Diameters of endomorphism monoids of chains
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913455259254784 |
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| author | East, James Gould, Victoria Miller, Craig Quinn-Gregson, Thomas |
| author_facet | East, James Gould, Victoria Miller, Craig Quinn-Gregson, Thomas |
| contents | The left and right diameters of a monoid are topological invariants defined in terms of suprema of lengths of derivation sequences with respect to finite generating sets for the universal left or right congruences. We compute these parameters for the endomorphism monoid $End(C)$ of a chain $C$. Specifically, if $C$ is infinite then the left diameter of $End(C)$ is 2, while the right diameter is either 2 or 3, with the latter equal to 2 precisely when $C$ is a quotient of $C{\setminus}\{z\}$ for some endpoint $z$. If $C$ is finite then so is $End(C),$ in which case the left and right diameters are 1 (if $C$ is non-trivial) or 0. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diameters of endomorphism monoids of chains East, James Gould, Victoria Miller, Craig Quinn-Gregson, Thomas Rings and Algebras Combinatorics The left and right diameters of a monoid are topological invariants defined in terms of suprema of lengths of derivation sequences with respect to finite generating sets for the universal left or right congruences. We compute these parameters for the endomorphism monoid $End(C)$ of a chain $C$. Specifically, if $C$ is infinite then the left diameter of $End(C)$ is 2, while the right diameter is either 2 or 3, with the latter equal to 2 precisely when $C$ is a quotient of $C{\setminus}\{z\}$ for some endpoint $z$. If $C$ is finite then so is $End(C),$ in which case the left and right diameters are 1 (if $C$ is non-trivial) or 0. |
| title | Diameters of endomorphism monoids of chains |
| topic | Rings and Algebras Combinatorics |
| url | https://arxiv.org/abs/2408.00416 |