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Representing some families of monotone maps by principal lattice congruences

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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.

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  1. University of Szeged, Bolyai Institute, Szeged, Aradi vértanúk tere 1, Hungary, 6720

    Gábor Czédli

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  1. Gábor Czédli
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Correspondence to Gábor Czédli.

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Presented by B. Davey.

Dedicated to George Grätzer on the occasion of his eightieth birthday

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Czédli, G. Representing some families of monotone maps by principal lattice congruences. Algebra Univers. 77, 51–77 (2017). https://doi.org/10.1007/s00012-016-0419-7

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