Page 186 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Theorem 5.4.8. The following seven conditions on a skew lattice S are equivalent.
i) S is strictly categorical.
ii) Given both a > b > c and a > bʹ > c in S with b D bʹ, b = bʹ must follow.
iii) Given both a ≥ b ≥ c and a ≥ bʹ ≥ c in S with b D bʹ, b = bʹ must follow.
iv) S has no subalgebra isomorphic to either of the following 4-element
skew chains:
a a
b – – L b′. b – – R b′ .
c c
v) Given a > b in S, the interval subalgebra [a, b] = {x ∈ S⎪a ≥ x ≥ b} is a
sublattice.
vi) Given any a ∈ S, [a]↑ = {x ∈ S⎪x ≥ a} is a normal subalgebra of S and
[a]↓ = {x ∈ S⎪a ≥ x} is a conormal subalgebra of S.
vii) S is categorical and given any skew chain A > B > C of D-classes in S,
for each coset bijection ϕ: A → C, unique coset bijections ψ: A → B and
χ: B → C exist such that ϕ = χψ.
Proof. Theorem 5.4.7(iii) gives us (i) ⇒ (ii). Conversely, if S satisfies (ii) then no subalgebra of
S can be one of the forbidden algebras in Theorem 5.4.5, making S categorical. We next show
that given x, y ∈ B, there exist u and v in B such that x –A u –C y and x –C v –A y. This guarantees
that in B, every A-coset meets every C-coset. Indeed, pick a ∈ A and c ∈ C so that
a > x > c. Note that a > a∧(c∨y∨c)∧a, c∨(a∧y∧a)∨c > c. But by assumption x is the unique
element in B between a and c under >. Thus
a∧(c∨y∨c)∧a = x = c∨(a∧y∧a)∨c
so that both x –A c∨y∨c –C y and x –C a∧y∧a –A y in B, which gives (ii) ⇒ (i). Next let S be
categorical with A > B > C as stated in (vii). The unique factorization in (vii) occurs precisely
when (ii) holds, making (ii) and (vii) equivalent. Finally, (iii) – (vi) are easily seen to be
equivalent variants of (ii) .£
Corollary 5.4.9. Strictly categorical skew lattices form a variety of skew lattices.
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Theorem 5.4.8. The following seven conditions on a skew lattice S are equivalent.
i) S is strictly categorical.
ii) Given both a > b > c and a > bʹ > c in S with b D bʹ, b = bʹ must follow.
iii) Given both a ≥ b ≥ c and a ≥ bʹ ≥ c in S with b D bʹ, b = bʹ must follow.
iv) S has no subalgebra isomorphic to either of the following 4-element
skew chains:
a a
b – – L b′. b – – R b′ .
c c
v) Given a > b in S, the interval subalgebra [a, b] = {x ∈ S⎪a ≥ x ≥ b} is a
sublattice.
vi) Given any a ∈ S, [a]↑ = {x ∈ S⎪x ≥ a} is a normal subalgebra of S and
[a]↓ = {x ∈ S⎪a ≥ x} is a conormal subalgebra of S.
vii) S is categorical and given any skew chain A > B > C of D-classes in S,
for each coset bijection ϕ: A → C, unique coset bijections ψ: A → B and
χ: B → C exist such that ϕ = χψ.
Proof. Theorem 5.4.7(iii) gives us (i) ⇒ (ii). Conversely, if S satisfies (ii) then no subalgebra of
S can be one of the forbidden algebras in Theorem 5.4.5, making S categorical. We next show
that given x, y ∈ B, there exist u and v in B such that x –A u –C y and x –C v –A y. This guarantees
that in B, every A-coset meets every C-coset. Indeed, pick a ∈ A and c ∈ C so that
a > x > c. Note that a > a∧(c∨y∨c)∧a, c∨(a∧y∧a)∨c > c. But by assumption x is the unique
element in B between a and c under >. Thus
a∧(c∨y∨c)∧a = x = c∨(a∧y∧a)∨c
so that both x –A c∨y∨c –C y and x –C a∧y∧a –A y in B, which gives (ii) ⇒ (i). Next let S be
categorical with A > B > C as stated in (vii). The unique factorization in (vii) occurs precisely
when (ii) holds, making (ii) and (vii) equivalent. Finally, (iii) – (vi) are easily seen to be
equivalent variants of (ii) .£
Corollary 5.4.9. Strictly categorical skew lattices form a variety of skew lattices.
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