Invertibility in the Misère Multiverse

Abstract

Understanding invertibility in restricted misère play has been challenging; in particular, the possibility of non-conjugate inverses posed difficulties. Advances have been made in a few specific universes, but a general theorem was elusive. We prove that every universe has the conjugate property, and also give a characterisation of the invertible elements of each universe. We then explore when a universe can have non-trivial invertible elements, leaving a slew of open problems to be further investigated.