On factorisations of Left dead ends
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916411990867968 |
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| author | Davies, Alfie |
| author_facet | Davies, Alfie |
| contents | 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 misère theory. The algebraic structure of misère 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: $\overline{1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17871 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On factorisations of Left dead ends Davies, Alfie Combinatorics Commutative Algebra 91A46 (Primary) 06F05, 20M13, 20M14 (Secondary) 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 misère theory. The algebraic structure of misère 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: $\overline{1}$. |
| title | On factorisations of Left dead ends |
| topic | Combinatorics Commutative Algebra 91A46 (Primary) 06F05, 20M13, 20M14 (Secondary) |
| url | https://arxiv.org/abs/2409.17871 |