Skip to main content
SpringerLink
Log in

Boolean sets, skew Boolean algebras and a non-commutative Stone duality

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

We describe right-hand skew Boolean algebras in terms of a class of presheaves of sets over Boolean algebras called Boolean sets, and prove a duality theorem between Boolean sets and étalé spaces over Boolean spaces.

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. Doctor H. P.: The categories of Boolean lattices, Boolean rings and Boolean spaces. Canad. Math. Bull. 7, 245–252 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  2. Kudryavtseva G.: A refinement of Stone duality to skew Boolean algebras. Algebra Universalis 67, 397–416 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kudryavtseva G.: A dualizing object approach to noncommutative Stone duality. J. Austral. Math. Soc. 95, 383–403 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kudryavtseva G., Lawson M. V.: The structure of generalized inverse semigroups. Semigroup Forum 89, 19–216 (2014)

    Article  MathSciNet  Google Scholar 

  5. Kudryavtseva, G., Lawson, M. V.: Etale actions and non-commutative Stone duality. (in preparation)

  6. Lawson, M. V.: Inverse semigroups: the theory of partial symmetries. World Scientific, NJ (1998)

  7. Lawson M. V.: A non-commutative generalization of Stone duality. J. Austral. Math. Soc. 88, 385–404 (2010)

    Article  MATH  Google Scholar 

  8. Lawson M. V.: Non-commutative Stone duality: inverse semigroups, topological groupoids and \({C^\bigstar}\)-algebras. Internat. J. Algebra Comput. 22, 1250058 (2012)

    Article  MathSciNet  Google Scholar 

  9. Lawson M. V., Lenz D. H.: Pseudogroups and their etale groupoids. Adv. Math. 244, 117–170 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Leech J.: Skew Boolean Algebras. Algebra Universalis 27, 497–506 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. Leech J.: Recent developments in the theory of skew lattices. Semigroup Forum 52, 7–24 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic. Springer, New York (1994)

  13. Stone M.H.: Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41, 375–481 (1937)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. Ganna Kudryavtseva

    Present address: University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova cesta 2, SI-1000, Ljubljana, Slovenia

Authors and Affiliations

  1. Faculty of Computer and Information Science, University of Ljubljana, Tržaška cesta 25, SI-1001, Ljubljana, Slovenia

    Ganna Kudryavtseva

  2. Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, SI-1000, Ljubljana, Slovenia

    Ganna Kudryavtseva

  3. Institute Jožef Stefan, Jamova cesta 39, SI-1000, Ljubljana, Slovenia

    Ganna Kudryavtseva

  4. Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, Scotland

    Mark V. Lawson

Authors
  1. Ganna Kudryavtseva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark V. Lawson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark V. Lawson.

Additional information

Presented by M. Jackson.

The first author was partially supported by ARRS grant P1-0288, and the second by EPSRC grant EP/I033203/1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kudryavtseva, G., Lawson, M.V. Boolean sets, skew Boolean algebras and a non-commutative Stone duality. Algebra Univers. 75, 1–19 (2016). https://doi.org/10.1007/s00012-015-0361-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-015-0361-0

2010 Mathematics Subject Classification

Key words and phrases

Search

Navigation