On sums of $\mathscr{P}$-free forms under misère play
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916781211254784 |
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| author | Davies, Alfie Miller, Sarah Milley, Rebecca |
| author_facet | Davies, Alfie Miller, Sarah Milley, Rebecca |
| contents | 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 misère monoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On sums of $\mathscr{P}$-free forms under misère play Davies, Alfie Miller, Sarah Milley, Rebecca Combinatorics 91A46 (Primary) 06F05, 20M14, 20K27 (Secondary) 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 misère monoids. |
| title | On sums of $\mathscr{P}$-free forms under misère play |
| topic | Combinatorics 91A46 (Primary) 06F05, 20M14, 20K27 (Secondary) |
| url | https://arxiv.org/abs/2506.05257 |