Knuth's non-associative "group" on ${\mathcal P}(\mathbb{N})$
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914643021135872 |
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| author | van der Zypen, Dominic |
| author_facet | van der Zypen, Dominic |
| contents | Donald Knuth introduced in The Art of Computer Programming (Vol 4a) a fast approximation to the addition of integers (given in binary) in terms of bit-wise operations by $a + b \; \approx \; a \oplus b \oplus ((a\land b) \ll 1).$ Generalizing this to infinite bit-strings we get a binary operation on ${\mathcal P}(\mathbb{N})$, the power-set of $\mathbb{N}$ (which we identify with the collection of infinite bit-strings). We show that this operation is ``group-like'' in that it has a neutral element, inverses, but it is not associative. There are a lot of questions left, which the author has not been able to answer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08635 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Knuth's non-associative "group" on ${\mathcal P}(\mathbb{N})$ van der Zypen, Dominic General Mathematics Donald Knuth introduced in The Art of Computer Programming (Vol 4a) a fast approximation to the addition of integers (given in binary) in terms of bit-wise operations by $a + b \; \approx \; a \oplus b \oplus ((a\land b) \ll 1).$ Generalizing this to infinite bit-strings we get a binary operation on ${\mathcal P}(\mathbb{N})$, the power-set of $\mathbb{N}$ (which we identify with the collection of infinite bit-strings). We show that this operation is ``group-like'' in that it has a neutral element, inverses, but it is not associative. There are a lot of questions left, which the author has not been able to answer. |
| title | Knuth's non-associative "group" on ${\mathcal P}(\mathbb{N})$ |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2401.08635 |