Page 34 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 34
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Thanks to various dualities, coherent quasi-orderings are defined as follows.
1) If (N, ∨, ∧) is a quasilattice, its natural quasiordering ≺ is just ∧≺, or the dual of ∨≺.
2) If (N, ∨, ∧) is a paralattice, its natural partial ordering ≤ is ∧≤, or the dual of ∨≤.
3) If (N, ∨, ∧) is a skew(*) lattice, its left and right quasiorderings, ≺L and ≺R, are ∧≺L
and ∧≺R respectively, or their duals of the appropriate ∨-quasi-orderings.
Congruences
A congruence on a noncommutative lattice N is an equivalence relation θ on N such that
for all a, b, c ∈N,
a θ b implies a∧c θ b∧c, c∧a θ c∧b, a∨c θ b∨c and c∨a θ c∨b.
In accord with standard notation, Δ denotes the least congruence (equality) and ∇ denotes the
greatest congruence. Besides Δ and ∇, two other congruences of interest are:
The least lattice congruence is the smallest congruence λ on N such that N/λ is a lattice.
N/λ is thus the maximal lattice image of N. On the interval [λ, ∇], θ∩supi(θi) = supi (θ∩θi),
since [λ, ∇] is lattice isomorphic with Con(N/λ), the congruence lattice of the lattice N/λ.
On the other hand, the least rectangular congruence, is the least congruence ρ on N for
which N/ρ is rectangular. Clearly ρ is the congruence generated from the relation ∨≤ ∪ ∧≤.
Theorem 1.3.5. If (N, ∨, ∧) is a noncommutative lattice with D∨ and D∧ the
D-congruences for ∨ and ∧, then λ is the congruence on N generated from the relation D∨ ∪ D∧.
Proof. If δ is the congruence on N generated from D∨ ∪ D∧, then N/δ is commutative in both ∨
and ∧. Since N satisfies absorption identities inducing B1 and C1 on any commutative image, N/δ
is a lattice. Thus λ ⊆ δ. Since δ ⊆ λ is clear, λ = δ follows. £
Corollary 1.3.6. Let (N, ∨, ∧) be a noncommutative lattice for which D∨ = D∧. Then
λ = D, the common D-equivalence for both operations, N is a quasilattice, N/D is the maximal
lattice image of N and all the D-classes are the maximal antilattices in N. Conversely, for all
quasilattices, D∨ = D∧.
Proof. If D∨ = D∧, then λ is clearly the common D-equivalence. By the Clifford-McLean
Theorem for bands, N/D is the maximal lattice image of N and the D-classes are maximal anti-
lattices in N. Hence a D a ∧ (b ∨ a ∨ b) ∧ a in N. But since a (∧)≥ a ∧ (b ∨ a ∨ b) ∧ a in N,
equality follows: a = a ∧ (b ∨ a ∨ b) ∧ a. Similarly, a = a ∨ (b ∧ a ∧ b) ∨ a so that N is a quasi-
lattice. The converse is clear. £
32
Thanks to various dualities, coherent quasi-orderings are defined as follows.
1) If (N, ∨, ∧) is a quasilattice, its natural quasiordering ≺ is just ∧≺, or the dual of ∨≺.
2) If (N, ∨, ∧) is a paralattice, its natural partial ordering ≤ is ∧≤, or the dual of ∨≤.
3) If (N, ∨, ∧) is a skew(*) lattice, its left and right quasiorderings, ≺L and ≺R, are ∧≺L
and ∧≺R respectively, or their duals of the appropriate ∨-quasi-orderings.
Congruences
A congruence on a noncommutative lattice N is an equivalence relation θ on N such that
for all a, b, c ∈N,
a θ b implies a∧c θ b∧c, c∧a θ c∧b, a∨c θ b∨c and c∨a θ c∨b.
In accord with standard notation, Δ denotes the least congruence (equality) and ∇ denotes the
greatest congruence. Besides Δ and ∇, two other congruences of interest are:
The least lattice congruence is the smallest congruence λ on N such that N/λ is a lattice.
N/λ is thus the maximal lattice image of N. On the interval [λ, ∇], θ∩supi(θi) = supi (θ∩θi),
since [λ, ∇] is lattice isomorphic with Con(N/λ), the congruence lattice of the lattice N/λ.
On the other hand, the least rectangular congruence, is the least congruence ρ on N for
which N/ρ is rectangular. Clearly ρ is the congruence generated from the relation ∨≤ ∪ ∧≤.
Theorem 1.3.5. If (N, ∨, ∧) is a noncommutative lattice with D∨ and D∧ the
D-congruences for ∨ and ∧, then λ is the congruence on N generated from the relation D∨ ∪ D∧.
Proof. If δ is the congruence on N generated from D∨ ∪ D∧, then N/δ is commutative in both ∨
and ∧. Since N satisfies absorption identities inducing B1 and C1 on any commutative image, N/δ
is a lattice. Thus λ ⊆ δ. Since δ ⊆ λ is clear, λ = δ follows. £
Corollary 1.3.6. Let (N, ∨, ∧) be a noncommutative lattice for which D∨ = D∧. Then
λ = D, the common D-equivalence for both operations, N is a quasilattice, N/D is the maximal
lattice image of N and all the D-classes are the maximal antilattices in N. Conversely, for all
quasilattices, D∨ = D∧.
Proof. If D∨ = D∧, then λ is clearly the common D-equivalence. By the Clifford-McLean
Theorem for bands, N/D is the maximal lattice image of N and the D-classes are maximal anti-
lattices in N. Hence a D a ∧ (b ∨ a ∨ b) ∧ a in N. But since a (∧)≥ a ∧ (b ∨ a ∨ b) ∧ a in N,
equality follows: a = a ∧ (b ∨ a ∨ b) ∧ a. Similarly, a = a ∨ (b ∧ a ∧ b) ∨ a so that N is a quasi-
lattice. The converse is clear. £
32
