Page 135 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press

IV: Skew Boolean Algebras

[2.2]

Theorem 4.2.6. The free left-handed skew Boolean algebra LSBAn on {x1, …, xn} is a
direct sum of the primitive algebras PL{L|M} where {L|M} ranges over partitions {L|M} of

{x1, …, xn} where L ≠ ∅. Thus:

⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
LSBAn ≅ 0 ⎠⎟ × 1 ⎟⎠ × 3 ⎝⎜ 2 ⎟⎠ × 4 ⎝⎜ 3 ⎠⎟ × ... × (n + n ⎠⎟ .
1⎝⎜ 2⎝⎜ 1)L⎜⎝
L L

Dually, the free right-handed skew Boolean algebra RSBAn on {x1, …, xn} is a direct
sum of the primitive algebras PR{L|M} where {L|M} shares the same range. Thus:

⎛n⎞ ⎛n⎞ ⎛n⎞ ⎛n⎞ ⎛ n⎞
⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟
1⎜ ⎟ 2⎜⎝⎜1 ⎟ ⎜ 2 ⎟⎠⎟ ⎜ ⎟ (n + 1)R⎜⎜⎝ n⎠⎟⎟
RSBAn ≅ ⎜⎝ 0 ⎠⎟ × ⎟⎠ × 3 ⎜⎝ × 4 ⎝⎜ 3 ⎟⎠ × ... × .

R R

Finally, the free skew Boolean algebra SBAn on {x1, …, xn} is a direct sum of the
primitive algebras PL{L|M}•PR{L|M} where {L|M} again shares the same range. Thus:

⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞
1⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜⎝ 0 ⎟⎠ ×⎝⎜⎜1 ⎟⎟⎠ 3L • 3R ⎜⎝⎜ 2⎠⎟⎟ × ⎜⎝ 3 ⎠⎟ ⎜⎝ n ⎠⎟
( ) ( ) ( )SBAn
≅ × 2 4L • 4R × ... × (n + 1)L• (n + 1)R . £

Corollary 4.2.7. For all n ≥ 1:

⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞
n⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
i) |LSBAn| = 2⎝⎜1 ⎟⎠ 2 ⎠⎟ 4 ⎝⎜ 3 ⎟⎠  ( + n ⎟⎠ = |RSBAn|.
3⎝⎜ 1)⎝⎜

⎛ n⎞ ⎛ n⎞ ⎛ n⎞ ⎛ n⎞
ii) |SBAn| = 2⎝⎜⎜1 ⎠⎟⎟ 5⎝⎜⎜ 2⎟⎠⎟10⎜⎝⎜ 3⎟⎠⎟  (n2 + 1)⎝⎜⎜ n⎟⎠⎟ .

Moreover, if αL(n), αR(n) and α(n) denote the number of atoms in LSBAn, RSBAn and
SBAn, respectively, then:

iii) αL(n) = ⎛ n⎞ 1 + ⎛ n⎞ 2 + ... + ⎛ n ⎞ (n − 1) + ⎛ n⎞ n = αR(n) .
⎝⎜1 ⎠⎟ ⎜⎝ 2⎟⎠ ⎜⎝ n− 1⎠⎟ ⎝⎜ n⎠⎟

iv) α(n) = ⎛ n⎞ 1 + ⎛ n⎞ 4 +  + ⎛ n ⎞ (n − 1)2 + ⎛ n⎞ n2 . £
⎝⎜1 ⎠⎟ ⎝⎜ 2⎟⎠ ⎝⎜ n− 1⎟⎠ ⎜⎝ n⎠⎟

Standard combinatorial arguments give the following simplifications:

Corollary 4.2.8. Given αL(n), αR(n) and α(n) as above:

αL(n) = αR(n) = n2n–1 and α(n) = n(n+1)2n–2, so that α(n) = n+1 αL(n).
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