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arXiv · 2506.05257

On sums of $\mathscr{P}$-free forms under mis\`ere play

Alfie Davies·Sarah Miller·Rebecca Milley
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Abstract

Milley and Renault proved an interesting characterisation of invertible elements in the dead-ending universe: they are the games with no subpositions of outcome $\mathscr{P}$ (the '$\mathscr{P}$-free' games). We generalise their approach to obtain a stronger result and show in particular that the set of $\mathscr{P}$-free blocking games is closed under addition, which yields that every $\mathscr{P}$-free blocking game is invertible modulo the blocking universe. This has consequences for the invertible subgroups of various other mis\`ere monoids.

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Alfie Davies, Sarah Miller, Rebecca Milley. 2025-06-05. On sums of $\mathscr{P}$-free forms under mis\`ere play. https://arxiv.org/abs/2506.05257

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