Page 205 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 205
V: Further Topics in Skew Lattices
[A1: An] = [A1: A2][A2: An] = [A1: A2][A2: A3][A3: An] =…= [A1: A2][A2: A3]…[An–1: An]. £
Returning to skew diamonds, the following restatement of Theorem 2.4.10 parallels to
some extent the theorem above:
Theorem 5.7.5. In a symmetric skew diamond {J > A, B > M}, [M: J] = [M: A] [M: B]
and [J: M] = [J: A][J : B] . £
Lemma 5.7.6. Given finite D-classes X > Y in a skew lattice, |X|[Y: X] = |Y|[X: Y], or put
otherwise, [Y: X] = |Y| [X: Y] .
|X|
Proof. |X|[Y: X] and |Y|[X: Y] expand to [X: Y] ωXY [Y: X] and to [Y: X] ωXY [X: Y]. £
This next result of relevance is from Pita da Costa’s dissertation.
Theorem 5.7.7. Given a finite cancellative skew diamond {J > A, B > M}, |A| |B| = |J| |M|.
Proof. One has |A| = [A : J] = [M : B] = |M| . The first and third equalities come from the
|J| [J : A] [B : M] |B|
previous lemma. The middle equality is from [B: M] = [J: A] and [M: B] = [A: J] in Theorem
5.7.1. The equality now follows by cross-multiplying. £
This outcome fails in the four simply cancellative NS7 variants and both symmetric NC5
variants. (But see Corollary 2.4.11.) One thus has:
Corollary 5.7.8. A skew lattice S is cancellative if and only if it is quasi-distributive and
all of its skew diamonds are cancellative. The latter occurs if and only if |A| |B| = |J| |M| holds in
all finite skew diamonds {J > A, B > M} of S. £
This situation is sharpened if the skew diamond {J > A, B > M} is pointed in that |J| = 1
or |M| = 1. If J has a unique element, it is often denoted by 1; if it is M, the single point is often
denoted by 0. NSL7 ,0 , NSR7 ,0 , NSL7 ,1 and NSR7 ,1 are pointed, while NCL5 and NCR5 are
doubly pointed. Indeed, a skew diamond is simply cancellative if and only if all doubly pointed
skew diamond subalgebras are sublattices (thus eliminating any possible NC5 subalgebras).
What about full cancellation? The next two Theorems are from the 2011 paper of Cvetko-Vah,
Kinyon, Leech and Spinks.
Theorem 5.7.9. A quasi-distributive skew lattice is cancellative if and only if all pointed
skew diamonds in it factor as products of primitive skew lattices.
Proof. Given a skew lattice S, the condition on pointed skew diamonds in it excludes copies of
NCL5 , NCR5 , NSL7 ,0 , NSR7 ,0 , NSL7 ,1 and NSR7 ,1 from being subalgebras since in each of
these six cases the order of the join or meet classes is inconsistent with such a factorization. This
203
[A1: An] = [A1: A2][A2: An] = [A1: A2][A2: A3][A3: An] =…= [A1: A2][A2: A3]…[An–1: An]. £
Returning to skew diamonds, the following restatement of Theorem 2.4.10 parallels to
some extent the theorem above:
Theorem 5.7.5. In a symmetric skew diamond {J > A, B > M}, [M: J] = [M: A] [M: B]
and [J: M] = [J: A][J : B] . £
Lemma 5.7.6. Given finite D-classes X > Y in a skew lattice, |X|[Y: X] = |Y|[X: Y], or put
otherwise, [Y: X] = |Y| [X: Y] .
|X|
Proof. |X|[Y: X] and |Y|[X: Y] expand to [X: Y] ωXY [Y: X] and to [Y: X] ωXY [X: Y]. £
This next result of relevance is from Pita da Costa’s dissertation.
Theorem 5.7.7. Given a finite cancellative skew diamond {J > A, B > M}, |A| |B| = |J| |M|.
Proof. One has |A| = [A : J] = [M : B] = |M| . The first and third equalities come from the
|J| [J : A] [B : M] |B|
previous lemma. The middle equality is from [B: M] = [J: A] and [M: B] = [A: J] in Theorem
5.7.1. The equality now follows by cross-multiplying. £
This outcome fails in the four simply cancellative NS7 variants and both symmetric NC5
variants. (But see Corollary 2.4.11.) One thus has:
Corollary 5.7.8. A skew lattice S is cancellative if and only if it is quasi-distributive and
all of its skew diamonds are cancellative. The latter occurs if and only if |A| |B| = |J| |M| holds in
all finite skew diamonds {J > A, B > M} of S. £
This situation is sharpened if the skew diamond {J > A, B > M} is pointed in that |J| = 1
or |M| = 1. If J has a unique element, it is often denoted by 1; if it is M, the single point is often
denoted by 0. NSL7 ,0 , NSR7 ,0 , NSL7 ,1 and NSR7 ,1 are pointed, while NCL5 and NCR5 are
doubly pointed. Indeed, a skew diamond is simply cancellative if and only if all doubly pointed
skew diamond subalgebras are sublattices (thus eliminating any possible NC5 subalgebras).
What about full cancellation? The next two Theorems are from the 2011 paper of Cvetko-Vah,
Kinyon, Leech and Spinks.
Theorem 5.7.9. A quasi-distributive skew lattice is cancellative if and only if all pointed
skew diamonds in it factor as products of primitive skew lattices.
Proof. Given a skew lattice S, the condition on pointed skew diamonds in it excludes copies of
NCL5 , NCR5 , NSL7 ,0 , NSR7 ,0 , NSL7 ,1 and NSR7 ,1 from being subalgebras since in each of
these six cases the order of the join or meet classes is inconsistent with such a factorization. This
203
