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Δ1-completions of a Poset

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.

References

  1. Almeida, A.: Canonical extensions and relational representations of lattices with negation. Stud. Log. 91(2), 171–199 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  2. Banaschewski, B.: Hüllensysteme und Erweiterung von Quasi-Ordnungen. Z. Math. Log. Grundl. Math. 2, 117–130 (1956)

    MathSciNet  MATH  Article  Google Scholar 

  3. Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeille completion. Arch. Math. (Basel) 18, 369–377 (1967)

    MathSciNet  MATH  Article  Google Scholar 

  4. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press (2001)

  5. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press (2002)

  6. Dedekind, R.: Stetigkeit und Irrationale Zahlen, Authorised Translation Entitled Essays in the Theory of Numbers. Chicago Open Court Publisher (1901)

  7. Dunn, J.M., Hardegree, G.M.: Algebraic Methods in Philosophical Logic. Oxford University Press, New York (2001)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  9. Erné, M.: Adjunctions and standard constructions for partially ordered sets. Contrib. Gen. Algebra 2, 77–106 (1983)

    Google Scholar 

  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)

    Google Scholar 

  11. Gehrke, M.: Generalized Kripke frames. Stud. Log. 84, 241–275 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  12. Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238, 345–371 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  13. Gehrke, M., Harding, J., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  14. Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Jpn. 40, 207–215 (1994)

    MATH  Google Scholar 

  15. Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  16. Gehrke, M., Priestley, H.A.: Duality for double quasioperator algebras via their canonical extensions. Stud. Log. 68, 31–68 (2007)

    MathSciNet  Article  Google Scholar 

  17. Gehrke, M., Priestley, H.A.: Canonical extensions and completions of posets and lattices. Rep. Math. Log. 48, 133–152 (2008)

    MathSciNet  Google Scholar 

  18. Gehrke, M., Jansana, R., Palmigiano, A.: Canonical extensions for congruential logics with the deduction theorem. Ann. Pure Appl. Logic 161, 1502–1519 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  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)

  21. Jónsson, B., Tarski, A.: Boolean algebras with operators, I. Am. J. Math. 73, 891–939 (1951)

    MATH  Article  Google Scholar 

  22. Jónsson, B., Tarski, A.: Boolean algebras with operators, II. Am. J. Math. 74, 127–162 (1952)

    MATH  Article  Google Scholar 

  23. Hartung, G.: A topological representation of lattices. Algebra Univers. 29, 273–299 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  24. MacNeille, H.M.: Partially ordered sets. Trans. Am. Math. Soc. 42, 416–460 (1937)

    MathSciNet  Article  Google Scholar 

  25. Urquhart, A.: A topological representation theory for lattices. Algebra Univers. 8, 45–58 (1978)

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Mai Gehrke.

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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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Keywords

  • Completions of a poset
  • Canonical extensions