Abstract
This note gives a complete characterization of when the ordinal sum of two lattices (the lattice obtained by placing the second lattice on top of the first) is projective. This characterization applies not only to the class of all lattices, but to any variety of lattices, and in particular, to the class of distributive lattices. Lattices L with the property that every epimorphism onto L has an isotone section are also characterized.
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References
Balbes, R.: Generating sets for catalytic and projective distributive lattices. Algebra Univer. 30, 262–268 (1993)
Balbes, R., Horn, A.: Projective distributive lattices. Pacific J. Math. 33, 421–435 (1970)
Bergman, G. M., Grätzer, G.: Isotone maps on lattices. Algebra Univer. 68(1–2), 17–37 (2012)
Freese, R., Ježek, J., Nation, J. B.: Free Lattices. Amer. Math. Soc., Providence. Mathematical Surveys and Monographs, vol. 42 (1995)
Freese, R., Nation, J. B.: Projective lattices. Pac. J. Math. 75, 93–106 (1978)
Jónsson, B., Nation, J. B.: A report on sublattices of a free lattice. In: Contributions to universal algebra, pp. 233–257. North-Holland Publishing Co., Amsterdam. Coll. Math. Soc. János Bolyai, vol. 17 (1977)
Kostinsky, A.: Projective lattices and bounded homomorphisms. Pac. J. Math. 40, 111–119 (1972)
McKenzie, R.: Equational bases and non-modular lattice varieties. Trans. Amer. Math. Soc. 174, 1–43 (1972)
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Freese, R., Nation, J.B. Projective Ordinal Sums of Lattices and Isotone Sections. Order 32, 245–254 (2015). https://doi.org/10.1007/s11083-014-9329-5
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DOI: https://doi.org/10.1007/s11083-014-9329-5