| Autor: |
J. B. Nation |
| Jazyk: |
angličtina |
| Rok vydání: |
2024 |
| Předmět: |
|
| Zdroj: |
Cubo, Vol 26, Iss 3, Pp 443-473 (2024) |
| Druh dokumentu: |
article |
| ISSN: |
0719-0646 |
| DOI: |
10.56754/0719-0646.2603.443 |
| Popis: |
We show that there are finite distributive lattices that are not the congruence lattice of any finite semidistributive lattice. For \(0 \leq k \leq 2\), the distributive lattice \((\mathbf{B}_k)_{++} = \mathbf{2} + \mathbf{B}_k\), where \(\mathbf{B}_k\) denotes the boolean lattice with \(k\) atoms, is not the congruence lattice of any finite semidistributive lattice. Neither can these lattices be a filter in the congruence lattice of a finite semidistributive lattice. However, each \((\mathbf{B}_k)_{++}\) with \(k \geq 3\) is the congruence lattice of a finite semidistributive lattice, say \(\mathbf{L}_k\). These lattices \(\mathbf{L}_k\) cannot be bounded (in the sense of McKenzie), as no \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) is the congruence lattice of a finite bounded lattice. A companion paper shows that every \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) can be represented as the congruence lattice of an infinite semidistributive lattice. We also find sufficient conditions for a finite distributive lattice to be representable as the congruence lattice of a finite bounded (and hence semidistributive) lattice. |
| Databáze: |
Directory of Open Access Journals |
| Externí odkaz: |
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