Page 79 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 79
II: Skew Lattices
Proof. The basic coset structure insures that θ is at least a bijection. To see that θ as defined is
an isomorphism, first observe that for x, xʹ in Ib and y, yʹ in B, regularity gives:
θ(x, y) ∨ θ(xʹ, yʹ) = y∨x∨y ∨ yʹ∨xʹ∨yʹ = y∨x∨xʹ∨yʹ = y∨yʹ∨x∨xʹ∨y∨yʹ = θ(i∨j, b∨bʹ).
Thus θ is at least a ∨-isomorphism of skew lattices, in which case it is also a full isomorphism
between corresponding D-classes, where u∧v = v∧u holds. Suppose say x = b. Then we have
both θ(b, y) ∧ θ(xʹ, yʹ) = y ∧ yʹ∨xʹ∨yʹ = y∧yʹ since yʹ is the unique image of yʹ∨xʹ∨yʹ in B, while
θ(b∧xʹ, y∧yʹ) = (y∧yʹ)∨(b∧xʹ)∨(y∨yʹ) = y∨yʹ since b∧xʹ = b. The case where xʹ = b is similar. £
This simple result can be extended several ways. We begin with normal skew lattices
possessing D-classes that are minimal with respect to the partial ordering of D-classes.
Proposition 2.6.2. Let S be a normal skew lattice with a minimal D-class B. Pick b ∈ B
and let T be the subalgebra b∨S∨b of S given as {b∨x∨b⎪x ∈ S} or equivalently {x ∈ S⎪x ≥ b}.
Then b is the zero element of T, and an isomorphism θ: T × B → S is given by θ(x, y) = y∨x∨y.
Proof. That θ: T × B → S is a bijection, that between corresponding D-classes an isomorphism,
follows from the lemma. Given (x, y), (xʹ, yʹ) in T × B, that θ(x, y) ∨ θ(xʹ, b) = θ(x∨xʹ, y∨b) is
seen exactly as above so that θ is at least a ∨-isomorphism. Next, observe that
θ(x, y) ∧ θ(xʹ, yʹ) = (y∨x∨y) ∧ (yʹ∨xʹ∨ yʹ) ≤ (yʹ∨xʹ∨ yʹ) ∨ (y∨x∨y) = yʹ∨xʹ∨x∨y
since u∧v ≤ v∨u for skew lattices in general. θ(x∧xʹ, y∧ yʹ) = (y∧ yʹ)∨(x∧xʹ)∨(y∧ yʹ), on the other
hand, is D-related to x∧xʹ and thus to (y∨x∨y)∧(yʹ∨xʹ∨ yʹ). But (y∧yʹ)∨(x∧xʹ)∨(y∧ yʹ) ≤ yʹ∨xʹ∨x∨y
also. Indeed:
(y∧yʹ)∨(x∧xʹ)∨(y∧yʹ)∨(yʹ∨xʹ∨x∨y) = (y∧yʹ)∨(x∧xʹ)∨(yʹ∨xʹ∨x∨y) = (y∧yʹ)∨(x∧xʹ)∨(xʹ∨x∨y)
= (y∧yʹ)∨(xʹ∨x∨y) = (y∧yʹ)∨yʹ∨(xʹ∨x∨y) = yʹ∨xʹ∨x∨y
by a combination of absorption and regularity. Likewise:
(yʹ∨xʹ∨x∨y) ∨ (y∧yʹ)∨(x∧xʹ)∨(y∧yʹ) = yʹ∨xʹ∨x∨y.
Since θ(x, y)∧θ(xʹ, yʹ) is D-equivalent to θ(x∧xʹ, y∧ yʹ) in a normal skew lattice with a common
upper bound in (S, ≥), they are equal, i.e., θ(x, y) ∧ θ(xʹ, yʹ) = θ(x∧xʹ, y∧ yʹ). £
As an application, let A > B > C be a normal skew chain. Then first, this skew chain is
isomorphic to the product of a skew chain Aʹ > Bʹ > {0} with C. But Aʹ > Bʹ in turn is
isomorphic to the product of a skew chain Aʺ > {0} with Bʹ. Thus A > B > C essentially is
Aʺ×Bʹ×C > Bʹ×C > C with all downward projections and upward coset injections being the
coordinate-wise functions: p(a, b, c) projects down to (b, c) or even further down to c, while say
ϕa–1(b, c) = (a, b, c). In general we have:
77
Proof. The basic coset structure insures that θ is at least a bijection. To see that θ as defined is
an isomorphism, first observe that for x, xʹ in Ib and y, yʹ in B, regularity gives:
θ(x, y) ∨ θ(xʹ, yʹ) = y∨x∨y ∨ yʹ∨xʹ∨yʹ = y∨x∨xʹ∨yʹ = y∨yʹ∨x∨xʹ∨y∨yʹ = θ(i∨j, b∨bʹ).
