Page 67 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 67
II: Skew Lattices
Proof. The theorem holds when S is primitive. Next, assume S is bounded with maximal class A
and minimal class Z. Then Bd(S) = A∪Z decomposes into components Ai∪Zi. We say that an
element x ∈S belongs to component Ai∪Zi if the latter is the unique boundary component such
that there exists u ∈A and v ∈Z such that u ≥ x ≥ v. For any y ∈ S such that either x ≤ y or x ≥ y it
is clear that x and y belong to the same boundary component. Hence the inclusion Bd(S) ⊆ S
indices a bijection between the classes of components. Next, let x belong to Ai∪Zi and y belong
to Aj∪Zj. Pick u ∈ Ai and w ∈ Aj such that u ≥ x∨y∨x and w ≥ y∨x∨y. By the previous theorem,
u∨w ρ (x∨y∨x)∨( y∨x∨y) = x∨y. Thus x∨y lies in the component of S containing
(Ai∪Zi) ∨ (Aj∪Zj). Likewise, x∧y lies in the component containing (Ai∪Zi) ∧ (Aj∪Zj). The
bounded case of the theorem now follows from the primitive case. The general case follows from
the fact that every skew lattice is the directed union of its intervals. £
Recall that a noncommutative lattice splits if it factors as the product of a lattice and an
antilattice (here a rectangular skew lattice). Since every component of a skew lattice S meets
every D-class of S we have the following corollary.
Corollary 2.4.7. Given a skew lattice S, S/(D∩ρ) is its maximal split image. £
Orthogonal D-classes and the behavior of skew diamonds
Let A, B and C be D-classes in a skew lattice S such that both A and B are comparable
with C. By a class component of A [or B] in C is meant the intersection of a component of A∪C
[or B∪C] with C. We say that A and B are orthogonal in C if each class component of A in C
lies in a unique coset of B in C and likewise each class component of B in C lies in a unique coset
of A in C where
A and B being orthogonal in C is equivalent to asserting that the image in C of each x in
A lies in a unique coset of B in C and likewise the image in C of each y in B lies in a unique coset
of A in C. That is, each x in A is covered in C by a unique coset of B in C and dually each y in B
is covered in C by a unique coset of A in C.
Lemma 2.4.8. Given D-classes A, B and C in a skew lattice S, if A and B are
orthogonal in C, then each coset of A in C has nonempty intersection with each coset of B in C;
all such coset intersections, moreover, have a common cardinality.
Proof. Indeed let A1 and A2 be cosets of A in C, let B1 and B2 be cosets of B in C, and let ϕ1 and
ϕ2 be coset bijections of A1 and A2 into a common coset of A in C. (If C lies above A1 and A2
then ϕ1 and ϕ2 are inverses of the downward bijections.) The bijection ϕ2−1ϕ1 and its inverse
ϕ1−1ϕ2 keep individual elements in the same class component of A in C. Orthogonality implies
that both bijections restrict to an inverse pair of bijections of A1∩B1 with A2∩B1. Similarly
A2∩B1 is in 1-1 correspondence with A2∩B2, so the assertion is verified. £
65
Proof. The theorem holds when S is primitive. Next, assume S is bounded with maximal class A
and minimal class Z. Then Bd(S) = A∪Z decomposes into components Ai∪Zi. We say that an
element x ∈S belongs to component Ai∪Zi if the latter is the unique boundary component such
that there exists u ∈A and v ∈Z such that u ≥ x ≥ v. For any y ∈ S such that either x ≤ y or x ≥ y it
is clear that x and y belong to the same boundary component. Hence the inclusion Bd(S) ⊆ S
indices a bijection between the classes of components. Next, let x belong to Ai∪Zi and y belong
to Aj∪Zj. Pick u ∈ Ai and w ∈ Aj such that u ≥ x∨y∨x and w ≥ y∨x∨y. By the previous theorem,
u∨w ρ (x∨y∨x)∨( y∨x∨y) = x∨y. Thus x∨y lies in the component of S containing
(Ai∪Zi) ∨ (Aj∪Zj). Likewise, x∧y lies in the component containing (Ai∪Zi) ∧ (Aj∪Zj). The
bounded case of the theorem now follows from the primitive case. The general case follows from
the fact that every skew lattice is the directed union of its intervals. £
Recall that a noncommutative lattice splits if it factors as the product of a lattice and an
antilattice (here a rectangular skew lattice). Since every component of a skew lattice S meets
every D-class of S we have the following corollary.
Corollary 2.4.7. Given a skew lattice S, S/(D∩ρ) is its maximal split image. £
Orthogonal D-classes and the behavior of skew diamonds
Let A, B and C be D-classes in a skew lattice S such that both A and B are comparable
with C. By a class component of A [or B] in C is meant the intersection of a component of A∪C
[or B∪C] with C. We say that A and B are orthogonal in C if each class component of A in C
lies in a unique coset of B in C and likewise each class component of B in C lies in a unique coset
of A in C where
A and B being orthogonal in C is equivalent to asserting that the image in C of each x in
A lies in a unique coset of B in C and likewise the image in C of each y in B lies in a unique coset
of A in C. That is, each x in A is covered in C by a unique coset of B in C and dually each y in B
is covered in C by a unique coset of A in C.
Lemma 2.4.8. Given D-classes A, B and C in a skew lattice S, if A and B are
orthogonal in C, then each coset of A in C has nonempty intersection with each coset of B in C;
all such coset intersections, moreover, have a common cardinality.
Proof. Indeed let A1 and A2 be cosets of A in C, let B1 and B2 be cosets of B in C, and let ϕ1 and
ϕ2 be coset bijections of A1 and A2 into a common coset of A in C. (If C lies above A1 and A2
then ϕ1 and ϕ2 are inverses of the downward bijections.) The bijection ϕ2−1ϕ1 and its inverse
ϕ1−1ϕ2 keep individual elements in the same class component of A in C. Orthogonality implies
that both bijections restrict to an inverse pair of bijections of A1∩B1 with A2∩B1. Similarly
A2∩B1 is in 1-1 correspondence with A2∩B2, so the assertion is verified. £
65
