Page 195 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 195
V: Further Topics in Skew Lattices
coset bijections within certain configurations of D-classes. (See the final section of [Kinyon,
Leech, Pita Costa].) For our purposes, however, their importance lies in the following “umbrella”
result and in the fact that both are direct consequences of symmetry.
Theorem 5.5.10. A quasi-distributive, linearly distributive skew lattice is ∧-distributive if
and only if it satisfies (5.5.10). Likewise, it is ∨-distributive if and only if it satisfies (5.5.11). In
general, a skew lattice is distributive if and only if it is quasi-distributive, linearly distributive and
satisfies both (5.5.10) and (5.5.11).
Proof. Clearly, (5.2.1) and (5.2.2) imply respectively (5.5.10) and (5.5.11). Conversely, suppose
that we have a skew lattice that is quasi-distributive and linearly distributive, and also satisfies
(5.5.10). In the left-handed case we have:
(x ∧ y) ∨ (x ∧ z) = x ∧ [(y ∧ x) ∨ (z ∧ x)] by (5.5.3L)
by (5.5.9)
= x ∧ [y ∨ (z ∧ x)] by left-handedness
= x ∧ [(z ∧ x) ∨ y ∨ (z ∧ x)] by (5.5.9)
= x ∧ [z ∨ y ∨ (z ∧ x)]
= x ∧ (y ∨ z) by (5.5.10L).
Thus (5.2.1L) holds. Likewise, the right-handed case must hold. The general case for (5.5.10)
implying (5.2.1) now follows as usual. The argument equating (5.5.11) with ∨-distributivity,
given that the skew lattice is both quasi-distributive and linearly distributive, is dual. £
A consequence of the above results is the following theorem. Recall that a skew lattice is
lower symmetric if x∨y = y∨x implies x∧y = y∧x. Dually, a skew lattice is upper symmetric if
x∧y = y∧x implies x∨y = y∨x. Recall that both types of partial symmetry were characterized
respectively by (5.1.3) x∧y∧(x∨y) = (y∨x)∧y∧x and by (5.1.4) x∨y∨(x∧y) = (y∧x)∨y∨x.
Theorem 5.5.11. An upper symmetric skew lattice is lower distributive if and only if it is
both quasi-distributive and linearly distributive; dually, a lower symmetric skew lattice is upper
distributive if and only if it is both quasi-distributive and linearly distributive. Thus a symmetric
skew lattice is distributive if and only if it is both quasi-distributive and linearly distributive.
Proof. The first statement follows from Lemma 5.5.9 and the following proposition. The second
statement follows by duality and the third follows in the usual way from the first two. £
Proposition 5.5.12. Upper symmetric skew lattices satisfy (5.5.10). Dually, lower
symmetric skew lattices satisfy (5.5.11).
Proof. We need only prove the first assertion. We begin with the case of a left-handed skew
lattice S, organizing the proof in this case in the following steps.
i) For all x, y, z in S, x ∨ y ≥ [x ∨ (y∧z)] ∧ [z ∨ (y∧z)].
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coset bijections within certain configurations of D-classes. (See the final section of [Kinyon,
Leech, Pita Costa].) For our purposes, however, their importance lies in the following “umbrella”
result and in the fact that both are direct consequences of symmetry.
Theorem 5.5.10. A quasi-distributive, linearly distributive skew lattice is ∧-distributive if
and only if it satisfies (5.5.10). Likewise, it is ∨-distributive if and only if it satisfies (5.5.11). In
general, a skew lattice is distributive if and only if it is quasi-distributive, linearly distributive and
satisfies both (5.5.10) and (5.5.11).
Proof. Clearly, (5.2.1) and (5.2.2) imply respectively (5.5.10) and (5.5.11). Conversely, suppose
that we have a skew lattice that is quasi-distributive and linearly distributive, and also satisfies
(5.5.10). In the left-handed case we have:
(x ∧ y) ∨ (x ∧ z) = x ∧ [(y ∧ x) ∨ (z ∧ x)] by (5.5.3L)
by (5.5.9)
= x ∧ [y ∨ (z ∧ x)] by left-handedness
= x ∧ [(z ∧ x) ∨ y ∨ (z ∧ x)] by (5.5.9)
= x ∧ [z ∨ y ∨ (z ∧ x)]
= x ∧ (y ∨ z) by (5.5.10L).
Thus (5.2.1L) holds. Likewise, the right-handed case must hold. The general case for (5.5.10)
implying (5.2.1) now follows as usual. The argument equating (5.5.11) with ∨-distributivity,
given that the skew lattice is both quasi-distributive and linearly distributive, is dual. £
A consequence of the above results is the following theorem. Recall that a skew lattice is
lower symmetric if x∨y = y∨x implies x∧y = y∧x. Dually, a skew lattice is upper symmetric if
x∧y = y∧x implies x∨y = y∨x. Recall that both types of partial symmetry were characterized
respectively by (5.1.3) x∧y∧(x∨y) = (y∨x)∧y∧x and by (5.1.4) x∨y∨(x∧y) = (y∧x)∨y∨x.
Theorem 5.5.11. An upper symmetric skew lattice is lower distributive if and only if it is
both quasi-distributive and linearly distributive; dually, a lower symmetric skew lattice is upper
distributive if and only if it is both quasi-distributive and linearly distributive. Thus a symmetric
skew lattice is distributive if and only if it is both quasi-distributive and linearly distributive.
Proof. The first statement follows from Lemma 5.5.9 and the following proposition. The second
statement follows by duality and the third follows in the usual way from the first two. £
Proposition 5.5.12. Upper symmetric skew lattices satisfy (5.5.10). Dually, lower
symmetric skew lattices satisfy (5.5.11).
Proof. We need only prove the first assertion. We begin with the case of a left-handed skew
lattice S, organizing the proof in this case in the following steps.
i) For all x, y, z in S, x ∨ y ≥ [x ∨ (y∧z)] ∧ [z ∨ (y∧z)].
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