Page 77 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 77
II: Skew Lattices
projection of J × B upon B. (ii) k is a set of monomorphisms kj: B → A called injections such that
the compositions ϕ–1kj are precisely the canonical injections b → (j, b) of B into the various {j} ×
B. Clearly the inverse injections kj –1 jointly decompose the projection K. Any factorization ϕ for
which (i) and (ii) hold is said to be compatible with the projection.
Example 2.5.2. A = ⎡• • • • • •⎤ → B = [• • •] with J= ⎡• ••⎤⎦⎥.
⎢ • • ⎥ ⎢⎣•
⎣⎢• • •
• ⎦⎥
K is the union of the four obvious isomorphisms of each of the four displayed quadrants of A
onto B. The four injections are the four embeddings of B upon each of the four quadrants given
by the inverse isomorphisms. £
Rectangular skew lattices and projective pairs form a category. Indeed, if (K, k): A → B
and (L, l): B → C are projections, then the composite projection is (L, l)(K, k) = (LK, kl): A → C
upon setting kl = {kol⎮k ∈ k, l ∈ l}. This composition is well-defined and associative. Moreover
if ϕ: J × B ≅ A is a compatible factorization for (K, k): A → B and ψ: Jʹ × C ≅ B is a compatible
factorization for (L, l): B → C, then a compatible factorization for (LK, kl): A → C is given by
J × J′ × C ⎯1⎯J×⎯ψ→ J × B ⎯ϕ⎯→ A where k jl j′ (c) = k j (ψ(j′, c)) = ϕ(j, ψ(j′, c)) for c ∈C.
Lemma 2.5.5. Let A > B be two D-classes in a normal skew lattice, let K: A → B be the
unique projection of A onto B defined implicitly by x ≥ K(x) and let k be the let of all coset
bijections from B onto cosets of A. Together the pair (K, k) forms a projective pair. (We call
this pair the natural projective pair form A to B.)
Proof. For any b in B, a factorization is given by ϕ: b∨A∨b × B → A, where ϕ(x, y) = y∨x∨y.
Here b∨A∨b is the image set {b∨a∨b⎪a ∈A} = {a ∈A⎪a ≥ b} of b in A that naturally
parametrizes the cosets of B in A. Clearly ϕ is a bijection. Given the rectangular situation one
need only show that ϕ is a ∨-homomorphism; but (y∨x∨y)∨(yʹ∨xʹ∨yʹ) = (y∨yʹ)∨(x∨xʹ)∨(y∨yʹ) is
given by regularity, since x, xʹ ≻ y, yʹ. £
A rectangular functor on a lattice T is a functor K from (T, ≥) to the category RP of
rectangular skew lattices and projective pairs. K is separable if K(s) ∩ K(t) = ∅ for all s ≠ t ∈ T.
For s ≥ t ∈ T, the projection from K(s) to K(t) is denoted by K(s, t) and its injections (upward
coset bijections) by k(s, t, i) with i parametrizing the various injections.
Lemma 2.5.6. If S is a normal skew lattice with T = S/D, then K: T → RP defined as in
the previous lemma is a rectangular functor.
Proof. This follows from the previous lemma and the fact that any normal skew lattice is strictly
categorical. £
75
projection of J × B upon B. (ii) k is a set of monomorphisms kj: B → A called injections such that
the compositions ϕ–1kj are precisely the canonical injections b → (j, b) of B into the various {j} ×
B. Clearly the inverse injections kj –1 jointly decompose the projection K. Any factorization ϕ for
which (i) and (ii) hold is said to be compatible with the projection.
Example 2.5.2. A = ⎡• • • • • •⎤ → B = [• • •] with J= ⎡• ••⎤⎦⎥.
⎢ • • ⎥ ⎢⎣•
⎣⎢• • •
• ⎦⎥
K is the union of the four obvious isomorphisms of each of the four displayed quadrants of A
onto B. The four injections are the four embeddings of B upon each of the four quadrants given
by the inverse isomorphisms. £
Rectangular skew lattices and projective pairs form a category. Indeed, if (K, k): A → B
and (L, l): B → C are projections, then the composite projection is (L, l)(K, k) = (LK, kl): A → C
upon setting kl = {kol⎮k ∈ k, l ∈ l}. This composition is well-defined and associative. Moreover
if ϕ: J × B ≅ A is a compatible factorization for (K, k): A → B and ψ: Jʹ × C ≅ B is a compatible
factorization for (L, l): B → C, then a compatible factorization for (LK, kl): A → C is given by
J × J′ × C ⎯1⎯J×⎯ψ→ J × B ⎯ϕ⎯→ A where k jl j′ (c) = k j (ψ(j′, c)) = ϕ(j, ψ(j′, c)) for c ∈C.
Lemma 2.5.5. Let A > B be two D-classes in a normal skew lattice, let K: A → B be the
unique projection of A onto B defined implicitly by x ≥ K(x) and let k be the let of all coset
bijections from B onto cosets of A. Together the pair (K, k) forms a projective pair. (We call
this pair the natural projective pair form A to B.)
Proof. For any b in B, a factorization is given by ϕ: b∨A∨b × B → A, where ϕ(x, y) = y∨x∨y.
Here b∨A∨b is the image set {b∨a∨b⎪a ∈A} = {a ∈A⎪a ≥ b} of b in A that naturally
parametrizes the cosets of B in A. Clearly ϕ is a bijection. Given the rectangular situation one
need only show that ϕ is a ∨-homomorphism; but (y∨x∨y)∨(yʹ∨xʹ∨yʹ) = (y∨yʹ)∨(x∨xʹ)∨(y∨yʹ) is
given by regularity, since x, xʹ ≻ y, yʹ. £
A rectangular functor on a lattice T is a functor K from (T, ≥) to the category RP of
rectangular skew lattices and projective pairs. K is separable if K(s) ∩ K(t) = ∅ for all s ≠ t ∈ T.
For s ≥ t ∈ T, the projection from K(s) to K(t) is denoted by K(s, t) and its injections (upward
coset bijections) by k(s, t, i) with i parametrizing the various injections.
Lemma 2.5.6. If S is a normal skew lattice with T = S/D, then K: T → RP defined as in
the previous lemma is a rectangular functor.
Proof. This follows from the previous lemma and the fact that any normal skew lattice is strictly
categorical. £
75
