Algebra universalis

, Volume 77, Issue 1, pp 51–77 | Cite as

Representing some families of monotone maps by principal lattice congruences

  • Gábor Czédli


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 \({A \subseteq B}\) of L, congruence generation defines a natural map Princ(A) \({\longrightarrow}\) 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.

Key words and phrases

principal congruence lattice congruence ordered set order poset quasi-colored lattice preordering quasiordering monotone map categorified lattice functor lattice category lifting diagrams 

2010 Mathematics Subject Classification



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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.University of Szeged, Bolyai InstituteSzegedHungary

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