On factorisations of Left dead ends

TL;DR

This paper explores the algebraic structure of Left dead ends in misère combinatorial game theory, establishing foundational properties, bounds on factorizations, and identifying unique factorization elements.

Contribution

It introduces the study of the monoid of Left dead ends, proving its key properties, and develops tools for factorization analysis, including the concept of flexibility.

Findings

The monoid of Left dead ends is reduced, pocancellative, and an FF-monoid.

Sharp bounds are provided on the lengths of factorizations.

Large families of games with unique factorizations are identified, including a prime element.

Abstract

Losing a game is difficult. Recent work of Larsson, Nowakowski, and Santos, as well as that of Siegel, has opened up enticing lines of research into partizan mis\`ere theory. The algebraic structure of mis\`ere monoids is not yet well understood. A natural object to study is a universe of games, which is a set of games satisfying various closure properties. We prove that a universe is determined by its Left ends, thus motivating a study of such games, which was also noted by Siegel for dead ending universes in particular. We initiate this study by investigating the atomic structure of the monoid of Left dead ends, showing that it is reduced, pocancellative, and an FF-monoid. We give sharp bounds on the lengths of factorisations, and also build tools to find atoms. Some results require the introduction of a new concept called flexibility, which is a measure of the distance from a game to integers. We also discover large families of games admitting unique factorisations; this includes the uncovering of a prime element: $\overline{1}$.