Page 73 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 73
II: Skew Lattices
Proof. Pick b ∈ Bj. Then for all x ∈ Ai, ϕ(x) = x∧b∧x. Thus given x, y in Ai we have
ϕ(x∧y) = (x∧y)∧b∧(x∧y) = x∧b∧y = x∧b∧b∧y = x∧bx∧y∧b∧y = ϕ(x) ∧ ϕ(y),
due to regularity and Corollary 1.2.8. But since Ai and Bj are rectangular algebras satisfying
x∨y = y∧x, the proposition follows. £
In the left [right] rectangular case this proposition is trivially true since it is easily seen
that any bijection between left [right] rectangular skew lattices must be an isomorphism.
2.5 Partial skew lattices and coset projections
Every nonrectangular skew lattice S is the union of its maximal primitive skew lattices
and the latter jointly determine its structure. Indeed one could view primitive skew lattices as
“lego pieces” that when appropriately “snapped together” produce entire skew lattices. To pursue
this perspective, we need a new concept. A partial skew lattice on a quasi-ordered set (S, ≻) is a
4-tuple (S, , ∨, ∧) where ∨ and ∧ are partial binary operations defined for pairs of elements
comparable under ≻ in such a way that the union U of any chain of ≻-equivalence classes forms a
skew lattice under ∨ and ∧ whose natural quasi-order coincides with ≻ over U. In particular,
each equivalence class A under ∨ and ∧ is a rectangular skew lattice and each comparable pair of
such classes, say A > B, forms a primitive skew lattice. That ∨ and ∧ are associative on totally
quasi-ordered subsets is called linear associativity.
Lemma 2.5.1. Let (S, , ∨, ∧) be a partial skew lattice and let m ≺ x, y ≺ p in S. Then
both (x ∨ p) ∨ y = x ∨ (p ∨ y) and (x ∧ m) ∧ y = x ∧ (m ∧ y). (This form of associativity, where x
and y need not be comparable, is called extremal associativity.)
Proof. (x ∨ p) ∨ (p ∨ y) reduces to both (x ∨ p) ∨ y and x ∨ (p ∨ y). Thus the latter must be equal.
Similar remarks hold for (x ∧ m) ∧ y and x ∧ (m ∧ y). £
Every skew lattice (S; ∨, ∧) induces a canonical partial skew lattice (S; ≻, ∨, ∧) upon
restricting the given binary operations, ∨ and ∧, to ≻-related pairs. When is a given partial skew
lattice (S, , ∨, ∧) canonical for some skew lattice? Two conditions are obviously necessary.
I. The ≻-equivalence classes (proto-D-classes of mutually ≻-related elements) must
have both join and meet classes.
II. Each pair of equivalence classes must be orthogonal in both their join and meet
classes.
Given both conditions, both ∨ and ∧ can be extended to full binary operations as follows.
III. Set x∨y = (x∨q∨x)∨(y∨p∨y) for any p, q in the join class of x and y such that p ≥ x
and q ≥ y.
71
Proof. Pick b ∈ Bj. Then for all x ∈ Ai, ϕ(x) = x∧b∧x. Thus given x, y in Ai we have
ϕ(x∧y) = (x∧y)∧b∧(x∧y) = x∧b∧y = x∧b∧b∧y = x∧bx∧y∧b∧y = ϕ(x) ∧ ϕ(y),
due to regularity and Corollary 1.2.8. But since Ai and Bj are rectangular algebras satisfying
x∨y = y∧x, the proposition follows. £
In the left [right] rectangular case this proposition is trivially true since it is easily seen
that any bijection between left [right] rectangular skew lattices must be an isomorphism.
2.5 Partial skew lattices and coset projections
Every nonrectangular skew lattice S is the union of its maximal primitive skew lattices
and the latter jointly determine its structure. Indeed one could view primitive skew lattices as
“lego pieces” that when appropriately “snapped together” produce entire skew lattices. To pursue
this perspective, we need a new concept. A partial skew lattice on a quasi-ordered set (S, ≻) is a
4-tuple (S, , ∨, ∧) where ∨ and ∧ are partial binary operations defined for pairs of elements
comparable under ≻ in such a way that the union U of any chain of ≻-equivalence classes forms a
skew lattice under ∨ and ∧ whose natural quasi-order coincides with ≻ over U. In particular,
each equivalence class A under ∨ and ∧ is a rectangular skew lattice and each comparable pair of
such classes, say A > B, forms a primitive skew lattice. That ∨ and ∧ are associative on totally
quasi-ordered subsets is called linear associativity.
Lemma 2.5.1. Let (S, , ∨, ∧) be a partial skew lattice and let m ≺ x, y ≺ p in S. Then
both (x ∨ p) ∨ y = x ∨ (p ∨ y) and (x ∧ m) ∧ y = x ∧ (m ∧ y). (This form of associativity, where x
and y need not be comparable, is called extremal associativity.)
Proof. (x ∨ p) ∨ (p ∨ y) reduces to both (x ∨ p) ∨ y and x ∨ (p ∨ y). Thus the latter must be equal.
Similar remarks hold for (x ∧ m) ∧ y and x ∧ (m ∧ y). £
Every skew lattice (S; ∨, ∧) induces a canonical partial skew lattice (S; ≻, ∨, ∧) upon
restricting the given binary operations, ∨ and ∧, to ≻-related pairs. When is a given partial skew
lattice (S, , ∨, ∧) canonical for some skew lattice? Two conditions are obviously necessary.
I. The ≻-equivalence classes (proto-D-classes of mutually ≻-related elements) must
have both join and meet classes.
II. Each pair of equivalence classes must be orthogonal in both their join and meet
classes.
Given both conditions, both ∨ and ∧ can be extended to full binary operations as follows.
III. Set x∨y = (x∨q∨x)∨(y∨p∨y) for any p, q in the join class of x and y such that p ≥ x
and q ≥ y.
71
