Page 207 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 207
V: Further Topics in Skew Lattices
g are reciprocal ∨-isomorphisms that must restrict to isomorphisms between corresponding D-
classes (since x∧y = y∨x on D-classes), giving, e.g., m∨A∨m ≅ mʹ∨A∨mʹ. Thus f induces
isomorphisms between pointed primitive algebras,
m∨A∨m ∪ {m} ≅ mʹ∨A∨mʹ ∪ {mʹ} and m∨B∨m ∪ {m} ≅ mʹ∨B∨mʹ ∪ {mʹ}.
If S is also cancellative, then the previous theorem gives m∨S∨m ≅ mʹ∨S∨mʹ.
Theorem 5.7.10. If {J > A, B > M} is a cancellative skew diamond, then all pointed
skew diamonds m∨S∨m ⊆ S for m ∈M, are isomorphic. Dually, all pointed skew diamonds
j∧S∧j ⊆ S for j ∈J are isomorphic. £
Summing up much of the discussion about cancallative skew lattices we have:
Theorem 5.7.11. For a skew lattice S the following are equivalent:
i) S is cancellative.
ii) S is quasi-distributive and all [finite]skew diamonds in S are cancellative.
iii) S is quasi-distributive and symmetric, with all [finite]skew diamonds in it being strictly
categorical.
iv) S is quasi-distributive and all [finite]pointed skew diamonds in S factor as a product of
two primitive skew lattices.
v) S is quasi-distributive and |A||B| = |J||M| in any finite skew diamond {J > A, B > M} in S.
While neither the classes of distributive skew lattices or cancellative skew lattices
includes the other, all skew diamonds in cancellative skew lattices are distributive (being strictly
categorical), and all skew chains in distributive skew lattices are cancellative (being true in
general).
Historical remarks
The results in Section 5.1, on symmetry come from [Cvetko-Vah, Kinyon, Leech and
Spinks, 2011]. The results on comparing distributive identities in Section 2 are due to [Spinks,
1998 and 2000] and [Cvetko-Vah, 2006]. The material in Section 3 on cancellation is mostly
from [Cvetko-Vah, Kinyon et al, 2011] again, while the results in Section 4 on categorical
behavior are from [Kinyon and Leech, 2013]. The material in Sections 5 and 6 on distributivity
and its consquences comes from [Kinyon, Leech and Pita Costa, 2014?]. The various counting
results in the final section are from the dissertations of Pita Costa [2012] and Cvetko-Vah [2005]
as well as [Cvetko-Vah, Kinyon et al, 2011].
205
g are reciprocal ∨-isomorphisms that must restrict to isomorphisms between corresponding D-
classes (since x∧y = y∨x on D-classes), giving, e.g., m∨A∨m ≅ mʹ∨A∨mʹ. Thus f induces
isomorphisms between pointed primitive algebras,
m∨A∨m ∪ {m} ≅ mʹ∨A∨mʹ ∪ {mʹ} and m∨B∨m ∪ {m} ≅ mʹ∨B∨mʹ ∪ {mʹ}.
If S is also cancellative, then the previous theorem gives m∨S∨m ≅ mʹ∨S∨mʹ.
Theorem 5.7.10. If {J > A, B > M} is a cancellative skew diamond, then all pointed
skew diamonds m∨S∨m ⊆ S for m ∈M, are isomorphic. Dually, all pointed skew diamonds
j∧S∧j ⊆ S for j ∈J are isomorphic. £
Summing up much of the discussion about cancallative skew lattices we have:
Theorem 5.7.11. For a skew lattice S the following are equivalent:
i) S is cancellative.
ii) S is quasi-distributive and all [finite]skew diamonds in S are cancellative.
iii) S is quasi-distributive and symmetric, with all [finite]skew diamonds in it being strictly
categorical.
iv) S is quasi-distributive and all [finite]pointed skew diamonds in S factor as a product of
two primitive skew lattices.
v) S is quasi-distributive and |A||B| = |J||M| in any finite skew diamond {J > A, B > M} in S.
While neither the classes of distributive skew lattices or cancellative skew lattices
includes the other, all skew diamonds in cancellative skew lattices are distributive (being strictly
categorical), and all skew chains in distributive skew lattices are cancellative (being true in
general).
Historical remarks
The results in Section 5.1, on symmetry come from [Cvetko-Vah, Kinyon, Leech and
Spinks, 2011]. The results on comparing distributive identities in Section 2 are due to [Spinks,
1998 and 2000] and [Cvetko-Vah, 2006]. The material in Section 3 on cancellation is mostly
from [Cvetko-Vah, Kinyon et al, 2011] again, while the results in Section 4 on categorical
behavior are from [Kinyon and Leech, 2013]. The material in Sections 5 and 6 on distributivity
and its consquences comes from [Kinyon, Leech and Pita Costa, 2014?]. The various counting
results in the final section are from the dissertations of Pita Costa [2012] and Cvetko-Vah [2005]
as well as [Cvetko-Vah, Kinyon et al, 2011].
205
