Acta Mathematica Hungarica

, Volume 147, Issue 1, pp 12–18

# Representing a monotone map by principal lattice congruences

• G. Czédli
Article

## Abstract

For a lattice L, let Princ (L) denote the ordered set of principal congruences of L. In a pioneering paper, G. Grätzer proved that bounded ordered sets (in other words, posets with 0 and 1) are, up to isomorphism, exactly the Princ (L) of bounded lattices L. Here we prove that for each 0-separating boundpreserving monotone map ψ between two bounded ordered sets, there are a lattice L and a sublattice K of L such that, in essence, ψ is the map from Princ (K) to Princ (L) that sends a principal congruence to the congruence it generates in the larger lattice.

## Key words and phrases

principal congruence lattice congruence ordered set order poset quasi-colored lattice preordering quasiordering monotone map

06B10

## References

1. 1.
G. Czédli, The ordered set of principal congruences of a countable lattice, algebra universalis, to appear; see also at http://www.math.u-szeged.hu/~czedli/
2. 2.
Czédli G.: Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices. Algebra Universalis 67, 313–345 (2012)
3. 3.
G. Czédli, Representing some families of monotone maps by principal lattice congruences, Algebra Universalis, submitted in April, 2015.Google Scholar
4. 4.
K. P. Bogart, R. Freese and J. P. S. Kung (editors), The Dilworth Theorems. Selected papers of Robert P. Dilworth, Birkhäuser Boston, Inc. (Boston, MA, 1990), xxvi+465 pp. ISBN: 0-8179-3434-7Google Scholar
5. 5.
G. Grätzer, The Congruences of a Finite Lattice. A Proof-by-picture Approach, Birkhäuser (Boston, 2006).Google Scholar
6. 6.
Grätzer G.: The order of principal congruences of a bounded lattice. Algebra Universalis 70, 95–105 (2013)
7. 7.
Grätzer G., Lakser H., Schmidt E. T.: Congruence lattices of finite semimodular lattices. Canad. Math. Bull. 41, 290–297 (1998)
8. 8.
Huhn A. P.: On the representation of distributive algebraic lattices. III. Acta Sci. Math. (Szeged) 53, 11–18 (1989)
9. 9.
Růžička P.: Free trees and the optimal bound in Wehrung’s theorem. Fund. Math. 198, 217–228 (2008)
10. 10.
Schmidt E. T.: The ideal lattice of a distributive lattice with 0 is the congruence lattice of a lattice. Acta Sci. Math. (Szeged) 43, 153–168 (1981)
11. 11.
Wehrung F.: A solution to Dilworth’s congruence lattice problem. Adv. Math. 216, 610–625 (2007)