Page 93 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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III: Quasilattices, Paralattices and their Congruences
From the explicitly split case, Q = T × A, (iv) is easily seen. Conversely, since x∨y = y∨x implies
x = y in each D-class, (iv) implies that each ρ-class uniquely meets each D-class, that is,
(iv) implies (ii). But (ii) in turn implies (v). Indeed, let (θ1, θ2) ∈ [D, ∇] × [Δ, D]. Upon setting
θ = (ρ∧θ1)∨θ2, the distributive properties of D yield (θoD, θ∩D) = (θ1, θ2). Hence the indicated
injective homomorphism is also surjective and (v) follows. Finally, given (v), a congruence θ
exists such that θoD = ∇ and θo D = Δ and (iii) holds. £
Every split quasilattice is a refined quasilattice. Within the variety of refined
quasilattices, split quasilattices are characterized as follows.
Theorem 3.1.4. Split quasilattices form the subvariety of refined quasilattices S for
which (S, ∨) and (S, ∧) are normal, in that u∨x∨y∨v = u∨y∨x∨v and u∧x∧y91∧v = u∧y∧x∧v hold,
Proof. Both identities hold for lattices as well as for rectangular quasilattices, and hence also for
split quasilattices. Conversely, suppose that both identities hold on a refined quasilattice. Then
x∨y = y∨x iff x∧y = y∧x. For x∨y = y∨x implies x∨y ≥ y, x so that
x∧y = (x∨y)∧x∧y∧(x∨y) = (x∨y)∧y∧x∧(x∨y) = y∧x.
The other direction is seen in similar fashion. Next, define a binary relation ρ0 by xρ0y if
x∨y = y∨x (or equivalently, x∧y = y∧x). Clearly ρ0 is both reflexive and symmetric. From
x ρ0 y ρ0 z, x∨y∨z = y∨x∨z∨y = y∨z∨x∨y = z∨y∨x follows. Denoting this common value by u we
get u ≥ x, z and x∧z = u∧x∧z∧u = u∧z∧x∧u = z∧x. Hence ρ0 is also transitive and thus an
equivalence. By the identities in the theorem statement, ρ0 is seen to be a congruence. Since
≤ ⊆ ρ0, we get ρ ⊆ ρ0. Conversely, given x ρ0 y, from x ≤ x∨y ≥ y first x ρ y and then ρ0 = ρ
follows. Hence Theorem 3.1.3(iv) is satisfied and N splits. £
Corollary 3.1.5. A noncommutative lattice Q is a split quasilattice if and only if
ρ∩δ = Δ. In general, Q/ρ∩δ is the maximal split quasilattice image of any quasilattice Q.
Proof. Q/ρ∩δ is isomorphic to a subalgebra of the product Q/ρ × Q/δ where Q/ρ is an antilattice
and Q/δ is a lattice. Thus Q/ρ∩δ splits also and is indeed isomorphic to Q/ρ × Q/δ under the map
x(ρ∩δ) → (xρ, xδ). £
We have seen that Con(Q) is a subdirect product of [Δ, D] and [D, ∇]. If {Di⎮i ∈ I} is
the set of all D-classes of Q, then an embedding D*: [Δ, D] → ∏iCon(Di) of complete lattices is
given by
D*(θ) = 〈θ ∩ Di × Di)⎮i ∈ I〉
Since ∏iCon(Di) is in turn embedded in Con(∏iDi), this suggests that [Δ, D] shares similarities
with congruence lattices of rectangular quasilattices. On the other hand, the interval [ρ, ∇] is
isomorphic to the congruence lattice Con(N/ρ) of the greatest rectangular image Q/ρ of Q. This
leads us to inquire how [Δ, D] and [ρ,∇] are related.
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From the explicitly split case, Q = T × A, (iv) is easily seen. Conversely, since x∨y = y∨x implies
x = y in each D-class, (iv) implies that each ρ-class uniquely meets each D-class, that is,
(iv) implies (ii). But (ii) in turn implies (v). Indeed, let (θ1, θ2) ∈ [D, ∇] × [Δ, D]. Upon setting
θ = (ρ∧θ1)∨θ2, the distributive properties of D yield (θoD, θ∩D) = (θ1, θ2). Hence the indicated
injective homomorphism is also surjective and (v) follows. Finally, given (v), a congruence θ
exists such that θoD = ∇ and θo D = Δ and (iii) holds. £
Every split quasilattice is a refined quasilattice. Within the variety of refined
quasilattices, split quasilattices are characterized as follows.
Theorem 3.1.4. Split quasilattices form the subvariety of refined quasilattices S for
which (S, ∨) and (S, ∧) are normal, in that u∨x∨y∨v = u∨y∨x∨v and u∧x∧y91∧v = u∧y∧x∧v hold,
Proof. Both identities hold for lattices as well as for rectangular quasilattices, and hence also for
split quasilattices. Conversely, suppose that both identities hold on a refined quasilattice. Then
x∨y = y∨x iff x∧y = y∧x. For x∨y = y∨x implies x∨y ≥ y, x so that
x∧y = (x∨y)∧x∧y∧(x∨y) = (x∨y)∧y∧x∧(x∨y) = y∧x.
The other direction is seen in similar fashion. Next, define a binary relation ρ0 by xρ0y if
x∨y = y∨x (or equivalently, x∧y = y∧x). Clearly ρ0 is both reflexive and symmetric. From
x ρ0 y ρ0 z, x∨y∨z = y∨x∨z∨y = y∨z∨x∨y = z∨y∨x follows. Denoting this common value by u we
get u ≥ x, z and x∧z = u∧x∧z∧u = u∧z∧x∧u = z∧x. Hence ρ0 is also transitive and thus an
equivalence. By the identities in the theorem statement, ρ0 is seen to be a congruence. Since
≤ ⊆ ρ0, we get ρ ⊆ ρ0. Conversely, given x ρ0 y, from x ≤ x∨y ≥ y first x ρ y and then ρ0 = ρ
follows. Hence Theorem 3.1.3(iv) is satisfied and N splits. £
Corollary 3.1.5. A noncommutative lattice Q is a split quasilattice if and only if
ρ∩δ = Δ. In general, Q/ρ∩δ is the maximal split quasilattice image of any quasilattice Q.
Proof. Q/ρ∩δ is isomorphic to a subalgebra of the product Q/ρ × Q/δ where Q/ρ is an antilattice
and Q/δ is a lattice. Thus Q/ρ∩δ splits also and is indeed isomorphic to Q/ρ × Q/δ under the map
x(ρ∩δ) → (xρ, xδ). £
We have seen that Con(Q) is a subdirect product of [Δ, D] and [D, ∇]. If {Di⎮i ∈ I} is
the set of all D-classes of Q, then an embedding D*: [Δ, D] → ∏iCon(Di) of complete lattices is
given by
D*(θ) = 〈θ ∩ Di × Di)⎮i ∈ I〉
Since ∏iCon(Di) is in turn embedded in Con(∏iDi), this suggests that [Δ, D] shares similarities
with congruence lattices of rectangular quasilattices. On the other hand, the interval [ρ, ∇] is
isomorphic to the congruence lattice Con(N/ρ) of the greatest rectangular image Q/ρ of Q. This
leads us to inquire how [Δ, D] and [ρ,∇] are related.
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