On a problem of R. Pöschel on locally invertible monoids (Q810019)

A submonoid M of \({}^ AA\) is said to be locally invertible if for every \(f\in M\) and every finite \(g\subseteq f\) there is \(h\in M\) such that \(h\circ g=id\). For a subset F of \({}^ AA\), the set of all \(f\in^ AA\) such that every finite \(g\subseteq f\) can be extended to a map in F is said to be a local closure of F, denoted by Loc(F). Let P(\(\lambda\)) denote the following statement: ``Every locally invertible submonoid M of \(^{\lambda}\lambda\) is contained in Loc(Sym(\(\lambda\))\(\cap Loc(M))'' \). For finite \(\lambda\) this is trivially true; and J. Kollar proved P(\(\omega\)). R. Pöschel asked whether P(\(\lambda\)) holds for an uncountable \(\lambda\). The main result of the paper under review: \(2^{\kappa}=\kappa^+\) implies \(\neg P(2^{\kappa})\). It is also shown that, for countable monoids M and \(\lambda <2^{\omega}\), Martin's axiom \(MA_{\lambda}\) implies P(\(\lambda\)). It is not known whether \(P(2^{\omega})\) is consistent with ZFC.