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\(\Delta_1\)-completions of a poset. (English) Zbl 1317.06002

A \(\Delta_1\)-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. In this paper, the authors completely classify \(\Delta_1\)-completions of a poset \(P\) in terms of certain polarities \((\mathcal F,\mathcal I,R)\) where \(\mathcal F\) is a closure system of up-sets of \(P\) and \(\mathcal I\) is a closure system of down-sets of \(P\) and \(R\) is a relation from \(\mathcal F\) to \(\mathcal I\) satisfying four conditions. The relation essentially specifies which meets of down-sets are below which joins of up-sets in the completion. Further, the authors prove that the compact \(\Delta_1\)-completions may be described just by a collection of filters and a collection of ideals, taken as parameters. The compact \(\Delta_1\)-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, the authors use parametric description of \(\Delta_1\)-completions to compare the canonical extension to other compact \(\Delta_1\)-completions identifying its relative merits.

MSC:

06A06 Partial orders, general
06B23 Complete lattices, completions
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[1] Almeida, A.: Canonical extensions and relational representations of lattices with negation. Stud. Log. 91(2), 171–199 (2009) · Zbl 1172.03032
[2] Banaschewski, B.: Hüllensysteme und Erweiterung von Quasi-Ordnungen. Z. Math. Log. Grundl. Math. 2, 117–130 (1956) · Zbl 0073.26904
[3] Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeille completion. Arch. Math. (Basel) 18, 369–377 (1967) · Zbl 0157.34101
[4] Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press (2001) · Zbl 0988.03006
[5] Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press (2002) · Zbl 1002.06001
[6] Dedekind, R.: Stetigkeit und Irrationale Zahlen, Authorised Translation Entitled Essays in the Theory of Numbers. Chicago Open Court Publisher (1901) · JFM 32.0185.01
[7] Dunn, J.M., Hardegree, G.M.: Algebraic Methods in Philosophical Logic. Oxford University Press, New York (2001) · Zbl 1014.03002
[8] Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symb. Log. 70(3), 713–740 (2005) · Zbl 1101.03021
[9] Erné, M.: Adjunctions and standard constructions for partially ordered sets. Contrib. Gen. Algebra 2, 77–106 (1983) · Zbl 0533.06001
[10] 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) · Zbl 1067.06003
[11] Gehrke, M.: Generalized Kripke frames. Stud. Log. 84, 241–275 (2006) · Zbl 1115.03013
[12] Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238, 345–371 (2001) · Zbl 0988.06003
[13] Gehrke, M., Harding, J., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005) · Zbl 1077.03009
[14] Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Jpn. 40, 207–215 (1994) · Zbl 0855.06009
[15] Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005) · Zbl 1077.03009
[16] Gehrke, M., Priestley, H.A.: Duality for double quasioperator algebras via their canonical extensions. Stud. Log. 68, 31–68 (2007) · Zbl 1127.06009
[17] Gehrke, M., Priestley, H.A.: Canonical extensions and completions of posets and lattices. Rep. Math. Log. 48, 133–152 (2008) · Zbl 1147.06005
[18] Gehrke, M., Jansana, R., Palmigiano, A.: Canonical extensions for congruential logics with the deduction theorem. Ann. Pure Appl. Logic 161, 1502–1519 (2010) · Zbl 1238.03051
[19] 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)
[20] Johnstone, P.T.: Stone Spaces. Cambridge University Press (1982) · Zbl 0499.54001
[21] Jónsson, B., Tarski, A.: Boolean algebras with operators, I. Am. J. Math. 73, 891–939 (1951) · Zbl 0045.31505
[22] Jónsson, B., Tarski, A.: Boolean algebras with operators, II. Am. J. Math. 74, 127–162 (1952) · Zbl 0049.15801
[23] Hartung, G.: A topological representation of lattices. Algebra Univers. 29, 273–299 (1992) · Zbl 0790.06005
[24] MacNeille, H.M.: Partially ordered sets. Trans. Am. Math. Soc. 42, 416–460 (1937) · JFM 63.0833.04
[25] Urquhart, A.: A topological representation theory for lattices. Algebra Univers. 8, 45–58 (1978) · Zbl 0382.06010
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