Page 103 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 103
III: Quasilattices, Paralattices and their Congruences
Proof. Clearly skew* lattices are regular paralattices. Thus the join of both varieties lies in the
variety of regular paralattices. One the other hand, the larger variety is generated by its four
subvarieties of flat algebras by Theorem 3.3.4. But these four subvarieties are either subvarieties
of the varieties of skew lattices or the varieties of skew* lattices. The “join” assertion follows.
The final remark follows again from the isomorphism of Theorem 3.3.4. £
More generally:
Proposition 3.4.6. Given a refined quasilattice, both R(∨) and L(∨) are congruences with
respect to ∨, while R(∧) and L(∧) are congruences with respect to ∧.
Proof. First observe that ≤ is surjective between comparable D-classes: given classes, A ≤ B,
then for all a ∈ A and b ∈ B, there exist bA ∈ A and aB ∈ B such that bA ≤ b and a ≤ aB. Indeed
we may choose bA = b∧a∧b and aB = a∨b∨a. It follows from Theorem 1.2.19 that both ∨ and ∧
satisfy the identity xyxzx = xyzx. Since ≤ is surjective, u and v exist in the D-class of x such that
x∧y ≤ u and z∧x ≤ v. Because u, x and v lie in a common D-class, u∧101x∧101v = u∧101v and
thus
x∧y∧x∧z∧x = (x∧y)∧u∧x∧v∧(z∧x) = (x∧y)∧u∧v∧(z∧x) = x∧y∧z∧x.
From the theory of bands, the identity x∧y∧x∧z∧x = x∧y∧z∧x implies R(∧) and L(∧) are
∧-congruences. Similarly one shows that R(∨) and L(∨) are ∨-congruences. £
From refined quasilattices to skew lattices and back again
Many results established for skew lattices extend to refined quasilattices. In particular the
following extensions of Theorem 2.2.1 and its immediate consequences hold, with the proofs
being essentially the same.
Theorem 3.4.7. Let (S, ∨, ∧) be a refined quasilattice with D-classes A and B. If J and
M are the respective join and meet classes of A and B in S/D, then
J = {a∨b⎮a ∈ A, b ∈ B & a∨b = b∨a} and M = {a∧b⎮a ∈ A, b ∈ B & a∧b = b∧a}. £
Corollary 3.4.8. Given a refined quasilattice (N, ∨, ∧) and an infinite cardinal number
ℵα, then the union of all D-classes of order [equal to or] less than ℵα is a subalgebra of N. In
particular, the union of all finite D-classes is a subalgebra of N. £
101
Proof. Clearly skew* lattices are regular paralattices. Thus the join of both varieties lies in the
variety of regular paralattices. One the other hand, the larger variety is generated by its four
subvarieties of flat algebras by Theorem 3.3.4. But these four subvarieties are either subvarieties
of the varieties of skew lattices or the varieties of skew* lattices. The “join” assertion follows.
The final remark follows again from the isomorphism of Theorem 3.3.4. £
More generally:
Proposition 3.4.6. Given a refined quasilattice, both R(∨) and L(∨) are congruences with
respect to ∨, while R(∧) and L(∧) are congruences with respect to ∧.
Proof. First observe that ≤ is surjective between comparable D-classes: given classes, A ≤ B,
then for all a ∈ A and b ∈ B, there exist bA ∈ A and aB ∈ B such that bA ≤ b and a ≤ aB. Indeed
we may choose bA = b∧a∧b and aB = a∨b∨a. It follows from Theorem 1.2.19 that both ∨ and ∧
satisfy the identity xyxzx = xyzx. Since ≤ is surjective, u and v exist in the D-class of x such that
x∧y ≤ u and z∧x ≤ v. Because u, x and v lie in a common D-class, u∧101x∧101v = u∧101v and
thus
x∧y∧x∧z∧x = (x∧y)∧u∧x∧v∧(z∧x) = (x∧y)∧u∧v∧(z∧x) = x∧y∧z∧x.
From the theory of bands, the identity x∧y∧x∧z∧x = x∧y∧z∧x implies R(∧) and L(∧) are
∧-congruences. Similarly one shows that R(∨) and L(∨) are ∨-congruences. £
From refined quasilattices to skew lattices and back again
Many results established for skew lattices extend to refined quasilattices. In particular the
following extensions of Theorem 2.2.1 and its immediate consequences hold, with the proofs
being essentially the same.
Theorem 3.4.7. Let (S, ∨, ∧) be a refined quasilattice with D-classes A and B. If J and
M are the respective join and meet classes of A and B in S/D, then
J = {a∨b⎮a ∈ A, b ∈ B & a∨b = b∨a} and M = {a∧b⎮a ∈ A, b ∈ B & a∧b = b∧a}. £
Corollary 3.4.8. Given a refined quasilattice (N, ∨, ∧) and an infinite cardinal number
ℵα, then the union of all D-classes of order [equal to or] less than ℵα is a subalgebra of N. In
particular, the union of all finite D-classes is a subalgebra of N. £
101
