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

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.