Invertibility modulo dead-ending no-P-universes

Abstract: In normal version of combinatorial game theory, all games are invertible, whereas only the empty game is invertible in mis`ere version. For this reason, several restricted universes were earlier considered for their study, in which more games are invertible. We here study combinatorial games in mis`ere version, in particular universes where no player would like to pass their turn In these universes, we prove that having one extra condition makes all games become invertible. We then focus our attention on a specific quotient, called Q_Z, and show that all sums of universes whose quotient is Q_Z also have Q_Z as their quotient.