Four generators of an equivalence lattice with consecutive block counts
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929551601303552 |
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| author | Czédli, Gábor |
| author_facet | Czédli, Gábor |
| contents | The block count of an equivalence $μ\in$ Equ$(A)$ is the number blnum$(μ)$ of blocks of (the partition corresponding to) $μ$. We say that $X=\{μ_1,μ_2,μ_3,μ_4\}$ is a four-element generating set of Equ$(A)$ with consecutive block counts if $X$ generates Equ$(A)$ and blnum$(μ_{i+1})$ = blnum$(μ_{1})+i$ for $i\in\{1,2,3\}$. We prove that if the number of elements of a finite set $A$ is six or at least eight, then Equ$(A)$ has a four-element generating set with consecutive block counts. Also, we present a historical remark on the connection between equivalence lattices and quasiorder lattices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_15328 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Four generators of an equivalence lattice with consecutive block counts Czédli, Gábor Rings and Algebras 06B99, 06C10 The block count of an equivalence $μ\in$ Equ$(A)$ is the number blnum$(μ)$ of blocks of (the partition corresponding to) $μ$. We say that $X=\{μ_1,μ_2,μ_3,μ_4\}$ is a four-element generating set of Equ$(A)$ with consecutive block counts if $X$ generates Equ$(A)$ and blnum$(μ_{i+1})$ = blnum$(μ_{1})+i$ for $i\in\{1,2,3\}$. We prove that if the number of elements of a finite set $A$ is six or at least eight, then Equ$(A)$ has a four-element generating set with consecutive block counts. Also, we present a historical remark on the connection between equivalence lattices and quasiorder lattices. |
| title | Four generators of an equivalence lattice with consecutive block counts |
| topic | Rings and Algebras 06B99, 06C10 |
| url | https://arxiv.org/abs/2410.15328 |