A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE
Let \(\lfloor x \rfloor\) and \(\lceil x\rceil \) denote the lower integer part and the upper integer part of a real number \(x\), respectively. Our main goal is to construct four partitions of a finite set \(A\) with \(n\geq 7\) elements such that each of the four partitions has exactly \(\lceil n/...
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Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2025-07-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/842 |
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| author | Gábor Czédli |
| author_facet | Gábor Czédli |
| author_sort | Gábor Czédli |
| collection | DOAJ |
| description | Let \(\lfloor x \rfloor\) and \(\lceil x\rceil \) denote the lower integer part and the upper integer part of a real number \(x\), respectively. Our main goal is to construct four partitions of a finite set \(A\) with \(n\geq 7\) elements such that each of the four partitions has exactly \(\lceil n/2\rceil \) blocks and any other partition of \(A\) can be obtained from the given four by forming joins and meets in a finite number of steps. We do the same with \(\lceil n/2\rceil-1\) instead of \(\lceil n/2\rceil\), too. To situate the paper within lattice theory, recall that the partition lattice \(\mathrm{Eq}(A)\) of a set \(A\) consists of all partitions (equivalently, of all equivalence relations) of \(A\). For a natural number \(n\), \([n]\) and \(\mathrm{Eq}(n)\) will stand for \(\{1,2,\dots,n\}\) and \(\mathrm{Eq}([n])\), respectively. In 1975, Heinrich Strietz proved that, for any natural number \(n\geq 3\), \(\mathrm{Eq} (n)\) has a four-element generating set; half a dozen papers have been devoted to four-element generating sets of partition lattices since then. We give a simple proof of his just-mentioned result. We call a generating set \(X\) of \(\mathrm{Eq}(n)\) horizontal if each member of \(X\) has the same height, denoted by \(h(X)\), in \(\mathrm{Eq} (n)\); no such generating sets have been known previously. We prove that for each natural number \(n\ge 4\), \(\mathrm{Eq}(n)\) has two four-element horizontal generating sets \(X\) and \(Y\) such that \(h(Y)=h(X) +1\); for \(n\geq 7\), \(h(X)= \lfloor n/2 \rfloor\). |
| format | Article |
| id | doaj-art-e5c3bf4b8c3b4081875da401ef86d404 |
| institution | Kabale University |
| issn | 2414-3952 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-e5c3bf4b8c3b4081875da401ef86d4042025-08-20T03:34:33ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522025-07-0111110.15826/umj.2025.1.004235A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICEGábor Czédli0Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 SzegedLet \(\lfloor x \rfloor\) and \(\lceil x\rceil \) denote the lower integer part and the upper integer part of a real number \(x\), respectively. Our main goal is to construct four partitions of a finite set \(A\) with \(n\geq 7\) elements such that each of the four partitions has exactly \(\lceil n/2\rceil \) blocks and any other partition of \(A\) can be obtained from the given four by forming joins and meets in a finite number of steps. We do the same with \(\lceil n/2\rceil-1\) instead of \(\lceil n/2\rceil\), too. To situate the paper within lattice theory, recall that the partition lattice \(\mathrm{Eq}(A)\) of a set \(A\) consists of all partitions (equivalently, of all equivalence relations) of \(A\). For a natural number \(n\), \([n]\) and \(\mathrm{Eq}(n)\) will stand for \(\{1,2,\dots,n\}\) and \(\mathrm{Eq}([n])\), respectively. In 1975, Heinrich Strietz proved that, for any natural number \(n\geq 3\), \(\mathrm{Eq} (n)\) has a four-element generating set; half a dozen papers have been devoted to four-element generating sets of partition lattices since then. We give a simple proof of his just-mentioned result. We call a generating set \(X\) of \(\mathrm{Eq}(n)\) horizontal if each member of \(X\) has the same height, denoted by \(h(X)\), in \(\mathrm{Eq} (n)\); no such generating sets have been known previously. We prove that for each natural number \(n\ge 4\), \(\mathrm{Eq}(n)\) has two four-element horizontal generating sets \(X\) and \(Y\) such that \(h(Y)=h(X) +1\); for \(n\geq 7\), \(h(X)= \lfloor n/2 \rfloor\).https://umjuran.ru/index.php/umj/article/view/842partition lattice, equivalence lattice, minimum-sized generating set, horizontal generating set, four-element generating set. |
| spellingShingle | Gábor Czédli A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE Ural Mathematical Journal partition lattice, equivalence lattice, minimum-sized generating set, horizontal generating set, four-element generating set. |
| title | A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE |
| title_full | A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE |
| title_fullStr | A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE |
| title_full_unstemmed | A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE |
| title_short | A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE |
| title_sort | pair of four element horizontal generating sets of a partition lattice |
| topic | partition lattice, equivalence lattice, minimum-sized generating set, horizontal generating set, four-element generating set. |
| url | https://umjuran.ru/index.php/umj/article/view/842 |
| work_keys_str_mv | AT gaborczedli apairoffourelementhorizontalgeneratingsetsofapartitionlattice AT gaborczedli pairoffourelementhorizontalgeneratingsetsofapartitionlattice |