## $$\Delta_1$$-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
Full Text:

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.