Skip to main content
Log in

Distributivity in skew lattices

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

Distributive skew lattices satisfying \(x\wedge (y\vee z)\wedge x = (x\wedge y\wedge x) \vee (x\wedge z\wedge x)\) and its dual are studied, along with the larger class of linearly distributive skew lattices, whose totally preordered subalgebras are distributive. Linear distributivity is characterized by the behavior of the natural partial order \(\ge \) on elements in chains of comparable \({\mathcal {D}}\)-classes, \(A>B>C\), with particular attention given to midpoints \(b\) of chains \(a > b > c\) where \(a \in A\), \(b \in B\) and \(c \in C\). Since distributive skew lattices are linearly distributive and have distributive maximal lattice images (but not conversely in general), we give criteria that guarantee that skew lattices with both properties are distributive. In particular symmetric skew lattices (where \(x\wedge y = y\wedge x\) if and only if \(x\vee y = y\vee x\)) that have both properties are distributive.

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

Access this article

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. Bauer, A., Cvetko-Vah, K.: Stone duality for skew Boolean intersection algebras. Houst. J. Math. 39, 73–109 (2013)

    MATH  MathSciNet  Google Scholar 

  2. Bauer, A., Cvetko-Vah, K., Gehrke, M., van Gool, S.J., Kudryavtseva, G.: A non-commutative Priestley duality. Topol. Appl. 160, 1423–1438 (2013)

    Article  MATH  Google Scholar 

  3. Bignall, R., Leech, J.: Skew boolean algebras and discriminator varieties. Algebra Universalis 33, 387–398 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bignall, R., Spinks, M.: Propositional skew Boolean logic. In: Proceedings on 26th International Symposium on Multiple-valued Logic, pp. 43–48. IEEE Computer Society Press, (1996)

  5. Birkhoff, G.: Lattice Theory, vol. 25. American Mathematical Society, Providence (1967)

    MATH  Google Scholar 

  6. McCune, W.: Mace4/Prover9, Version Dec 2007, www.cs.unm.edu/~mccune/mace4

  7. Cvetko-Vah, K.: Skew lattices of matrices in rings. Algebra Universalis 53, 471–479 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cvetko-Vak, K.: Skew lattices in rings. PhD Thesis. University of Ljubljana (2005)

  9. Cvetko-Vah, K.: Internal decompositions of skew lattices. Commun. Algebra 35, 243–247 (2006)

    Article  MathSciNet  Google Scholar 

  10. Cvetko-Vah, K.: A new proof of Spinks’ theorem. Semigroup Forum 73, 267–272 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cvetko-Vah, K., Kinyon, M., Leech, J., Spinks, M.: Cancellation in skew lattices. Order 28, 9–32 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cvetko-Vah, K., Leech, J.: Associativity of the \(\nabla \) operation on bands in rings. Semigroup Forum 76, 32–50 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cvetko-Vah, K., Leech, J.: Rings whose idempotents are multiplicatively closed. Commun. Algebra 40, 3288–3307 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cvetko-Vah, K., Leech, J.: On maximal idempotent-closed subrings of \(M_{n}(F)\). Int. J. Algebra Comput. 21(7), 1097–1110 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kinyon, M., Leech, J.: Categorical skew lattices. Order 30, 763–777 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Laslo, G., Leech, J.: Green’s relations on noncommutative lattices. Acta Sci. Math. 68, 501–533 (2002). (Szeged)

    MATH  MathSciNet  Google Scholar 

  18. Leech, J.: Skew lattices in rings. Algebra Universalis 26, 48–72 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  19. Leech, J.: Normal skew lattices. Semigroup Forum 44, 1–8 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  20. Leech, J.: Skew boolean algebras. Algebra Universalis 27, 497–506 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  21. Leech, J.: The geometric structure of skew lattices. Trans. Am. Math. Soc. 335, 823–842 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Leech, J.: Small skew lattices in rings. Semigroup Forum 70, 307–311 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Leech, J., Spinks, M.: Skew Boolean algebras generated from generalized Boolean algebras. Algebra Universalis 58, 287–302 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Petrich, M.: Lectures in Semigroups. Akademie-Verlag, Berlin (1977)

    Google Scholar 

  26. Pita Costa, J.: Coset laws for categorical skew lattices. Algebra Universalis 68, 75–89 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pita Costa, J.: On the coset structure of skew lattices. PhD Thesis, University of Ljubljana (2012)

  28. Spinks, M.: Automated deduction in non-commutative lattice theory, Report 3/98, Monash University, Gippsland School of Computing and Information Technology (1998)

  29. Spinks, M.: On middle distributivity for skew lattices. Semigroup Forum 61, 341–345 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. Spinks, M., Veroff, R.: Axiomatizing the skew Boolean propositional calculus. J. Autom. Reason. 37, 3–20 (2006)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors express their appreciation to the referee for a careful reading of the manuscript. The referee’s helpful suggestions have resulted in an improved presentation. The author JPC would like to acknowledge that his work was funded by the EU project TOPOSYS (FP7-ICT-318493-STREP).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jonathan Leech.

Additional information

Communicated by László Márki.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kinyon, M., Leech, J. & Pita Costa, J. Distributivity in skew lattices. Semigroup Forum 91, 378–400 (2015). https://doi.org/10.1007/s00233-015-9722-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-015-9722-4

Keywords

Navigation