Implication semilattice of 990 quasigroup equational laws
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908926798200832 |
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| author | Floch, Bruno Le |
| author_facet | Floch, Bruno Le |
| contents | In his quest to disprove a claim by Peirce that all lattices are distributive, Ernst Schröder considered 135 years ago a list of 990 equational laws on quasigroups, analogous to associativity, such as $(x // y) * z = (y // x) \backslash\backslash z$. A quasigroup is a non-associative analogue of groups, specifically a set equipped with multiplication and right/left conjugate-division operations that are compatible. Each equation of interest identifies two three-variable expressions built from these operations. I determine all $114$ equivalence classes of their conjunctions, and all implications between them. This includes as a small corner the five-element non-distributive lattice identified by Schröder. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29909 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Implication semilattice of 990 quasigroup equational laws Floch, Bruno Le Rings and Algebras In his quest to disprove a claim by Peirce that all lattices are distributive, Ernst Schröder considered 135 years ago a list of 990 equational laws on quasigroups, analogous to associativity, such as $(x // y) * z = (y // x) \backslash\backslash z$. A quasigroup is a non-associative analogue of groups, specifically a set equipped with multiplication and right/left conjugate-division operations that are compatible. Each equation of interest identifies two three-variable expressions built from these operations. I determine all $114$ equivalence classes of their conjunctions, and all implications between them. This includes as a small corner the five-element non-distributive lattice identified by Schröder. |
| title | Implication semilattice of 990 quasigroup equational laws |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2603.29909 |