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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.02244 |
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| _version_ | 1866913974426009600 |
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| author | Brooke-Taylor, Andrew D. Cramer, Scott Edwards, Sheila K. Miller |
| author_facet | Brooke-Taylor, Andrew D. Cramer, Scott Edwards, Sheila K. Miller |
| contents | The set-theoretic large cardinal axiom known as I3 posits the existence of a non-trivial rank-to-rank embedding from an initial segment of the universe of sets into itself. Laver showed that the algebra generated by a single such embedding under the operation of application is in fact the free left distributive (LD) algebra on one generator. This and associated theorems using the set-theoretic structure of the embeddings yielded numerous results about general LD algebras under the assumption of I3, only some of which have since been proven without the use of such a strong axiom. A natural question is whether, under the assumption of I3, one can obtain a free LD algebra of embeddings on more than one generator. Here we show that, under an assumption only just above I3 in the large cardinal hierarchy (namely, I2), we indeed obtain a two-generated free left distributive algebra of rank-to-rank embeddings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02244 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A free two-generated left distributive algebra of elementary embeddings Brooke-Taylor, Andrew D. Cramer, Scott Edwards, Sheila K. Miller Logic 03E55 (Primary) 08B20, 20N02 (Secondary) The set-theoretic large cardinal axiom known as I3 posits the existence of a non-trivial rank-to-rank embedding from an initial segment of the universe of sets into itself. Laver showed that the algebra generated by a single such embedding under the operation of application is in fact the free left distributive (LD) algebra on one generator. This and associated theorems using the set-theoretic structure of the embeddings yielded numerous results about general LD algebras under the assumption of I3, only some of which have since been proven without the use of such a strong axiom. A natural question is whether, under the assumption of I3, one can obtain a free LD algebra of embeddings on more than one generator. Here we show that, under an assumption only just above I3 in the large cardinal hierarchy (namely, I2), we indeed obtain a two-generated free left distributive algebra of rank-to-rank embeddings. |
| title | A free two-generated left distributive algebra of elementary embeddings |
| topic | Logic 03E55 (Primary) 08B20, 20N02 (Secondary) |
| url | https://arxiv.org/abs/2508.02244 |