Page 39 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 39
I: Preliminaries
rectangular skew lattice, from x ∧ (x ∨ y) = x, it follows that x ∨ y = u ∧ x for some u. Similarly,
from (x ∨ y) ∧ y = y, it follows that x ∨ y = y ∧ v for some v. Combining we get
x ∨ y = (x ∨ y) ∧ (x ∨ y) = y ∧ v ∧ u ∧ x = y ∧ x.
Conversely, given an algebra (S; ∧, ∨) for which both (S; ∧) and (S; ∨) are rectangular bands and
x∨y = y∧x holds on S, it is easily seen that all four relevant absorption identities must hold,
making (S; ∧, ∨) a rectangular skew lattice. We thus have:
Proposition 1.3.14. The variety of rectangular bands is term equivalent to the variety of
rectangular skew lattices. Thus a map between rectangular bands f: A → B is a homomorphism
of bands if and only if it is a homomorphism between their induced rectangular skew lattices.
Likewise an equivalence on a rectangular band is a band congruence if and only if it is a
congruence on the derived skew lattice. £
We continue by considering the case of right zero bands (xy = y), and their derived right
rectangular skew lattices (where x∧y = y = y∨x). The following result is trivial.
Proposition 1.3.15. Given right zero bands B and Bʹ, each function f: B → Bʹ is a
homomorphism, and thus each equivalence relation on B is a congruence. (Similar remarks hold
also for left zero bands where xy = x.) £
We next consider the case of a factored rectangular band B = L × R, where L and R are
left zero and right zero bands, respectively. (All rectangular bands are isomorphic to such a
factorization.)
Theorem 1.3.16. Given a factored rectangular band B = L × R, and a left zero band Lʹ.
Then every homomorphism from B to Lʹ has the form f(l, r) = λ(l) where λ is any function from L
to Lʹ. Dually, every homomorphism from B to a right zero band Rʹ has the form f(l, r) = ρ(r)
where ρ is any function from R to Rʹ. Finally, given a second factored rectangular band
Bʹ = Lʹ × Rʹ, all homomorphisms from B to Bʹ are of the form f(l, r) = (λ(l), ρ(r)) where λ is any
function from L to Lʹ and ρ is any function from R to Rʹ.
Proof. Given a homomorphism from f: L × R → Lʹ, since homomorphisms send R-classes to R-
classes, f is constant on each R-class {l} × R and thus must factor through the left factor L,
leading to a chain of homomorphisms L × R → L → Lʹ whose composite f must be a
homomorphism. Here L × R → L is just the projection onto L. The second map is the induced
map λ: L → Lʹ that must be a homomorphism. £
Corollary 1.3.17. Given a factored rectangular band B = L × R, each congruence θ on B
has the form (l, r) θ (lʹ, rʹ) iff l θL lʹ and r θR rʹ for any pair of equivalences θL and θR on L and R
respectively. £
37
rectangular skew lattice, from x ∧ (x ∨ y) = x, it follows that x ∨ y = u ∧ x for some u. Similarly,
from (x ∨ y) ∧ y = y, it follows that x ∨ y = y ∧ v for some v. Combining we get
x ∨ y = (x ∨ y) ∧ (x ∨ y) = y ∧ v ∧ u ∧ x = y ∧ x.
Conversely, given an algebra (S; ∧, ∨) for which both (S; ∧) and (S; ∨) are rectangular bands and
x∨y = y∧x holds on S, it is easily seen that all four relevant absorption identities must hold,
making (S; ∧, ∨) a rectangular skew lattice. We thus have:
Proposition 1.3.14. The variety of rectangular bands is term equivalent to the variety of
rectangular skew lattices. Thus a map between rectangular bands f: A → B is a homomorphism
of bands if and only if it is a homomorphism between their induced rectangular skew lattices.
Likewise an equivalence on a rectangular band is a band congruence if and only if it is a
congruence on the derived skew lattice. £
We continue by considering the case of right zero bands (xy = y), and their derived right
rectangular skew lattices (where x∧y = y = y∨x). The following result is trivial.
Proposition 1.3.15. Given right zero bands B and Bʹ, each function f: B → Bʹ is a
homomorphism, and thus each equivalence relation on B is a congruence. (Similar remarks hold
also for left zero bands where xy = x.) £
We next consider the case of a factored rectangular band B = L × R, where L and R are
left zero and right zero bands, respectively. (All rectangular bands are isomorphic to such a
factorization.)
Theorem 1.3.16. Given a factored rectangular band B = L × R, and a left zero band Lʹ.
Then every homomorphism from B to Lʹ has the form f(l, r) = λ(l) where λ is any function from L
to Lʹ. Dually, every homomorphism from B to a right zero band Rʹ has the form f(l, r) = ρ(r)
where ρ is any function from R to Rʹ. Finally, given a second factored rectangular band
Bʹ = Lʹ × Rʹ, all homomorphisms from B to Bʹ are of the form f(l, r) = (λ(l), ρ(r)) where λ is any
function from L to Lʹ and ρ is any function from R to Rʹ.
Proof. Given a homomorphism from f: L × R → Lʹ, since homomorphisms send R-classes to R-
classes, f is constant on each R-class {l} × R and thus must factor through the left factor L,
leading to a chain of homomorphisms L × R → L → Lʹ whose composite f must be a
homomorphism. Here L × R → L is just the projection onto L. The second map is the induced
map λ: L → Lʹ that must be a homomorphism. £
Corollary 1.3.17. Given a factored rectangular band B = L × R, each congruence θ on B
has the form (l, r) θ (lʹ, rʹ) iff l θL lʹ and r θR rʹ for any pair of equivalences θL and θR on L and R
respectively. £
37
