Page 198 - 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
congruence classes, the components, are thus subalgebras of B. Given a component Bʹ of B, a
sub-skew chain is given by A > Bʹ > C. Since a∧(c∨b∨c)∧a is the same for all b in a common C-
coset and c∨(a∧b∧a)∨c is the same for all b in a common A-coset, we can extend Lemma 5.6.1:
Lemma 5.6.2. Given a distributive skew chain A > B > C, for any pair a > c with a ∈ A
and c ∈ C, each AC-component Bʹ in B has a unique midpoint b of a and c.
We next sharpen Lemma 5.5.3 as follows.
Lemma 5.6.3. Let S be a categorical skew chain consisting of D-classes A > B > C. If S
is left-handed, then (5.2.1L) holds if and only if a∧(b∨c) = (a∧b)∨c for all a ≻ b ≻ c where a > c.
Dually, if S is right-handed, then (5.2.1R) holds if and only if (c∨b)∧a = c∨(b∧a) for all a ≻ b ≻ c
where a > c. (These identities are left and right-handed cases of (5.6.1) above.)
Proof. We consider the left-handed case. If S is indeed distributive with a, b, c as stated in the
lemma, then a∧(b∨c) = (a∧b)∨(a∧c) = (a∧b)∨c, since a > c. Conversely, given just a ≻ b ≻ c in
the respective D-classes A > B > C, set cʹ = a∧c. Then a > cʹ and (a∧b) ∨ (a∧c) = (a∧b) ∨ cʹ.
Next, since c and cʹ lie in the same A-coset in C and S is categorical, both b∨c and b∨cʹ lie both
in a common C-coset in B and in a common A-coset in B so that a∧(b∨c) = a∧(b∨cʹ). Hence:
a∧(b∨c) = a∧(b∨cʹ) = (a∧b) ∨ cʹ = (a∧b) ∨ (a∧c)
The lemma now follows from Lemma 5.5.3 and left-right duality. £
Theorem 5.6.4. Given a skew chain A > B > C, the following condition are equivalent:
i) A > B > C is distributive
ii) For all a ∈ A, b ∈ B and c ∈ C with a > c, a∧(c ∨ b ∨ c)∧a = c∨(a ∧ b ∧ a)∨c.
iii) Given a ∈ A and c ∈ C with a > c, each component Bʹ of B contains a unique
.
midpoint b of a and c.
iv) For each component Bʹ of B, A > Bʹ > C is strictly categorical.
When these conditions hold, each coset bijection ϕ: A → C uniquely factors through each
component Bʹ of B in that unique coset bijections ψ: A → Bʹ and χ: Bʹ → C exist such that
ϕ = χψ under the usual composition of partial bijections.
Proof. Clearly (i) implies (ii). Given a > c in (ii), for each element x in B, both b1 = a∧(c∨x∨c)∧a
and b2 = c∨(a∧x∧a)∨c are midpoints of a and c in B Replacing x by any element in its C-coset,
does not change the b1-outcome. Likewise, replacing x by any element in its A-coset, does not
change the b2-outcome. Hence (ii) is equivalent to asserting that given a > c fixed, for all x in a
common AC-component Bʹ of B, both a∧(c∨x∨c)∧a and c∨(a∧x∧a)∨c produce the same output b
in Bʹ such that a > b > c. Conversely, for any b in Bʹ such that a > b > c we must have
a∧(c∨b∨c)∧a = b = c∨(a∧b∧a)∨c.
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congruence classes, the components, are thus subalgebras of B. Given a component Bʹ of B, a
sub-skew chain is given by A > Bʹ > C. Since a∧(c∨b∨c)∧a is the same for all b in a common C-
coset and c∨(a∧b∧a)∨c is the same for all b in a common A-coset, we can extend Lemma 5.6.1:
Lemma 5.6.2. Given a distributive skew chain A > B > C, for any pair a > c with a ∈ A
and c ∈ C, each AC-component Bʹ in B has a unique midpoint b of a and c.
We next sharpen Lemma 5.5.3 as follows.
Lemma 5.6.3. Let S be a categorical skew chain consisting of D-classes A > B > C. If S
is left-handed, then (5.2.1L) holds if and only if a∧(b∨c) = (a∧b)∨c for all a ≻ b ≻ c where a > c.
Dually, if S is right-handed, then (5.2.1R) holds if and only if (c∨b)∧a = c∨(b∧a) for all a ≻ b ≻ c
where a > c. (These identities are left and right-handed cases of (5.6.1) above.)
Proof. We consider the left-handed case. If S is indeed distributive with a, b, c as stated in the
lemma, then a∧(b∨c) = (a∧b)∨(a∧c) = (a∧b)∨c, since a > c. Conversely, given just a ≻ b ≻ c in
the respective D-classes A > B > C, set cʹ = a∧c. Then a > cʹ and (a∧b) ∨ (a∧c) = (a∧b) ∨ cʹ.
Next, since c and cʹ lie in the same A-coset in C and S is categorical, both b∨c and b∨cʹ lie both
in a common C-coset in B and in a common A-coset in B so that a∧(b∨c) = a∧(b∨cʹ). Hence:
a∧(b∨c) = a∧(b∨cʹ) = (a∧b) ∨ cʹ = (a∧b) ∨ (a∧c)
The lemma now follows from Lemma 5.5.3 and left-right duality. £
Theorem 5.6.4. Given a skew chain A > B > C, the following condition are equivalent:
i) A > B > C is distributive
ii) For all a ∈ A, b ∈ B and c ∈ C with a > c, a∧(c ∨ b ∨ c)∧a = c∨(a ∧ b ∧ a)∨c.
iii) Given a ∈ A and c ∈ C with a > c, each component Bʹ of B contains a unique
.
midpoint b of a and c.
iv) For each component Bʹ of B, A > Bʹ > C is strictly categorical.
When these conditions hold, each coset bijection ϕ: A → C uniquely factors through each
component Bʹ of B in that unique coset bijections ψ: A → Bʹ and χ: Bʹ → C exist such that
ϕ = χψ under the usual composition of partial bijections.
Proof. Clearly (i) implies (ii). Given a > c in (ii), for each element x in B, both b1 = a∧(c∨x∨c)∧a
and b2 = c∨(a∧x∧a)∨c are midpoints of a and c in B Replacing x by any element in its C-coset,
does not change the b1-outcome. Likewise, replacing x by any element in its A-coset, does not
change the b2-outcome. Hence (ii) is equivalent to asserting that given a > c fixed, for all x in a
common AC-component Bʹ of B, both a∧(c∨x∨c)∧a and c∨(a∧x∧a)∨c produce the same output b
in Bʹ such that a > b > c. Conversely, for any b in Bʹ such that a > b > c we must have
a∧(c∨b∨c)∧a = b = c∨(a∧b∧a)∨c.
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