Abstract
We investigate connections between the free lattice generated by a poset while preserving certain bounds and the canonical extension of a poset. Explicitly, we describe how the free lattice generated by a poset while preserving certain bounds can be constructed as a colimit of ‘intermediate structures’as they occur in the construction of a canonical extension of a poset.
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Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeille completion. Arch. Math. (Basel) 18, 369–377 (1967)
Dean, R.A.: Free lattices generated by partially ordered sets and preserving bounds. Canad. J. Math. 16, 136–148 (1964)
Dunn, J., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symb. Logic 70, 713–740 (2005)
Egrot, R.: Order polarities. J. Logic Comput. 30(3), 785–833 (2020)
Freese, R., Ježek, J., Nation, J.B.: Free lattices, volume 42 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1995)
Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238(1), 345–371 (2001)
Gehrke, M., Jansana, R., Palmigiano, A.: \({{\Delta }_1}\)-completions of a poset. Order 30(1), 39–64 (2013)
Hodges, W.: Model theory, volume 42 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1993)
Lakser, H.: Lattices freely generated by an order and preserving certain bounds. Algebra Universalis 67(2), 113–120 (2012)
Mac Lane, S.L.: Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics, 2nd edn. Springer-Verlag, New York (1998)
MacNeille, H.M.: Partially ordered sets. Trans. Amer. Math. Soc. 42(3), 416–460 (1937)
Morton, W.: Canonical extensions of posets. Algebra Universalis 72(2), 167–200 (2014)
Morton, W., van Alten, C.J.: Distributive and completely distributive lattice extensions of ordered sets. Internat. J. Algebra Comput. 28(3), 521–541 (2018)
Tunnicliffe, W.R.: The completion of a partially ordered set with respect to a polarization. Proc. London Math. Soc. 3(28), 13–27 (1974)
van Gool, S.J.: Methods for canonicity. Master’s thesis, Universiteit van Amsterdam (2009)
Whitman, P.M.: Free lattices. Ann. of Math. 2(42), 325–330 (1941)
Whitman, P.M.: Free lattices. II. Ann. Math. 2(43), 104–115 (1942)
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Egrot, R. Amalgamating Poset Extensions and Generating Free Lattices. Order 40, 47–62 (2023). https://doi.org/10.1007/s11083-022-09594-7
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DOI: https://doi.org/10.1007/s11083-022-09594-7