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, Volume 35, Issue 1, pp 1–22 | Cite as

Free Skew Boolean Intersection Algebras and Set Partitions

  • Ganna Kudryavtseva
Article
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Abstract

We show that atoms of the n-generated free left-handed skew Boolean intersection algebra are in a bijective correspondence with pointed partitions of non-empty subsets of \(\{1,2,\dots , n\}\). Furthermore, under the canonical inclusion into the k-generated free algebra, where kn, an atom of the n-generated free algebra decomposes into an orthogonal join of atoms of the k-generated free algebra in an agreement with the containment order on the respective partitions. As a consequence of these results, we describe the structure of finite free left-handed skew Boolean intersection algebras and express several their combinatorial characteristics in terms of Bell numbers and Stirling numbers of the second kind. We also look at the infinite case. For countably many generators, our constructions lead to the ‘partition analogue’ of the Cantor tree whose boundary is the ‘partition variant’ of the Cantor set.

Keywords

Skew Boolean intersection algebra Set partition Containment order Normal form Free algebra Partition tree Cantorian tree 

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References

  1. 1.
    Bauer, A., Cvetko-Vah, K.: Stone duality for skew Boolean algebras with intersections. Houston J. Math. 39(1), 73–109 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bignall, R. J., Leech, J. E.: Skew Boolean algebras and discriminator varieties. Algebra Universalis 33, 387–398 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burris, S., Sankappanavar, H.: A Course in Universal Algebra, Grad. Texts Math, vol. 78. Springer (1981)Google Scholar
  4. 4.
    Cvetko-Vah, K., Leech, J., Spinks, M.: Skew lattices and binary operations on functions. J. Applied Logic 11, 253–265 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cvetko-Vah, K., Salibra, A.: The connection of skew Boolean algebras and discriminator varieties to Church algebras. Algebra Universalis 73, 369–390 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Howie, J. M.: Introduction to Semigroup Theory. Academic Press, London (1976)zbMATHGoogle Scholar
  7. 7.
    Kimura, N.: The structure of idempotent semigroups, I. Pacific J. Math. 8, 257–275 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kudryavtseva, G.: A refinement of Stone duality to skew Boolean algebras. Algebra Universalis 67, 397–416 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kudryavtseva, G.: A dualizing object approach to non-commutative Stone duality. J. Aust. Math. Soc. 95, 383–403 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kudryavtseva, G., Lawson, M. V.: Boolean sets, skew Boolean algebras and a non-commutative stone duality. Algebra Universalis 75, 1–19 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kudryavtseva, G., Leech, J.: Free skew boolean algebras. Int. J. Algebra Comput. doi: 10.1142/S0218196716500569
  12. 12.
    Leech, J.: Skew boolean algebras. Algebra Universalis 27, 497–506 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Leech, J.: Normal skew lattices. Semigroup Forum 44, 1–8 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Leech, J.: Recent developments in the theory of skew lattices. Semigroup Forum 52, 7–24 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Leech, J., Spinks, M: Varieties of skew boolean algebras with intersections, to appear in J. Aust. Math. Soc.Google Scholar
  16. 16.
    Michon, G.: Les Cantors réguliers. C. R. Acad. Sci. Paris Sér. I Math. 19, 673–675 (1985)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Pearson, J., Bellissard, J.: Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets. J. Noncommut. Geom. 3, 447–480 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Weisstein, E. W: Bell polynomial, from mathworld - a wolfram web resource, available online at http://mathworld.wolfram.com/bellpolynomial.html

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Faculty of Civil and Geodetic EngineeringUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Physics and MechanicsInstitute of MathematicsLjubljanaSlovenia
  3. 3.Jožef Stefan InstituteLjubljanaSlovenia

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