Free skew Boolean algebras

Abstract

Abstract. We study the structure and properties of free skew Boolean algebras. For finite generating sets, these free algebras are finite and we give their representation as a product of primitive algebras and provide formulas for calculating their cardinality. We also characterize atomic elements and central elements and calculate the number of such elements. These results are used to study minimal generating sets of finite skew Boolean algebras. We also prove that the center of the free infinitely generated algebra is trivial and show that all free algebras have intersections.

Free Skew Boolean Intersection Algebras and Set Partitions

“…Thus all results obtained in Sections 3 and 4 of this paper for lefthanded algebras, admit extensions to general algebras. These extensions can be obtained similarly as in [11] where results on free SBAs are discussed and explicitly formulated also for right-handed and general algebras. We leave the details to an interested reader.…”

“…Then y 1 → (1, 1, 0, 1, 1, 1), y 2 → (2, 0, 1, 1, 2, 2), y 3 → (0, 2, 2, 2, 1, 2), and the obtained three elements generate S = 3 6 L as an SBIA. Note that S can not be generated by three generators as an SBA, but at least by four generators, see [11].…”

“…Let X be a countable set. It was shown in [11] that L SBA X /D, the free left-handed SBA over X, is isomorphic to GBA X . We now show, that in contrast to Corollary 29, also L SBIA X /D ≃ GBA X .…”

“…Free skew Boolean algebras have been recently systematically studied by Leech and the author in [11]. There it was shown that a free SBA over X can be looked at as a certain 'upgrade' of the free GBA over X.…”

“…In particular, in the case where the generating set X is finite of cardinality n, the atoms of the free left-handed SBA over X are in a bijective correspondence with pointed non-empty subsets of X, that is, pairs (A, a) where A is a non-empty subset of X and a ∈ A. It is shown in [11] that for any generating set X, the free SBA over X has intersections, however it is not free as an SBIA, as the intersection of any two generators equals 0. This raises the question to study the structure of free SBIAs.…”