# Representing a monotone map by principal lattice congruences

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## 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## Mathematics Subject Classification

06B10## Preview

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© Akadémiai Kiadó, Budapest, Hungary 2015