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

Free Skew Boolean Intersection Algebras and Set Partitions

Article

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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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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