Abstract
A join-completion of a poset is a completion for which each element is obtainable as a supremum, or join, of elements from the original poset. It is well known that the join-completions of a poset are in one-to-one correspondence with the closure systems on the lattice of up-sets of the poset. A Δ1-completion of a poset is a completion for which, simultaneously, each element is obtainable as a join of meets of elements of the original poset and as a meet of joins of elements from the original poset. We show that Δ1-completions are in one-to-one correspondence with certain triples consisting of a closure system of down-sets of the poset, a closure system of up-sets of the poset, and a binary relation between these two systems. Certain Δ1-completions, which we call compact, may be described just by a collection of filters and a collection of ideals, taken as parameters. The compact Δ1-completions of a poset include its MacNeille completion and all its join- and all its meet-completions. These completions also include the canonical extension of the given poset, a completion that encodes the topological dual of the poset when it has one. Finally, we use our parametric description of Δ1-completions to compare the canonical extension to other compact Δ1-completions identifying its relative merits.
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References
Almeida, A.: Canonical extensions and relational representations of lattices with negation. Stud. Log. 91(2), 171–199 (2009)
Banaschewski, B.: Hüllensysteme und Erweiterung von Quasi-Ordnungen. Z. Math. Log. Grundl. Math. 2, 117–130 (1956)
Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeille completion. Arch. Math. (Basel) 18, 369–377 (1967)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press (2001)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press (2002)
Dedekind, R.: Stetigkeit und Irrationale Zahlen, Authorised Translation Entitled Essays in the Theory of Numbers. Chicago Open Court Publisher (1901)
Dunn, J.M., Hardegree, G.M.: Algebraic Methods in Philosophical Logic. Oxford University Press, New York (2001)
Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symb. Log. 70(3), 713–740 (2005)
Erné, M.: Adjunctions and standard constructions for partially ordered sets. Contrib. Gen. Algebra 2, 77–106 (1983)
Erné, M.: Adjunctions and Galois connections: origins, history and development. In: Denecke, K., et al. (eds.) Galois Connections and Applications, pp. 1–138. Kluwer, Boston, MA (2004)
Gehrke, M.: Generalized Kripke frames. Stud. Log. 84, 241–275 (2006)
Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238, 345–371 (2001)
Gehrke, M., Harding, J., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005)
Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Jpn. 40, 207–215 (1994)
Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005)
Gehrke, M., Priestley, H.A.: Duality for double quasioperator algebras via their canonical extensions. Stud. Log. 68, 31–68 (2007)
Gehrke, M., Priestley, H.A.: Canonical extensions and completions of posets and lattices. Rep. Math. Log. 48, 133–152 (2008)
Gehrke, M., Jansana, R., Palmigiano, A.: Canonical extensions for congruential logics with the deduction theorem. Ann. Pure Appl. Logic 161, 1502–1519 (2010)
Haim, M.: Duality for lattices with operators: a modal logic approach. Master Dissertation MoL2000-02, ILLC. http://www.illc.uva.nl/Publications/reportlist.php?Series=MoL (2000)
Johnstone, P.T.: Stone Spaces. Cambridge University Press (1982)
Jónsson, B., Tarski, A.: Boolean algebras with operators, I. Am. J. Math. 73, 891–939 (1951)
Jónsson, B., Tarski, A.: Boolean algebras with operators, II. Am. J. Math. 74, 127–162 (1952)
Hartung, G.: A topological representation of lattices. Algebra Univers. 29, 273–299 (1992)
MacNeille, H.M.: Partially ordered sets. Trans. Am. Math. Soc. 42, 416–460 (1937)
Urquhart, A.: A topological representation theory for lattices. Algebra Univers. 8, 45–58 (1978)
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The research of the second author has been partially supported by SGR2005-00083 research grant of the research funding agency AGAUR of the Generalitat de Catalunya and by the MTM2008-01139 research grant of the Spanish Ministry of Education and Science.
The research of the third author has been supported by the VENI grant 639.031.726 of the Netherlands Organisation for Scientific Research (NWO).
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Gehrke, M., Jansana, R. & Palmigiano, A. Δ1-completions of a Poset. Order 30, 39–64 (2013). https://doi.org/10.1007/s11083-011-9226-0
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DOI: https://doi.org/10.1007/s11083-011-9226-0