Page 170 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 170
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
distinct displayed elements. We claim that this set is closed under ∨ and ∧. Each displayed D-
class is closed. We consider representatives of the less trivial remaining cases.
a∧(b∨a) ∨ (a∨b) = [a∧(b∨a) ∨ a]∨ b = a∧(b∨a) ∨ b = b∨a
since a∧(b∨a) R a and both a∧(b∨a) and b are ≤ b∨a. Likewise,
a∨[b∧(a∨b)] = a∨[b∧(a∨b)]∨(a∨b) = a∨(a∨b) = a∨b = [b∧(a∨b)]∨a∨b = [b∧(a∨b)]∨a
and
[a∧(b∨a)]∨[b∧(a∨b)] = [a∧(b∨a)]∨[b∧(a∨b)]∨a∨b = [a∧(b∨a)]∨a∨b = b∨a
by a prior calculation. Thus, T is as in the diagram and it is a copy of NSR7 ,0 . Dual remarks
involving NSL7 ,0 arise in the left-handed case.
Suppose next that S is any non-upper symmetric skew lattice with a, b ∈ S such that
a∧b = b∧a, but a∨b ≠ b∨a. Again, a and b generate a subalgebra Sʹ that forms a non-upper
symmetric subalgebra for which a∧b is the zero element. Moreover, one of Sʹ/R or Sʹ/L must be
non-upper symmetric and thus a copy of NSL7 ,0 or NSR7 ,0 . Whichever case it is, a copy also
exists in Sʹ by Theorem 2.2.9. The lower symmetric case follows by duality. Both cases combine
to characterize full symmetry. £
Upper and lower symmetry can each be parsed further in left-right fashion to give a four-
fold partition of symmetry that will prove useful in Section 5.3.
Theorem 5.1.3. Given a skew lattice S:
i) S/L being upper symmetric is equivalent to either of the following:
a) S contains no copy of NSR7 ,0 .
b) S satisfies x∨y∨x = (y∧x)∨y∨x.
ii) S/R being upper symmetric is equivalent to either of the following:
a) S contains no copy of NSL7 ,0 .
b) S satisfies x∨y∨x = x∨y∨(x∧y).
iii) S/R being lower symmetric is equivalent to either of the following:
a) S contains no copy of NSL7 ,1.
b) S satisfies x∧y∧x = (y∨x)∧y∧x.
iv) S/L being lower symmetric is equivalent to either of the following:
a) S contains no copy of NSR7 ,1 .
b) S satisfies x∧y∧x = x∧y∧(x∨y).
The above four conditions hence determine four varieties of skew lattices.
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distinct displayed elements. We claim that this set is closed under ∨ and ∧. Each displayed D-
class is closed. We consider representatives of the less trivial remaining cases.
a∧(b∨a) ∨ (a∨b) = [a∧(b∨a) ∨ a]∨ b = a∧(b∨a) ∨ b = b∨a
since a∧(b∨a) R a and both a∧(b∨a) and b are ≤ b∨a. Likewise,
a∨[b∧(a∨b)] = a∨[b∧(a∨b)]∨(a∨b) = a∨(a∨b) = a∨b = [b∧(a∨b)]∨a∨b = [b∧(a∨b)]∨a
and
[a∧(b∨a)]∨[b∧(a∨b)] = [a∧(b∨a)]∨[b∧(a∨b)]∨a∨b = [a∧(b∨a)]∨a∨b = b∨a
by a prior calculation. Thus, T is as in the diagram and it is a copy of NSR7 ,0 . Dual remarks
involving NSL7 ,0 arise in the left-handed case.
Suppose next that S is any non-upper symmetric skew lattice with a, b ∈ S such that
a∧b = b∧a, but a∨b ≠ b∨a. Again, a and b generate a subalgebra Sʹ that forms a non-upper
symmetric subalgebra for which a∧b is the zero element. Moreover, one of Sʹ/R or Sʹ/L must be
non-upper symmetric and thus a copy of NSL7 ,0 or NSR7 ,0 . Whichever case it is, a copy also
exists in Sʹ by Theorem 2.2.9. The lower symmetric case follows by duality. Both cases combine
to characterize full symmetry. £
Upper and lower symmetry can each be parsed further in left-right fashion to give a four-
fold partition of symmetry that will prove useful in Section 5.3.
Theorem 5.1.3. Given a skew lattice S:
i) S/L being upper symmetric is equivalent to either of the following:
a) S contains no copy of NSR7 ,0 .
b) S satisfies x∨y∨x = (y∧x)∨y∨x.
ii) S/R being upper symmetric is equivalent to either of the following:
a) S contains no copy of NSL7 ,0 .
b) S satisfies x∨y∨x = x∨y∨(x∧y).
iii) S/R being lower symmetric is equivalent to either of the following:
a) S contains no copy of NSL7 ,1.
b) S satisfies x∧y∧x = (y∨x)∧y∧x.
iv) S/L being lower symmetric is equivalent to either of the following:
a) S contains no copy of NSR7 ,1 .
b) S satisfies x∧y∧x = x∧y∧(x∨y).
The above four conditions hence determine four varieties of skew lattices.
168
