Page 136 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 136
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Proof. To see ⎛ n⎞ 1+ ⎛ n⎞ 2 + ...+ ⎛ n⎞ n = n2n–1, differentiate the binomial expansion of (1 + x)n
⎜⎝1 ⎟⎠ ⎝⎜ 2⎟⎠ ⎜⎝ n⎠⎟
and set x = 1. Setting x = 1 again in the second derivative of the binomial expansion of (1 + x)n
gives: ⎛ n⎞ 2 i1 + ⎛ n⎞ 3 i 2 + + ⎛ n⎞ n(n − 1) = n(n–1)2n–2.
⎝⎜ 2⎟⎠ ⎝⎜ 3⎠⎟ ⎜⎝ n⎠⎟
Adding the equality of the previous expansion to this and simplifying gives
⎛ n⎞ 1 + ⎛ n⎞ 4 + + ⎛ n⎞ n2 = n2n–1 + n(n–1)2n–2 = n(n+1)2n–2. £
⎝⎜1 ⎟⎠ ⎜⎝ 2⎟⎠ ⎜⎝ n⎠⎟
A short table of values follows with the sizes for n = 5 given to 4-digit accuracy.
n | L SBAn | α L (n) | SBAn | α (n)
2 12 4 20 6
3 864 12 10, 000 24
4 14, 929, 920 32 425 × 108 80
5 3.715 × 1016 80 3.017 × 1025 240
Since ⎛ n⎞ + ⎛ n⎞ + ... + ⎛ n⎞ = 2n – 1, we have:
⎝⎜ 1 ⎟⎠ ⎝⎜ 2⎟⎠ ⎝⎜ n⎠⎟
Corollary 4.2.9. A free (left-handed, right-handed or two-sided) skew Boolean algebra
on n generators has 2n – 1 primitive factors in its atomic decomposition. Thus any skew Boolean
algebra on n generators has at most that many. Any generalized Boolean algebra on n generators
thus has ≤ 2n – 1 atoms, and is free if and only if it has exactly that many. £
Theorems 4.2.2 and 4.2.6 also lead to:
Corollary 4.1.20. Every finite skew Boolean algebra can be embedded in a finite free
skew Boolean algebra. Every finite left-handed [right-handed] skew Boolean algebra is
isomorphic to a direct factor of a finite free left-handed [right-handed] skew Boolean algebra.
4.3 Connections with strongly distributive skew lattices
A skew lattice is embedded in a skew Boolean algebra (S; ∨, ∧, \, 0) if it is embedded in
its skew lattice reduct (S; ∨, ∧). Since such reducts are strongly distributive, theorems from the
previous section and Section 2.6 lead to the following results.
134
Proof. To see ⎛ n⎞ 1+ ⎛ n⎞ 2 + ...+ ⎛ n⎞ n = n2n–1, differentiate the binomial expansion of (1 + x)n
⎜⎝1 ⎟⎠ ⎝⎜ 2⎟⎠ ⎜⎝ n⎠⎟
and set x = 1. Setting x = 1 again in the second derivative of the binomial expansion of (1 + x)n
gives: ⎛ n⎞ 2 i1 + ⎛ n⎞ 3 i 2 + + ⎛ n⎞ n(n − 1) = n(n–1)2n–2.
⎝⎜ 2⎟⎠ ⎝⎜ 3⎠⎟ ⎜⎝ n⎠⎟
Adding the equality of the previous expansion to this and simplifying gives
⎛ n⎞ 1 + ⎛ n⎞ 4 + + ⎛ n⎞ n2 = n2n–1 + n(n–1)2n–2 = n(n+1)2n–2. £
⎝⎜1 ⎟⎠ ⎜⎝ 2⎟⎠ ⎜⎝ n⎠⎟
A short table of values follows with the sizes for n = 5 given to 4-digit accuracy.
n | L SBAn | α L (n) | SBAn | α (n)
2 12 4 20 6
3 864 12 10, 000 24
4 14, 929, 920 32 425 × 108 80
5 3.715 × 1016 80 3.017 × 1025 240
Since ⎛ n⎞ + ⎛ n⎞ + ... + ⎛ n⎞ = 2n – 1, we have:
⎝⎜ 1 ⎟⎠ ⎝⎜ 2⎟⎠ ⎝⎜ n⎠⎟
Corollary 4.2.9. A free (left-handed, right-handed or two-sided) skew Boolean algebra
on n generators has 2n – 1 primitive factors in its atomic decomposition. Thus any skew Boolean
algebra on n generators has at most that many. Any generalized Boolean algebra on n generators
thus has ≤ 2n – 1 atoms, and is free if and only if it has exactly that many. £
Theorems 4.2.2 and 4.2.6 also lead to:
Corollary 4.1.20. Every finite skew Boolean algebra can be embedded in a finite free
skew Boolean algebra. Every finite left-handed [right-handed] skew Boolean algebra is
isomorphic to a direct factor of a finite free left-handed [right-handed] skew Boolean algebra.
4.3 Connections with strongly distributive skew lattices
A skew lattice is embedded in a skew Boolean algebra (S; ∨, ∧, \, 0) if it is embedded in
its skew lattice reduct (S; ∨, ∧). Since such reducts are strongly distributive, theorems from the
previous section and Section 2.6 lead to the following results.
134
