Representing some families of monotone maps by principal lattice congruences

For a lattice L with 0 and 1, let Princ( L ) denote the set of principal congruences of L . Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Grätzer proved that every bounded ordered set is representable as Princ( L ); in fact, he constructed L as a lattice of length 5. For {0, 1}-...

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Bibliographic Details
Published in:Algebra universalis 2017-02, Vol.77 (1), p.51-77
Main Author: Czédli, Gábor
Format: Article
Language:English
Subjects:
Algebra
Batch processing
Congruences
Lattices (mathematics)
Mathematics
Mathematics and Statistics
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Summary:For a lattice L with 0 and 1, let Princ( L ) denote the set of principal congruences of L . Ordered by set inclusion, it is a bounded ordered set. In 2013, G. Grätzer proved that every bounded ordered set is representable as Princ( L ); in fact, he constructed L as a lattice of length 5. For {0, 1}-sublattices A ⊆ B of L , congruence generation defines a natural map Princ( A ) ⟶ Princ( B ). In this way, every family of {0, 1}-sublattices of L yields a small category of bounded ordered sets as objects and certain 0-separating {0, 1}-preserving monotone maps as morphisms such that every hom-set consists of at most one morphism. We prove the converse: every small category of bounded ordered sets with these properties is representable by principal congruences of selfdual lattices of length 5 in the above sense. As a corollary, we can construct a selfdual lattice L in G. Grätzer's above-mentioned result.
ISSN:0002-5240
1420-8911
DOI:10.1007/s00012-016-0419-7