Invertibility modulo dead-ending no-$\mathcal {P}$-universes

Abstract

In combinatorial game theory, under normal play convention, all games are invertible, whereas only the empty game is invertible in misère play. For this reason, several restricted universes of games were studied, in which more games are invertible. Here, we study combinatorial games under misère play, 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 ${\mathcal {Q}_{\mathbb {Z}}}$, and show that all sums of universes whose quotient is ${\mathcal {Q}_{\mathbb {Z}}}$ also have ${\mathcal {Q}_{\mathbb {Z}}}$ as their quotient.