Free Skew Boolean Intersection Algebras and Set Partitions

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, . . . , n}. Furthermore, under the canonical inclusion into the k-generated free algebra, where k ≥ n, 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 relation 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.

My journey into noncommutative lattices and their theory

“…Given n generators, these counts are respectively B n+1 − 1 and B n+2 − 2B n+1 . (Again, see [38,Theorem 28]. )…”

“…But instead of binomial coefficients n k−1 , the respective powers are given by Stirling numbers of the 2 nd kind, n+1 k . (See [38,Theorem 28].) Research on skew lattices and related subjects continues.…”

“…(The n th Bell number B n is the number of distinct partitions of an n-element set.) Interestingly, Bell numbers appear in Anya Kudryavtseva's paper [38] where they are used to count the number of atomic D-classes as well as the total number of atoms in a free left [right]-handed skew Boolean intersection algebra. Given n generators, these counts are respectively B n+1 − 1 and B n+2 − 2B n+1 .…”