Page 179 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 179
V: Further Topics in Skew Lattices
Theorem 5.3.10. Given a simply cancellative skew diamond {J > A, B > M}, the function
ξ: Comm2(A, B) → ω(J, M) defined by ξ(a, b) = (a∨b, a∧b) is a bijection. Conversely, given any
skew diamond {J > A, B > M} the map ξ as stated is always well-defined, but it is a bijection only
if the skew diamond is simply cancellative.
Proof. If {J > A, B > M} is not simply cancellative, then either a copy of NC5L or NCR5 occurs
which gives ξ a properly many-to-one instance. £
The above results in this section are from the 2011 paper of Cvetko-Vah, Kinyon, Leech
and Spinks.
We conclude this section with several observations:
Theorem 5.3.11. Skew lattices in rings are cancellative.
Proof. So let ab = ac and a∇b = a∇c. Expanding and making the obvious reductions reduces the
latter to
b + ba –bab = c + ca – cac = c + ca – cab.
Thus b(1 – a – ab) = c(1 – a – ab). But (1 – a – ab)2 = 1 – a + aba and so
b = b(1 – a + aba) = c(1 – a + aba).
Similarly c = c(1 – a + aca) = c(1 – a + aba) and b = c follows. Right cancellation is shown
similarly. £
Theorem 5.3.12. Strongly distributive skew lattices are cancellative.
Proof. Strongly distributive skew lattices are clearly quasi-distributive, and by Theorem 2.3.4
also symmetric. By the same theorem they are also normal, so that neither NCL5 nor NCR5 can
be a subalgebra. Hence none of the 5- or 7-element forbidden algebras can be a subalgebra,
making any strongly distributive skew lattice cancellative. £
Primitive skew lattices and skew chains in general are trivially cancellative: they are
clearly symmetric and quasi-distributive and it is impossible for either variant of NC5 to be a
subalgebra. Thus we have:
Proposition 5.3.13. Every skew lattice in the variety of skew lattices generated from the
class of all primitive skew lattices is cancellative. More generally, every skew lattice in the
variety of skew lattices generated from the class of all skew chains is cancellative. £
Queries: Are the varieties generated by these two classes of algebras the same? Is the
second variety in fact the variety of all cancellative skew lattices?
177
Theorem 5.3.10. Given a simply cancellative skew diamond {J > A, B > M}, the function
ξ: Comm2(A, B) → ω(J, M) defined by ξ(a, b) = (a∨b, a∧b) is a bijection. Conversely, given any
skew diamond {J > A, B > M} the map ξ as stated is always well-defined, but it is a bijection only
if the skew diamond is simply cancellative.
Proof. If {J > A, B > M} is not simply cancellative, then either a copy of NC5L or NCR5 occurs
which gives ξ a properly many-to-one instance. £
The above results in this section are from the 2011 paper of Cvetko-Vah, Kinyon, Leech
and Spinks.
We conclude this section with several observations:
Theorem 5.3.11. Skew lattices in rings are cancellative.
Proof. So let ab = ac and a∇b = a∇c. Expanding and making the obvious reductions reduces the
latter to
b + ba –bab = c + ca – cac = c + ca – cab.
Thus b(1 – a – ab) = c(1 – a – ab). But (1 – a – ab)2 = 1 – a + aba and so
b = b(1 – a + aba) = c(1 – a + aba).
Similarly c = c(1 – a + aca) = c(1 – a + aba) and b = c follows. Right cancellation is shown
similarly. £
Theorem 5.3.12. Strongly distributive skew lattices are cancellative.
Proof. Strongly distributive skew lattices are clearly quasi-distributive, and by Theorem 2.3.4
also symmetric. By the same theorem they are also normal, so that neither NCL5 nor NCR5 can
be a subalgebra. Hence none of the 5- or 7-element forbidden algebras can be a subalgebra,
making any strongly distributive skew lattice cancellative. £
Primitive skew lattices and skew chains in general are trivially cancellative: they are
clearly symmetric and quasi-distributive and it is impossible for either variant of NC5 to be a
subalgebra. Thus we have:
Proposition 5.3.13. Every skew lattice in the variety of skew lattices generated from the
class of all primitive skew lattices is cancellative. More generally, every skew lattice in the
variety of skew lattices generated from the class of all skew chains is cancellative. £
Queries: Are the varieties generated by these two classes of algebras the same? Is the
second variety in fact the variety of all cancellative skew lattices?
177
