On the set of principal congruences in a distributive congruence lattice of an algebra
Let Q be a subset of a finite distributive lattice D. An algebra A represents the inclusion Q ⊆ D by principal congruences if the congruence lattice of A is isomorphic to D and the ordered set of principal congruences of A corresponds to Q under this isomorphism. If there is such an algebra for ever...
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| Dokumentumtípus: | Cikk |
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2018
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| Sorozat: | Acta scientiarum mathematicarum
84 No. 3-4 |
| Kulcsszavak: | Algebra, Matematika |
| doi: | 10.14232/actasm-017-538-7 |
| Online Access: | http://acta.bibl.u-szeged.hu/56919 |
| Tartalmi kivonat: | Let Q be a subset of a finite distributive lattice D. An algebra A represents the inclusion Q ⊆ D by principal congruences if the congruence lattice of A is isomorphic to D and the ordered set of principal congruences of A corresponds to Q under this isomorphism. If there is such an algebra for every subset Q containing 0, 1, and all join-irreducible elements of D, then D is said to be fully (A1)-representable. We prove that every fully (A1)- representable finite distributive lattice is planar and it has at most one joinreducible coatom. Conversely, we prove that every finite planar distributive lattice with at most one join-reducible coatom is fully chain-representable in the sense of a recent paper of G. Grätzer. Combining the results of this paper with another result of the present author, it follows that every fully (A1)- representable finite distributive lattice is “fully representable” even by principal congruences of finite lattices. Finally, we prove that every chain-representable inclusion Q ⊆ D can be represented by the principal congruences of a finite (and quite small) algebra. |
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| Terjedelem/Fizikai jellemzők: | 357-375 |
| ISSN: | 0001-6969 |