(ef)^{-1}e^2f^2(ef)^{-1}e=f((ef)^{-1}ef(ef)^{-1})e=f(ef)^{-1}e.$$ Therefore, by the uniqueness of inverses, $f(ef)^{-1}e=(ef)^{-1}$. It follows that $$ (ef)^{-2}=(f(ef)^{-1}e)^2=f((ef)^{-1}ef(ef)^{-1})e=f(ef)^{-1}e=(ef)^{-1}, $$ i.e. $(ef)^{-1}$ is an idempotent and so $(ef)^{-1}=ef$. By symmetry, $fe$ is also an idempotent.

So, $$ (ef)(fe)(ef)=efef=ef,\ (fe)(ef)(fe)=fefe=fe, $$ and so $fe=(ef)^{-1}=ef$, as required.