Skip to main content
SpringerLink
Log in

Bohr Compactifications of Algebras and Structures

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

This paper provides a unifying framework for a range of categorical constructions characterised by universal mapping properties, within the realm of compactifications of discrete structures. Some classic examples fit within this broad picture: the Bohr compactification of an abelian group via Pontryagin duality, the zero-dimensional Bohr compactification of a semilattice, and the Nachbin order-compactification of an ordered set. The notion of a natural extension functor is extended to suitable categories of structures and such a functor is shown to yield a reflection into an associated category of topological structures. Our principal results address reconciliation of the natural extension with the Bohr compactification or its zero-dimensional variant. In certain cases the natural extension functor and a Bohr compactification functor are the same; in others the functors have different codomains but may agree on all objects. Coincidence in the stronger sense occurs in the zero-dimensional setting precisely when the domain is a category of structures whose associated topological prevariety is standard. It occurs, in the weaker sense only, for the class of ordered sets and, as we show, also for infinitely many classes of ordered structures. Coincidence results aid understanding of Bohr-type compactifications, which are defined abstractly. Ideas from natural duality theory lead to an explicit description of the natural extension which is particularly amenable for any prevariety of algebras with a finite, dualisable, generator. Examples of such classes—often varieties—are plentiful and varied, and in many cases the associated topological prevariety is standard.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories: The Joy of Cats. Wiley, New York (1990). Republished in: Reprints in Theory and Applications of Categories, no. 17 (2006), pp. 1–507. Available at http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf

    MATH  Google Scholar 

  2. Banaschewski, B.: Remarks on Dual Adjointness. In: Nordwestdeutsches Kategorienseminar, Tagung, Bremen, 1976. Math.-Arbeitspapiere 7, Teil A: Math. Forschungspapiere, pp 3–10. University of Bremen, Bremen (1976)

  3. Begum, S.N., Clark, D.M., Davey, B.A., Perkal, N.: Axiomatisation modulo Priestley, (preprint)

  4. Bergman, G.M.: An Invitation to General Algebra and Universal Constructions. Henry Helson, Berkeley (1998). Available at http://math.berkeley.edu/~gbergman/245

    MATH  Google Scholar 

  5. Bezhanishvili, G., Gehrke, M., Mines, R., Morandi, P.J.: Profinite completions and canonical extensions of Heyting algebras. Order 23, 143–161 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bezhanishvili, G., Mines, R., Morandi, P.J.: The Priestley separation axiom for scattered spaces. Order 19, 1–10 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bezhanishvili, G., Morandi, P.J.: Priestley rings and Priestley order-compactifications. Order 28, 399–413 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clark, D.M., Davey, B.A.: Natural Dualities for the Working Algebraist. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  9. Clark, D.M., Davey, B.A., Freese, R.S., Jackson, M.: Standard topological algebras: syntactic and principal congruences and profiniteness. Algebra Universalis 52, 343–376 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Clark, D.M., Davey, B.A., Haviar, M., Pitkethly, J.G., Talukder, M.R.: Standard topological quasi-varieties. Houston J. Math. 29, 859–887 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Clark, D.M., Davey, B.A., Jackson, M., Maróti, M., McKenzie, R.N.: Principal and syntactic congruences in congruence-distributive and congruence-permutable varieties. J. Aust. Math. Soc. 85, 59–74 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Clark, D.M., Davey, B.A., Jackson, M., Pitkethly, J.G.: The axiomatizability of topological prevarieties. Adv. Math. 218, 1604–1653 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Clark, D.M., Davey, B.A., Pitkethly, J.G., Rifqui, D.L.: Flat unars: the primal, the semi-primal and the dualisable. Algebra Universalis 63, 303–329 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Clark, D.M., Idziak, P.M., Sabourin, L.R., Szabó, C., Willard, R.: Natural dualities for quasivarieties generated by a finite commutative ring. Algebra Universalis 46, 285–320 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Davey, B.A.: Topological duality for prevarieties of universal algebras. In: Rota, G.-C. (ed.) Studies in Foundations and Combinatorics, Adv. in Math. Suppl. Stud. 1, pp 61–99. Academic Press, New York (1978)

  16. Davey, B.A.: Natural dualities for structures. Acta Univ. M. Belii Ser. Math. 13, 3–28 (2006). Available at http://actamath.savbb.sk/pdf/acta1301.pdf

    MathSciNet  MATH  Google Scholar 

  17. Davey, B.A., Gouveia, M.J., Haviar, M., Priestley, H.A.: Natural extensions and profinite completions of algebras. Algebra Universalis 66, 205–241 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Davey, B.A., Haviar, M., Priestley, H.A.: Boolean topological distributive lattices and canonical extensions. Appl. Categ. Structures 15, 225–241 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Davey, B.A., Haviar, M., Priestley, H.A.: Natural dualities in partnership. Appl. Categ. Structures 20, 583–602 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Davey, B.A., Haviar, M., Priestley, H.A.: Piggyback dualities revisited. Algebra Universalis. arXiv:1501.02512 [math.RA]

