Page 21 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 21
I: Preliminaries
Theorem 1.1.13. A Boolean lattice L is isomorphic the power lattice of some set if and
only if it is complete and atomic. If the latter holds, then upon denoting the set of all atoms in L
by AL, one has L ≅ 2AL under the map x → α(x). £
Proposition 1.1.14. For all x in a complete Boolean lattice L and all subsets Y of L:
(i) x ∧ sup(Y) = sup({x∧y ⎜y ∈ Y}) and x ∨ inf(Y) = inf({x∨y ⎜y ∈ Y}).
(ii) (sup Y)ʹ = inf({yʹ ⎜y ∈ Y}) and (inf Y)ʹ = sup({yʹ ⎜y ∈Y}).
(iii) x \ sup({x \ y ⎜y ∈ Y}) = inf(x \ Y) and x \ inf(Y) = sup({x \ y ⎜y ∈ Y}).
(iv) sup(Y) \ x = sup({y \ x ⎜y ∈ Y}) and inf(Y) \ x = inf({y \ x ⎜y ∈ Y}). £
Theorem 1.1.15. Given a Boolean algebra (L, ∧, ∨, 1, 0, ʹ), let a ∈ L be given. Then
both a↓ and a↑ are Boolean lattices and χ: L → a↓ × a↑ defined by χ(x) = (x∧a, x∨a) is an
isomorphism of Boolean lattices.
Proof. χ is at least a lattice embedding by Theorem 1.1.9. Next, let (u, v) ∈ a↓ × a↑ be given. If
x = (v \ a) ∨ u, then
a∧x = a∧[(v \ a) ∨ u] = (a ∧ (v \ a)) ∨ (a∧u) = 0 ∨ u = u
and
a∨x = a∨[(v \ a) ∨ u] = a ∨ (v ∧ aʹ) ∨ a = (a ∨ v) ∧ (a ∨ aʹ) = v∧1 = v.
Thus χ is also surjective and the theorem follows. £
2 denotes the Boolean lattice {1 > 0}. By mild abuse of notation, 2 also denotes the
Boolean algebra ({1, 0}; ∨, ∧, 1, 0, ʹ) again with 1 > 0. Put otherwise, 2 is the chain C1 reset in a
Boolean context. We have the following sequence of easy corollaries of Theorem 1.1.15 and
Corollary 1.10.
Corollary 1.1.16. Every finite Boolean lattice factors as a finite power of 2. £
Corollary1.1.17. The only nontrivial subdirectly irreducible Boolean algebra is 2. £
Corollary1.1.18. Every distributive lattice can be embedded into a Boolean lattice. £
By a generalized Boolean lattice is meant a lattice L with a minimal element 0 such that
each principal ideal x↓ of L is a Boolean lattice. Such a lattice is necessarily distributive;
moreover a difference operation on L is given by setting x \ y = x \ (y∧x) in the Boolean lattice x↓.
For Boolean lattices both differences agree. The relative DeMorgan identities also hold for
generalized Boolean lattices. Upon including \ in the signature, one has a generalized Boolean
algebra (L, ∧, ∨, \, 0). It is characterized by the identities for a distributive lattice with a minimal
element 0 together with the pair: (x∧y) ∧ (x\y) = 0 and (x∧y) ∨ (x\y) = x.
Every Boolean algebra (L, ∧, ∨, 1, 0, ʹ) possesses a generalized Boolean algebra reduct
(L, ∧, ∨, \, 0) with x \ y given as x∧yʹ. Conversely, any generalized Boolean algebra (L, ∧, ∨, \, 0)
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Theorem 1.1.13. A Boolean lattice L is isomorphic the power lattice of some set if and
only if it is complete and atomic. If the latter holds, then upon denoting the set of all atoms in L
by AL, one has L ≅ 2AL under the map x → α(x). £
Proposition 1.1.14. For all x in a complete Boolean lattice L and all subsets Y of L:
(i) x ∧ sup(Y) = sup({x∧y ⎜y ∈ Y}) and x ∨ inf(Y) = inf({x∨y ⎜y ∈ Y}).
(ii) (sup Y)ʹ = inf({yʹ ⎜y ∈ Y}) and (inf Y)ʹ = sup({yʹ ⎜y ∈Y}).
(iii) x \ sup({x \ y ⎜y ∈ Y}) = inf(x \ Y) and x \ inf(Y) = sup({x \ y ⎜y ∈ Y}).
(iv) sup(Y) \ x = sup({y \ x ⎜y ∈ Y}) and inf(Y) \ x = inf({y \ x ⎜y ∈ Y}). £
Theorem 1.1.15. Given a Boolean algebra (L, ∧, ∨, 1, 0, ʹ), let a ∈ L be given. Then
both a↓ and a↑ are Boolean lattices and χ: L → a↓ × a↑ defined by χ(x) = (x∧a, x∨a) is an
isomorphism of Boolean lattices.
Proof. χ is at least a lattice embedding by Theorem 1.1.9. Next, let (u, v) ∈ a↓ × a↑ be given. If
x = (v \ a) ∨ u, then
a∧x = a∧[(v \ a) ∨ u] = (a ∧ (v \ a)) ∨ (a∧u) = 0 ∨ u = u
and
a∨x = a∨[(v \ a) ∨ u] = a ∨ (v ∧ aʹ) ∨ a = (a ∨ v) ∧ (a ∨ aʹ) = v∧1 = v.
Thus χ is also surjective and the theorem follows. £
2 denotes the Boolean lattice {1 > 0}. By mild abuse of notation, 2 also denotes the
Boolean algebra ({1, 0}; ∨, ∧, 1, 0, ʹ) again with 1 > 0. Put otherwise, 2 is the chain C1 reset in a
Boolean context. We have the following sequence of easy corollaries of Theorem 1.1.15 and
Corollary 1.10.
Corollary 1.1.16. Every finite Boolean lattice factors as a finite power of 2. £
Corollary1.1.17. The only nontrivial subdirectly irreducible Boolean algebra is 2. £
Corollary1.1.18. Every distributive lattice can be embedded into a Boolean lattice. £
By a generalized Boolean lattice is meant a lattice L with a minimal element 0 such that
each principal ideal x↓ of L is a Boolean lattice. Such a lattice is necessarily distributive;
moreover a difference operation on L is given by setting x \ y = x \ (y∧x) in the Boolean lattice x↓.
For Boolean lattices both differences agree. The relative DeMorgan identities also hold for
generalized Boolean lattices. Upon including \ in the signature, one has a generalized Boolean
algebra (L, ∧, ∨, \, 0). It is characterized by the identities for a distributive lattice with a minimal
element 0 together with the pair: (x∧y) ∧ (x\y) = 0 and (x∧y) ∨ (x\y) = x.
Every Boolean algebra (L, ∧, ∨, 1, 0, ʹ) possesses a generalized Boolean algebra reduct
(L, ∧, ∨, \, 0) with x \ y given as x∧yʹ. Conversely, any generalized Boolean algebra (L, ∧, ∨, \, 0)
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