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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Format:  Article 
Published: 
2018

Series:  Acta scientiarum mathematicarum
84 No. 34 
Kulcsszavak:  Algebra, Matematika 
doi:  10.14232/actasm0175387 
Online Access:  http://acta.bibl.uszeged.hu/56919 
Summary:  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 joinirreducible 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 joinreducible coatom is fully chainrepresentable 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 chainrepresentable inclusion Q ⊆ D can be represented by the principal congruences of a finite (and quite small) algebra. 

Physical Description:  357375 
ISSN:  00016969 