Abstract
For bounded lattices L1 and L2, let \({f\colon L_1 \to L_2}\) be a lattice homomorphism. Then the map \({{\rm Princ}(f)\colon \rm {Princ}(\it L_1) \to {\rm Princ}(\it L_2)}\), defined by \({{\rm con}(x,y) \mapsto {\rm con}(f(x),f(y))}\), is a 0-preserving isotone map from the bounded ordered set Princ(L1) of principal congruences of L1 to that of L2. We prove that every 0-preserving isotone map between two bounded ordered sets can be represented in this way. Our result generalizes a 2016 result of G. Grätzer from \({\{0,1}\}\)-preserving isotone maps to 0-preserving isotone maps.
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This research was supported by NFSR of Hungary (OTKA), grant number K 115518.
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Czédli, G. Representing an isotone map between two bounded ordered sets by principal lattice congruences. Acta Math. Hungar. 155, 332–354 (2018). https://doi.org/10.1007/s10474-018-0844-5
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DOI: https://doi.org/10.1007/s10474-018-0844-5
Key words and phrases
- principal congruence
- lattice congruence
- ordered set
- poset
- quasi-colored lattice
- preordering
- quasiordering
- isotone map
- representation