Thus θ is at least a ∨-isomorphism of skew lattices, in which case it is also a full isomorphism
between corresponding D-classes, where u∧v = v∧u holds. Suppose say x = b. Then we have
both θ(b, y) ∧ θ(xʹ, yʹ) = y ∧ yʹ∨xʹ∨yʹ = y∧yʹ since yʹ is the unique image of yʹ∨xʹ∨yʹ in B, while
θ(b∧xʹ, y∧yʹ) = (y∧yʹ)∨(b∧xʹ)∨(y∨yʹ) = y∨yʹ since b∧xʹ = b. The case where xʹ = b is similar. £
This simple result can be extended several ways. We begin with normal skew lattices
possessing D-classes that are minimal with respect to the partial ordering of D-classes.
Proposition 2.6.2. Let S be a normal skew lattice with a minimal D-class B. Pick b ∈ B
and let T be the subalgebra b∨S∨b of S given as {b∨x∨b⎪x ∈ S} or equivalently {x ∈ S⎪x ≥ b}.
Then b is the zero element of T, and an isomorphism θ: T × B → S is given by θ(x, y) = y∨x∨y.
Proof. That θ: T × B → S is a bijection, that between corresponding D-classes an isomorphism,
follows from the lemma. Given (x, y), (xʹ, yʹ) in T × B, that θ(x, y) ∨ θ(xʹ, b) = θ(x∨xʹ, y∨b) is
seen exactly as above so that θ is at least a ∨-isomorphism. Next, observe that
θ(x, y) ∧ θ(xʹ, yʹ) = (y∨x∨y) ∧ (yʹ∨xʹ∨ yʹ) ≤ (yʹ∨xʹ∨ yʹ) ∨ (y∨x∨y) = yʹ∨xʹ∨x∨y
since u∧v ≤ v∨u for skew lattices in general. θ(x∧xʹ, y∧ yʹ) = (y∧ yʹ)∨(x∧xʹ)∨(y∧ yʹ), on the other
hand, is D-related to x∧xʹ and thus to (y∨x∨y)∧(yʹ∨xʹ∨ yʹ). But (y∧yʹ)∨(x∧xʹ)∨(y∧ yʹ) ≤ yʹ∨xʹ∨x∨y
also. Indeed:
(y∧yʹ)∨(x∧xʹ)∨(y∧yʹ)∨(yʹ∨xʹ∨x∨y) = (y∧yʹ)∨(x∧xʹ)∨(yʹ∨xʹ∨x∨y) = (y∧yʹ)∨(x∧xʹ)∨(xʹ∨x∨y)
= (y∧yʹ)∨(xʹ∨x∨y) = (y∧yʹ)∨yʹ∨(xʹ∨x∨y) = yʹ∨xʹ∨x∨y
by a combination of absorption and regularity. Likewise:
(yʹ∨xʹ∨x∨y) ∨ (y∧yʹ)∨(x∧xʹ)∨(y∧yʹ) = yʹ∨xʹ∨x∨y.
Since θ(x, y)∧θ(xʹ, yʹ) is D-equivalent to θ(x∧xʹ, y∧ yʹ) in a normal skew lattice with a common
upper bound in (S, ≥), they are equal, i.e., θ(x, y) ∧ θ(xʹ, yʹ) = θ(x∧xʹ, y∧ yʹ). £
As an application, let A > B > C be a normal skew chain. Then first, this skew chain is
isomorphic to the product of a skew chain Aʹ > Bʹ > {0} with C. But Aʹ > Bʹ in turn is
isomorphic to the product of a skew chain Aʺ > {0} with Bʹ. Thus A > B > C essentially is
Aʺ×Bʹ×C > Bʹ×C > C with all downward projections and upward coset injections being the
coordinate-wise functions: p(a, b, c) projects down to (b, c) or even further down to c, while say
ϕa–1(b, c) = (a, b, c). In general we have:
77
