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

TL;DR

This paper generalizes the characterization of invertible elements in the dead-ending universe of misère play, showing that $ ext{ extbackslash P}$-free blocking games form a closed, invertible subgroup with implications for other misère monoids.

Contribution

It extends Milley and Renault's work to prove $ ext{ extbackslash P}$-free blocking games are closed under addition and invertible within the universe.

Findings

$ ext{ extbackslash P}$-free blocking games form a closed set under addition.

Every $ ext{ extbackslash P}$-free blocking game is invertible modulo the blocking universe.

Implications for invertible subgroups in various misère monoids.

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.