Page 101 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 101
III: Quasilattices, Paralattices and their Congruences
partial order ≤ has natural joins. Thus, for any pair of elements x and y, their supremum x∨y
exists in (B, ≤). Upon denoting the band multiplication by ∧, one obtains a paralattice ( B; ∨, ∧)
where (B, ∨) is a semilattice, or equivalently, where D(∨) = Δ. Thus, in this case D will denote
D(∧).
Viewed as paralattices, ∨-bands form a subvariety characterized simply by x∨y = y∨x.
Flat ∨-bands are ∨-bands for which ∧ is either left regular (x∧y∧x = x∧y) or right-regular
(x∧y∧x = y∧x). Both types of flat ∨-bands form subvarieties. In terms of Bi and Cj identities of
Section 1.3, we have the following result and its left dual:
Theorem 3.4.1. A flat ∨-band satisfying a∧b∧a = b∧a (so that D∧ = R∧) is
characterized by the associativity of ∨ and ∧ together with the following four identities,
B1 and C2: a ∧ (a ∨ b) = a = (b ∧ a) ∨ a.
B3 and C3: a ∧ (b ∨ a) = a = a ∨ (b ∧ a).
Proof. These identities necessarily hold for such a ∨-band, as b∧a ≤ a ≤ a∨b = b∨a in the
coherent natural partial ordering. Conversely, any associative algebra (N; ∨, ∧) satisfying B1, B3,
C2 and C3 is at least a double band by Theorem 1.3.4. In this case B1 and C2 together assert that
∨≺R dualizes ∧≺L while B3 and C3 jointly assert that ∨≺L dualizes ∧≺L making ∨≺R = ∨≺L = ∨≤
and thus ∧≺L = ∧≤. Thus N is a flat ∨-band of the indicated type. £
Thanks to the dual (union) version of Theorem 1.3.13 we have:
Theorem 3.4.2. The congruence lattice of a ∨-band is distributive. £
How prevalent are ∨-bands among bands? Several established classes of bands turn out
to be ∨-bands.
Example 3.4.1. If B is a rectangular band and set S = B1, the extension obtained by
adjoining an identity element 1, then (B1, ≤) is a join semilattice characterized by 1 ≥ b for b in B.
1
•• • • • • •
Examples 3.4.2. Let B be the free (left, right or 2-sided) normal band BX (where
xyzw = xzyw) on alphabet X. Then B1 is a ∨-band. Indeed elements of B (in the 2-sided case)
look like aAc where A is any finite subset of X (thanks to middle commutativity). In (B, ≤),
aAc ≤ aʹAʹcʹ if and only if a = aʹ, c = cʹ and A ⊆ Aʹ. Thus any finite subset of B with a common
upper bound in (B, ≤) has a least upper bound:
99
partial order ≤ has natural joins. Thus, for any pair of elements x and y, their supremum x∨y
exists in (B, ≤). Upon denoting the band multiplication by ∧, one obtains a paralattice ( B; ∨, ∧)
where (B, ∨) is a semilattice, or equivalently, where D(∨) = Δ. Thus, in this case D will denote
D(∧).
Viewed as paralattices, ∨-bands form a subvariety characterized simply by x∨y = y∨x.
Flat ∨-bands are ∨-bands for which ∧ is either left regular (x∧y∧x = x∧y) or right-regular
(x∧y∧x = y∧x). Both types of flat ∨-bands form subvarieties. In terms of Bi and Cj identities of
Section 1.3, we have the following result and its left dual:
Theorem 3.4.1. A flat ∨-band satisfying a∧b∧a = b∧a (so that D∧ = R∧) is
characterized by the associativity of ∨ and ∧ together with the following four identities,
B1 and C2: a ∧ (a ∨ b) = a = (b ∧ a) ∨ a.
B3 and C3: a ∧ (b ∨ a) = a = a ∨ (b ∧ a).
Proof. These identities necessarily hold for such a ∨-band, as b∧a ≤ a ≤ a∨b = b∨a in the
coherent natural partial ordering. Conversely, any associative algebra (N; ∨, ∧) satisfying B1, B3,
C2 and C3 is at least a double band by Theorem 1.3.4. In this case B1 and C2 together assert that
∨≺R dualizes ∧≺L while B3 and C3 jointly assert that ∨≺L dualizes ∧≺L making ∨≺R = ∨≺L = ∨≤
and thus ∧≺L = ∧≤. Thus N is a flat ∨-band of the indicated type. £
Thanks to the dual (union) version of Theorem 1.3.13 we have:
Theorem 3.4.2. The congruence lattice of a ∨-band is distributive. £
How prevalent are ∨-bands among bands? Several established classes of bands turn out
to be ∨-bands.
Example 3.4.1. If B is a rectangular band and set S = B1, the extension obtained by
adjoining an identity element 1, then (B1, ≤) is a join semilattice characterized by 1 ≥ b for b in B.
1
•• • • • • •
Examples 3.4.2. Let B be the free (left, right or 2-sided) normal band BX (where
xyzw = xzyw) on alphabet X. Then B1 is a ∨-band. Indeed elements of B (in the 2-sided case)
look like aAc where A is any finite subset of X (thanks to middle commutativity). In (B, ≤),
aAc ≤ aʹAʹcʹ if and only if a = aʹ, c = cʹ and A ⊆ Aʹ. Thus any finite subset of B with a common
upper bound in (B, ≤) has a least upper bound:
99
