Atoms in four-element generating sets of partition lattices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929558530293760 |
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| author | Czédli, Gábor |
| author_facet | Czédli, Gábor |
| contents | Since Henrik Strietz's 1975 paper proving that the lattice Part($n$) of all partitions of an $n$-element finite set is four-generated, more than half a dozen papers have been devoted to four-element generating sets of this lattice. We prove that each element of Part($n$) with height one or two (in particular, each atom) belongs to a four-element generating set. Furthermore, our construction leads to a concise and easy proof of a 1996 result of the author stating that the lattice of partitions of a countably infinite set is four-generated as a complete lattice. In a recent paper "Generating Boolean lattices by few elements and exchanging session keys", see https://doi.org/10.30755/NSJOM.16637, the author establishes a connection between cryptography and small generating sets of some lattices, including Part($n$). Hence, it is worth pointing out that by combining a construction given here with a recent paper by the author, "Four-element generating sets with block count width at most two in partition lattices", available at https://tinyurl.com/czg-4gw2, we obtain many four-element generating sets of Part($n$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19650 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Atoms in four-element generating sets of partition lattices Czédli, Gábor Rings and Algebras 06B99, 06C10 Since Henrik Strietz's 1975 paper proving that the lattice Part($n$) of all partitions of an $n$-element finite set is four-generated, more than half a dozen papers have been devoted to four-element generating sets of this lattice. We prove that each element of Part($n$) with height one or two (in particular, each atom) belongs to a four-element generating set. Furthermore, our construction leads to a concise and easy proof of a 1996 result of the author stating that the lattice of partitions of a countably infinite set is four-generated as a complete lattice. In a recent paper "Generating Boolean lattices by few elements and exchanging session keys", see https://doi.org/10.30755/NSJOM.16637, the author establishes a connection between cryptography and small generating sets of some lattices, including Part($n$). Hence, it is worth pointing out that by combining a construction given here with a recent paper by the author, "Four-element generating sets with block count width at most two in partition lattices", available at https://tinyurl.com/czg-4gw2, we obtain many four-element generating sets of Part($n$). |
| title | Atoms in four-element generating sets of partition lattices |
| topic | Rings and Algebras 06B99, 06C10 |
| url | https://arxiv.org/abs/2410.19650 |