Page 94 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 94
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Consider the embedding χ: Con(Q) → [D, ∇] × [Δ, D] given by χ[θ] = (D ∨ θ, D ∩ θ).
Since ρoD = ∇, restricting χ to [ρ, ∇] yields an embedding χ0: [ ρ, ∇] → [Δ, D] of complete
lattices defined by χ0[θ] = D∩θ. When is χ0 an isomorphism? Precisely when ρ∩D = Δ. This
condition is clearly necessary as it states that Δ lies in the image of χ0. Given this condition, then
for all θ ∈ [Δ, D],
D ∩(θ ∨ρ) = (D ∩θ)∨(D ∩ρ) = θ∨Δ = θ,
which implies that χ0 is also surjective. But ρ∩D = Δ if and only if Q splits. Thus:
Theorem 3.1.6. The interval [ρ, ∇] of rectangular congruences on a quasilattice Q is
isomorphic to a complete sublattice of the interval [Δ, D] under the map θ → D∩θ. This
embedding is an isomorphism if and only if Q splits. £
To complete our basic picture of quasilattice congruences we give a partial complement
of the Clifford-McLean Theorem. In its proof, the terms ∨–morphism and ∧-morphism refer to
homomorphisms with respect to the stated operations.
Theorem 3.1.7. Given a lattice Λ with disjoint antilattices Dλ assigned to each λ ∈ Λ, a
quasilattice structure exists on Q = ∪λDλ such that the maximal antilattices in Q are precisely the
Dλ and N/D ≅ Λ. In addition, ∨ and ∧ can be defined so that [Δ, D] ≅ ∏λCon(Dλ).
Proof. Let Φ = {ϕ(λ, µ):Dλ → Dµ⎮λ ≤ µ} be an ascending family of ∨–morphisms such that for
all λ ∈ Λ, ϕ(λ, λ) is the identity map on Dλ and secondly, ϕ(µ, ν)ϕ(λ, µ) = ϕ(λ, ν) for all λ ≤ µ ≤ ν in
Λ. Similarly let Ψ = {ψ(µ, λ): Dµ → Dλ⎮λ ≤ µ} be a descending family of ∧–morphisms
satisfying dual properties. To define ∨, set x∨y = ϕ(λ, π)(x)∨ϕ(µ, π)(y) ∈ Dπ if x ∈ Dλ, y ∈ Dµ and
π = λ∨µ. Similarly, define ∧ on Q using Ψ in dual fashion. The operations ∨ and ∧ induced
from Φ and Ψ yield a quasilattice (Q; ∨, ∧) for which the Dλ are the maximal antilattices and for
which Q/D ≅ Λ as stated.
To complete the first assertion we need only exhibit at least one pair (Φ, Ψ). This is done
as follows. First, to each λ ∈ Λ pick some dλ in Dλ. Next, for each λ ∈ Λ, let ϕ(λ,λ) = ψ(λ,λ) be
the identity map on Dλ as required. Finally for each strict comparison λ < µ in Λ, let
ϕ(λ, µ): Dλ → Dµ be the constant map sending all Dλ to dµ in Dµ. Similarly define
ψ(µ, λ): Dµ → Dλ as the constant map sending all of Dµ onto dλ. As so defined, (Φ, Ψ) satisfies
all the required conditions. Moreover, for such a pair (Φ, Ψ), the embedding
D*: [Δ, D] → ∏λCon(Dλ) defined by D*(θ) = 〈θ ∩ Dλ×Dλ⎮λ ∈Λ〉 is onto. For given
〈θλ⎮λ ∈ Λ〉 ∈ ∏λCon(Dλ)〉, the union θ = ∪λθλ is at least an equivalence on Q. Thanks to the
pointed character of (Φ, Ψ) as defined, θ is a congruence inducing 〈θλ⎮λ∈Λ〉. £
92
Consider the embedding χ: Con(Q) → [D, ∇] × [Δ, D] given by χ[θ] = (D ∨ θ, D ∩ θ).
Since ρoD = ∇, restricting χ to [ρ, ∇] yields an embedding χ0: [ ρ, ∇] → [Δ, D] of complete
lattices defined by χ0[θ] = D∩θ. When is χ0 an isomorphism? Precisely when ρ∩D = Δ. This
condition is clearly necessary as it states that Δ lies in the image of χ0. Given this condition, then
for all θ ∈ [Δ, D],
D ∩(θ ∨ρ) = (D ∩θ)∨(D ∩ρ) = θ∨Δ = θ,
which implies that χ0 is also surjective. But ρ∩D = Δ if and only if Q splits. Thus:
Theorem 3.1.6. The interval [ρ, ∇] of rectangular congruences on a quasilattice Q is
isomorphic to a complete sublattice of the interval [Δ, D] under the map θ → D∩θ. This
embedding is an isomorphism if and only if Q splits. £
To complete our basic picture of quasilattice congruences we give a partial complement
of the Clifford-McLean Theorem. In its proof, the terms ∨–morphism and ∧-morphism refer to
homomorphisms with respect to the stated operations.
Theorem 3.1.7. Given a lattice Λ with disjoint antilattices Dλ assigned to each λ ∈ Λ, a
quasilattice structure exists on Q = ∪λDλ such that the maximal antilattices in Q are precisely the
Dλ and N/D ≅ Λ. In addition, ∨ and ∧ can be defined so that [Δ, D] ≅ ∏λCon(Dλ).
Proof. Let Φ = {ϕ(λ, µ):Dλ → Dµ⎮λ ≤ µ} be an ascending family of ∨–morphisms such that for
all λ ∈ Λ, ϕ(λ, λ) is the identity map on Dλ and secondly, ϕ(µ, ν)ϕ(λ, µ) = ϕ(λ, ν) for all λ ≤ µ ≤ ν in
Λ. Similarly let Ψ = {ψ(µ, λ): Dµ → Dλ⎮λ ≤ µ} be a descending family of ∧–morphisms
satisfying dual properties. To define ∨, set x∨y = ϕ(λ, π)(x)∨ϕ(µ, π)(y) ∈ Dπ if x ∈ Dλ, y ∈ Dµ and
π = λ∨µ. Similarly, define ∧ on Q using Ψ in dual fashion. The operations ∨ and ∧ induced
from Φ and Ψ yield a quasilattice (Q; ∨, ∧) for which the Dλ are the maximal antilattices and for
which Q/D ≅ Λ as stated.
To complete the first assertion we need only exhibit at least one pair (Φ, Ψ). This is done
as follows. First, to each λ ∈ Λ pick some dλ in Dλ. Next, for each λ ∈ Λ, let ϕ(λ,λ) = ψ(λ,λ) be
the identity map on Dλ as required. Finally for each strict comparison λ < µ in Λ, let
ϕ(λ, µ): Dλ → Dµ be the constant map sending all Dλ to dµ in Dµ. Similarly define
ψ(µ, λ): Dµ → Dλ as the constant map sending all of Dµ onto dλ. As so defined, (Φ, Ψ) satisfies
all the required conditions. Moreover, for such a pair (Φ, Ψ), the embedding
D*: [Δ, D] → ∏λCon(Dλ) defined by D*(θ) = 〈θ ∩ Dλ×Dλ⎮λ ∈Λ〉 is onto. For given
〈θλ⎮λ ∈ Λ〉 ∈ ∏λCon(Dλ)〉, the union θ = ∪λθλ is at least an equivalence on Q. Thanks to the
pointed character of (Φ, Ψ) as defined, θ is a congruence inducing 〈θλ⎮λ∈Λ〉. £
92
