Invertibility modulo dead-ending no-P-universes

TL;DR

This paper explores invertibility in misère combinatorial games within universes where players never pass, demonstrating that an additional condition ensures all games are invertible and analyzing the properties of a specific quotient Q_Z.

Contribution

It introduces a new condition in misère combinatorial games that guarantees invertibility of all games and studies the properties of the quotient Q_Z in this context.

Findings

All games become invertible under the new condition in certain universes.

The quotient Q_Z is preserved under sums of universes with the same quotient.

Universes with no passing moves exhibit unique invertibility properties.

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.