Page 169 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 169
V: Further Topics in Skew Lattices
j1 − j2 1
a1 − a2 b1 − b2 a1 − a2 b1 − b2
0 m1 − m2
The induced right-handed algebras are denoted respectively by NSR7 ,0 and NSR7 ,1 . Their left-
handed duals by are denoted NSL7 ,0 and NSL7 ,1. Cayley tables for NSR7 ,0 are given by:
∨ 0 an bn jn ∧ 0 an bn jn
0 0 an bn jn 0 00 0 0
am am am jm jm and am 0 an 0 an .
bm bm jm bm jm bm 0 0 bn bn
jm jm jm jm jm jm 0 an bn jn
All four algebras can be obtained from any one by ∨-∧ duality, L-R duality or a combination of
both. None is symmetric. Indeed, the tables above give a1∧b2 = 0 = b2∧a1 but a1∨b2 ≠ b2∨a1.
The following theorems appeared in 2011 in a paper of Cvetko-Vah, Kinyon, Leech and Spinks.
Theorem 5.1.2. A skew lattice is upper symmetric if and only if it contains no copy of
NSR7 ,0 or NSL7 ,0 . It is lower symmetric if and only if it contains no copy of NSR7 ,1 or NSL7 ,1 .
Finally, it is symmetric if and only if it contains no copy of any of these skew lattices.
Proof. Since neither NSR7 ,0 nor NSL7 ,0 is upper symmetric, any skew lattice containing a copy
of one of them cannot be upper symmetric. Conversely, consider first the case where S is a non-
upper symmetric, right-handed skew lattice. Thus a, b ∈ S exist such that a∧b = b∧a but
a∨b ≠ b∨a. Consider the subalgebra T generated by a and b. Since a∧b∧x = a∧b = x∧a∧b, for
x = a or b and hence for all x ∈ T, a∧b is the zero element of T. Moreover a and b must be in
incomparable R-classes for anti-symmetry to occur and thus one has the following configuration
with seven distinct elements in four D-classes within a right-handed skew diamond:
a∨b−b∨a
a − a ∧ (b ∨ a) b ∧ (a ∨ b) − b .
a∧b
For if a = a∧(b∨a), then b∨a ≥ a, b forcing a∨b = b∨a by Theorem 2.2.1, contradicting our
assumption on a and b. Thus a ≠ a∧(b∨a) and similarly b ≠ b∧(a∨b) giving us at least the seven
167
j1 − j2 1
a1 − a2 b1 − b2 a1 − a2 b1 − b2
0 m1 − m2
The induced right-handed algebras are denoted respectively by NSR7 ,0 and NSR7 ,1 . Their left-
handed duals by are denoted NSL7 ,0 and NSL7 ,1. Cayley tables for NSR7 ,0 are given by:
∨ 0 an bn jn ∧ 0 an bn jn
0 0 an bn jn 0 00 0 0
am am am jm jm and am 0 an 0 an .
bm bm jm bm jm bm 0 0 bn bn
jm jm jm jm jm jm 0 an bn jn
All four algebras can be obtained from any one by ∨-∧ duality, L-R duality or a combination of
both. None is symmetric. Indeed, the tables above give a1∧b2 = 0 = b2∧a1 but a1∨b2 ≠ b2∨a1.
The following theorems appeared in 2011 in a paper of Cvetko-Vah, Kinyon, Leech and Spinks.
Theorem 5.1.2. A skew lattice is upper symmetric if and only if it contains no copy of
NSR7 ,0 or NSL7 ,0 . It is lower symmetric if and only if it contains no copy of NSR7 ,1 or NSL7 ,1 .
Finally, it is symmetric if and only if it contains no copy of any of these skew lattices.
Proof. Since neither NSR7 ,0 nor NSL7 ,0 is upper symmetric, any skew lattice containing a copy
of one of them cannot be upper symmetric. Conversely, consider first the case where S is a non-
upper symmetric, right-handed skew lattice. Thus a, b ∈ S exist such that a∧b = b∧a but
a∨b ≠ b∨a. Consider the subalgebra T generated by a and b. Since a∧b∧x = a∧b = x∧a∧b, for
x = a or b and hence for all x ∈ T, a∧b is the zero element of T. Moreover a and b must be in
incomparable R-classes for anti-symmetry to occur and thus one has the following configuration
with seven distinct elements in four D-classes within a right-handed skew diamond:
a∨b−b∨a
a − a ∧ (b ∨ a) b ∧ (a ∨ b) − b .
a∧b
For if a = a∧(b∨a), then b∨a ≥ a, b forcing a∨b = b∨a by Theorem 2.2.1, contradicting our
assumption on a and b. Thus a ≠ a∧(b∨a) and similarly b ≠ b∧(a∨b) giving us at least the seven
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