A note on prehomomorphisms (Q5936178)

If \(G\) and \(S\) are inverse monoids, then a mapping \(\psi\colon G\to S\) is called a prehomomorphism if \(1\psi=1\), \((x\psi)^{-1}=x^{-1}\psi\), \((x\psi)(y\psi)\leq(xy)\psi\) for all \(x,y\in G\). It is shown that if \(G\) is a free group on the set \(X\), then any mapping from \(X\) into the inverse monoid \(S\) can be extended to a prehomomorphism \(\psi\colon G\to S\). This then leads to a short proof for the fact that every inverse monoid has an \(F\)-inverse monoid cover.