Page 182 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 182
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Lemma 5.4.3. Given a left-handed skew chain A > B > C, let a > c and aʹ > cʹ be input-
output pairs for a common coset bijection between A and C where a ≠ aʹ in A and c ≠ cʹ in C.
Upon setting A* = {a, aʹ}, B* = {x ∈ B⎪a > x > c or aʹ > x > cʹ} and C* = {c, cʹ}, one obtains a
sub-skew chain: A* > B* > C*. In particular,
i) aʹ > x > cʹ for x in B* implies: a > both a∧x and x∨c > c with a∧x –A* x –C* x∨c.
ii) a > x > c for x in B* implies: aʹ > both aʹ∧x and x∨cʹ > cʹ with aʹ∧x –A* x –C* x∨cʹ.
All A*-cosets and all C*-cosets in B* are of order 2. An A*C*-component in B* is either a
subset {b, bʹ} that is simultaneously an A* and C*-coset in B* or else it is a larger subset having
the alternating coset form:
… –A* • –C* • –A* • –C* • –A* • –C* …
Only the former can occur if the skew chain is categorical.
Proof. Being left-handed, we need only check the mixed outcomes, say a∧x, x∧a, c∨x and x∨c
where aʹ > x > cʹ for case (i). Trivially x∧a = x = c∨x. As for a∧x, a∧(a∧x) = a∧x = (a∧x)∧a, due
to left-handedness, so that a > a∧x; likewise c∧(a∧x) = c, while
(a∧x)∧c = a∧x∧a∧cʹ = a∧x∧cʹ = a∧cʹ = c
by left-handedness and the common coset bisection context. Hence a∧x > c also, so that a∧x is in
B*. The dual argument gives a > x∨c > c, so that x∨c ∈ B* also. Similarly (ii) holds and we
have a sub-skew chain.
Clearly the A*-cosets in B* either all have order 1 or all have order 2. If they have order
1, then a, aʹ > all elements in B*, and by transitivity, a, aʹ > both c, cʹ, so that a > c belongs to a
different coset bijection than aʹ > cʹ. Thus all A*-cosets in B* have order 2 and likewise all
C*-cosets in B* have order 2. In an A*C*-component in B*, if the first case does not occur, a
situation x –C* y –A* z with x, y and z distinct develops. Since A* and C*-cosets have size 2, it
extends in an alternating coset pattern in both directions, either doing so indefinitely or eventually
connecting to form a cycle of even length. £
This leads to:
Example 5.4.4: Consider the class of skew chains A > Bn > C for 1 ≤ n ≤ ω, where
A = {a1, a2},
Bn = {a1, a2, a3, … , a2n} or {… , b–2, b–1, b0, b1, b2, …} if n = ω and
C = {c1, c2}.
The natural partial ordering given by a1 > bodd > c1 and a2 > beven > c2. Both A and C are full
B-cosets as well as full cosets of each other. A- and C-cosets in B are given respectively by the
partitions:
180
Lemma 5.4.3. Given a left-handed skew chain A > B > C, let a > c and aʹ > cʹ be input-
output pairs for a common coset bijection between A and C where a ≠ aʹ in A and c ≠ cʹ in C.
Upon setting A* = {a, aʹ}, B* = {x ∈ B⎪a > x > c or aʹ > x > cʹ} and C* = {c, cʹ}, one obtains a
sub-skew chain: A* > B* > C*. In particular,
i) aʹ > x > cʹ for x in B* implies: a > both a∧x and x∨c > c with a∧x –A* x –C* x∨c.
ii) a > x > c for x in B* implies: aʹ > both aʹ∧x and x∨cʹ > cʹ with aʹ∧x –A* x –C* x∨cʹ.
All A*-cosets and all C*-cosets in B* are of order 2. An A*C*-component in B* is either a
subset {b, bʹ} that is simultaneously an A* and C*-coset in B* or else it is a larger subset having
the alternating coset form:
… –A* • –C* • –A* • –C* • –A* • –C* …
Only the former can occur if the skew chain is categorical.
Proof. Being left-handed, we need only check the mixed outcomes, say a∧x, x∧a, c∨x and x∨c
where aʹ > x > cʹ for case (i). Trivially x∧a = x = c∨x. As for a∧x, a∧(a∧x) = a∧x = (a∧x)∧a, due
to left-handedness, so that a > a∧x; likewise c∧(a∧x) = c, while
(a∧x)∧c = a∧x∧a∧cʹ = a∧x∧cʹ = a∧cʹ = c
by left-handedness and the common coset bisection context. Hence a∧x > c also, so that a∧x is in
B*. The dual argument gives a > x∨c > c, so that x∨c ∈ B* also. Similarly (ii) holds and we
have a sub-skew chain.
Clearly the A*-cosets in B* either all have order 1 or all have order 2. If they have order
1, then a, aʹ > all elements in B*, and by transitivity, a, aʹ > both c, cʹ, so that a > c belongs to a
different coset bijection than aʹ > cʹ. Thus all A*-cosets in B* have order 2 and likewise all
C*-cosets in B* have order 2. In an A*C*-component in B*, if the first case does not occur, a
situation x –C* y –A* z with x, y and z distinct develops. Since A* and C*-cosets have size 2, it
extends in an alternating coset pattern in both directions, either doing so indefinitely or eventually
connecting to form a cycle of even length. £
This leads to:
Example 5.4.4: Consider the class of skew chains A > Bn > C for 1 ≤ n ≤ ω, where
A = {a1, a2},
Bn = {a1, a2, a3, … , a2n} or {… , b–2, b–1, b0, b1, b2, …} if n = ω and
C = {c1, c2}.
The natural partial ordering given by a1 > bodd > c1 and a2 > beven > c2. Both A and C are full
B-cosets as well as full cosets of each other. A- and C-cosets in B are given respectively by the
partitions:
180
