Page 161 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 161
IV: Skew Boolean Algebras
non-isomorphic generalized Boolean algebras of order |Bj| ≤ max{ℵ0, |S/D|}. Since ω is limit-
preserving and each S in 〈3L〉 has a solution set, Freyd’s Adjoint Functor Theorem implies that ω
must have a left adjoint Ω. £
Thus given a skew Boolean algebra S, a homomorphism ηS: S → ω(Ω(S)) exists, called
the universal morphism from S to ω, such that for any homomorphism µ: S → ω(Bʹ) in 〈3L〉, a
unique homomorphism u: Ω(S) → Bʹ exists in 〈2〉 such that µ factors as uωoηS. Due to Corollary
4.5.10, each universal morphism ηS is an embedding. Thus, given a homomorphism f: S → Sʹ,
fΩ: Ω(S) → Ω(Sʹ) is the unique homomorphism in 〈2〉 making the following diagram commute.
S ⎯η⎯S⎯→ ω(Ω(S))
f ↓ ↓ ( f Ω )ω .
S′ ⎯η⎯S′⎯→ ω(Ω(S′))
What does ηS: S → Ω(S) look like for some typical left-handed skew Boolean algebras?
Theorem 4.5.17. Let P be a primitive left-handed skew Boolean algebra with P\{0} = X.
Then Ω(P) ≅ FBX, the generalized Boolean algebra reduct of the free Boolean algebra on X. In
particular, Ω(P) ≅ 22|X| if |X| is finite. (See [Leech and Spinks, 2008] Theorem 5.2.) £
In particular, Ω(2) ≅ 22 and Ω(3L) ≅ 24. One can show that Ω(S1 × S2) ≅ Ω(S1) × Ω(S2).
(See [Leech and Spinks, 2008] Theorem 5.3.) Ω(S) is thus easily determined when S/D is finite.
Twisted product constructions were introduced by Kalman [1958] as a means of building
De Morgan algebras from distributive lattices. The ω-construction given here is a variation of a
construction due to Pagliani [1998] for producing Nelson algebras from Heyting algebras. For
applications of twisted product constructions to both algebra and logic see Pagliani [1997].
Historical remarks
Skew Boolean algebras in some form seem to have been studied first by Robert Bignall
in his 1976 dissertation and then in a 1980 paper by his advisor, William Cornish. In 1990, a
paper on skew Boolean algebras appeared among Jonathan Leech’s early papers on skew lattices.
Bignall and Leech then published a joint paper on skew Boolean algebras with intersections in
1995. Much in Sections 4.1 and 4.4 appeared in these two papers. In Section 4.4, the material on
the lattice of varieties appeared in the 2017 paper by Leech and Spinks. The material in Section
4.2 on finite algebras and the free case in particular appeared in the 2016 paper by Ganya
Kudryatseva and Leech, as did the material in Section 4.4 on infinite free algebras. The material
in Section 4.3 appeared in a 2013 paper by Karin Cvetko-Vah, Matthew Spinks and Leech, but
the initial information about strongly distributive skew lattices appeared in Leech’s 1992 paper on
normal skew lattices. The construction in Section 4.5 was due to Spinks and developed in a joint
159
non-isomorphic generalized Boolean algebras of order |Bj| ≤ max{ℵ0, |S/D|}. Since ω is limit-
preserving and each S in 〈3L〉 has a solution set, Freyd’s Adjoint Functor Theorem implies that ω
must have a left adjoint Ω. £
Thus given a skew Boolean algebra S, a homomorphism ηS: S → ω(Ω(S)) exists, called
the universal morphism from S to ω, such that for any homomorphism µ: S → ω(Bʹ) in 〈3L〉, a
unique homomorphism u: Ω(S) → Bʹ exists in 〈2〉 such that µ factors as uωoηS. Due to Corollary
4.5.10, each universal morphism ηS is an embedding. Thus, given a homomorphism f: S → Sʹ,
fΩ: Ω(S) → Ω(Sʹ) is the unique homomorphism in 〈2〉 making the following diagram commute.
S ⎯η⎯S⎯→ ω(Ω(S))
f ↓ ↓ ( f Ω )ω .
S′ ⎯η⎯S′⎯→ ω(Ω(S′))
What does ηS: S → Ω(S) look like for some typical left-handed skew Boolean algebras?
Theorem 4.5.17. Let P be a primitive left-handed skew Boolean algebra with P\{0} = X.
Then Ω(P) ≅ FBX, the generalized Boolean algebra reduct of the free Boolean algebra on X. In
particular, Ω(P) ≅ 22|X| if |X| is finite. (See [Leech and Spinks, 2008] Theorem 5.2.) £
In particular, Ω(2) ≅ 22 and Ω(3L) ≅ 24. One can show that Ω(S1 × S2) ≅ Ω(S1) × Ω(S2).
(See [Leech and Spinks, 2008] Theorem 5.3.) Ω(S) is thus easily determined when S/D is finite.
Twisted product constructions were introduced by Kalman [1958] as a means of building
De Morgan algebras from distributive lattices. The ω-construction given here is a variation of a
construction due to Pagliani [1998] for producing Nelson algebras from Heyting algebras. For
applications of twisted product constructions to both algebra and logic see Pagliani [1997].
Historical remarks
Skew Boolean algebras in some form seem to have been studied first by Robert Bignall
in his 1976 dissertation and then in a 1980 paper by his advisor, William Cornish. In 1990, a
paper on skew Boolean algebras appeared among Jonathan Leech’s early papers on skew lattices.
Bignall and Leech then published a joint paper on skew Boolean algebras with intersections in
1995. Much in Sections 4.1 and 4.4 appeared in these two papers. In Section 4.4, the material on
the lattice of varieties appeared in the 2017 paper by Leech and Spinks. The material in Section
4.2 on finite algebras and the free case in particular appeared in the 2016 paper by Ganya
Kudryatseva and Leech, as did the material in Section 4.4 on infinite free algebras. The material
in Section 4.3 appeared in a 2013 paper by Karin Cvetko-Vah, Matthew Spinks and Leech, but
the initial information about strongly distributive skew lattices appeared in Leech’s 1992 paper on
normal skew lattices. The construction in Section 4.5 was due to Spinks and developed in a joint
159