  21. Davey, B.A., Jackson, M., Pitkethly, J.G., Talukder, M.R.: Natural dualities for semilattice-based algebras. Algebra Universalis 57, 463–490 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Davey, B.A., Nguyen, L., Pitkethly, J.G.: Counting relations on Ockham algebras. Algebra Universalis 74, 35–63 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Davey, B.A., Priestley, H.A.: Generalized piggyback dualities and applications to Ockham algebras. Houston J. Math. 13, 151–197 (1987)

    MathSciNet  MATH  Google Scholar 

  24. Davey, B.A., Priestley, H.A.: Canonical extensions and discrete dualities for finitely generated varieties of lattice-based algebras. Stud. Logica. 100, 137–161 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Davey, B.A., Talukder, M.R.: Dual categories for endodualisable Heyting algebras: optimization and axiomatization. Algebra Universalis 53, 331–355 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dikranjan, D., Ferrer, M.V., Hernández, S.: Dualities in topological groups. Sci. Math. Jpn. 72, 197–235 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Engelking, R.: General Topology. PWN—Polish Scientific Publishers, Warsaw (1977)

    MATH  Google Scholar 

  28. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press (2003)

  29. Goldberg, M.S.: Distributive Ockham algebras: free algebras and injectivity. Bull. Austral. Math. Soc. 24, 161–203 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gouveia, M.J., Priestley, H.A.: Profinite completions and canonical extensions of semilattice reducts of distributive lattices. Houston J. Math. 39, 1117–1136 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Gouveia, M.J., Priestley, H.A.: Profinite completions of semilattices and canonical extensions of semilattices and lattices. Order 31, 189–216 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hart, J.E., Kunen, K.: Bohr compactifications of discrete structures. Fund. Math. 160, 101–151 (1999)

    MathSciNet  MATH  Google Scholar 

  33. Hofmann, K.H., Mislove, M., Stralka, A.: The Pontryagin duality of compact O-dimensional semilattices and its applications. Lecture Notes in Mathematics 396, Springer (1974)

  34. Holm, P.: On the Bohr compactification. Math. Annalen 156, 34–46 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jackson, M.: Residual bounds for compact totally disconnected algebras. Houston J. Math. 34, 33–67 (2008)

    MathSciNet  MATH  Google Scholar 

  36. Jackson, M.: Natural dualities, nilpotence and projective planes. Algebra Universalis 74, 65–85 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Johnstone, P.T.: Stone Spaces. Cambridge University Press (1980)

  38. Johansen, S.M.: Natural dualities for three classes of relational structures. Algebra Universalis 63, 149–170 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Koppelberg, S.: Handbook of Boolean Algebras. In: Monk, J.D., Bonnet, R. (eds.) , vol. 1. North-Holland Publishing Co., Amsterdam (1989)

  40. Nachbin, L.: Topology and Order. Robert E. Krieger Publishing Co., Huntington (1976)

    MATH  Google Scholar 

  41. Nailana, K.R.: (Strongly) Zero-dimensional partially ordered spaces. Papers in honour of Bernhard Banaschewski (Cape Town, 1996), pp. 445–456. Kluwer, Dordrecht (2000) reprint Springer (2010)

  42. Numakura, K.: Theorems on compact totally disconnected semigroups and lattices. Proc. Amer. Math. Soc. 8, 623–626 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pitkethly, J.G., Davey, B.A.: Dualisability: Unary algebras and beyond. Advances in Mathematics 9. Springer (2005)

  44. Pontryagin, L.S.: Topological Groups, 2nd edn. Gordon & Breach, New York (1966)

    Google Scholar 

  45. Quackenbush, R., Szabó, C.: Strong duality for metacyclic groups. J. Aust. Math. Soc. 73, 377–392 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Semadeni, Z.: Banach Spaces of Continuous Functions. PWN—Polish Scientific Publishers, Warsaw (1971)

    MATH  Google Scholar 

  47. Stralka, A.: A partially ordered space which is not a Priestley space. Semigroup Forum 20, 293–297 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, La Trobe University, Victoria, 3086, Australia

    B. A. Davey

  2. Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01, Banská Bystrica, Slovak Republic

    M. Haviar

  3. Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford, OX2 6GG, UK

    H. A. Priestley

Authors
  1. B. A. Davey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Haviar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. A. Priestley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. A. Davey.

Additional information

The first author wishes to thank the Research Institute of M. Bel University in Banská Bystrica for its hospitality while working on this paper. The second author acknowledges support from Slovak grants APVV-0223-10, Mobility-ITMS 26110230082 and VEGA 1/0212/13.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davey, B.A., Haviar, M. & Priestley, H.A. Bohr Compactifications of Algebras and Structures. Appl Categor Struct 25, 403–430 (2017). https://doi.org/10.1007/s10485-016-9436-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-016-9436-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation