Page 40 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 40
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
The internal perspective. Given rectangular bands B and Bʹ, choose elements b ∈ B and
bʹ ∈ Bʹ as base points. Each x ∈ B is of the unique form x = lr where l ∈ Lb and r ∈ Rb. Indeed,
x factors as (xb)(bx) where xb ∈ Lb and bx ∈ Rb. Again, this factorization is unique relative to the
base point b. Clearly similar remarks hold for Bʹ and bʹ. Next, let λ: Lb → Lbʹ and ρ: Rb → Rbʹ
be functions, both of which are trivially homomorphisms between their restricted domains.
Finally define f: B → Bʹ by f(x) = λ(xb)ρ(bx), or equivalently, f(x) = λ(xb)bʹρ(bx).
Theorem 1.3.18. As defined, f is a homomorphism from B to Bʹ. Conversely, every
homomorphism from B to Bʹ arises in this manner. Finally, f is 1-1 or onto if and only if both λ
and ρ are. (Of course, the choice of b and bʹ is part of “this manner.”)
Proof. We first show that f as defined is a homomorphism from B to Bʹ.
whilst f(xy) = λ(xyb)bʹρ(bxy) = λ(xb)bʹρ(by) = λ(xb)ρ(by)
f(x)f(y) = λ(xb)bʹρ(bx)λ(yb)bʹρ(by) = λ(xb)ρ(by).
Conversely, if f: B → Bʹ is a homomorphism, then upon picking some b in B and setting bʹ = f(b),
for any x in B we get f(x) = f(xbbbx) = f(xb)bʹf(bx) = λ(xb)bʹρ(bx) where the functions λ and ρ are
the restrictions of f to Lb and Rb respectively. £
A primitive skew lattice is a skew lattice consisting of two rectangular algebras A > B,
where b∧a∧b = b or dually a∨b∨a = a, for all a ∈ A and b ∈ B. We are especially interested in
the case when B = {0}. We denote such an algebra by A0. Suppose we are given two such
algebras, A0 and B0. Then the following assertions follow from the behavior of 0 and the results
above
Proposition 1.3.19. Given primitive skew lattices A0 and B0, Hom(A0, B0) consists of (i)
all constant maps from A0 to B0 and (ii) all maps f from A0 to B0 for which f(0A) = (0B) and the
restriction f|A is a homomorphism from A to B. (Thus the nontrivial part of f is governed by the
conclusions of the preceding results.)
Corollary 1.3.20. Given a primitive skew lattice A0, the nontrivial congruences on A0
(where not all elements are related), are equivalences for which {0} forms a single equivalence
class and their restriction to A is a congruence on the subalgebra A.
38
The internal perspective. Given rectangular bands B and Bʹ, choose elements b ∈ B and
bʹ ∈ Bʹ as base points. Each x ∈ B is of the unique form x = lr where l ∈ Lb and r ∈ Rb. Indeed,
x factors as (xb)(bx) where xb ∈ Lb and bx ∈ Rb. Again, this factorization is unique relative to the
base point b. Clearly similar remarks hold for Bʹ and bʹ. Next, let λ: Lb → Lbʹ and ρ: Rb → Rbʹ
be functions, both of which are trivially homomorphisms between their restricted domains.
Finally define f: B → Bʹ by f(x) = λ(xb)ρ(bx), or equivalently, f(x) = λ(xb)bʹρ(bx).
Theorem 1.3.18. As defined, f is a homomorphism from B to Bʹ. Conversely, every
homomorphism from B to Bʹ arises in this manner. Finally, f is 1-1 or onto if and only if both λ
and ρ are. (Of course, the choice of b and bʹ is part of “this manner.”)
Proof. We first show that f as defined is a homomorphism from B to Bʹ.
whilst f(xy) = λ(xyb)bʹρ(bxy) = λ(xb)bʹρ(by) = λ(xb)ρ(by)
f(x)f(y) = λ(xb)bʹρ(bx)λ(yb)bʹρ(by) = λ(xb)ρ(by).
Conversely, if f: B → Bʹ is a homomorphism, then upon picking some b in B and setting bʹ = f(b),
for any x in B we get f(x) = f(xbbbx) = f(xb)bʹf(bx) = λ(xb)bʹρ(bx) where the functions λ and ρ are
the restrictions of f to Lb and Rb respectively. £
A primitive skew lattice is a skew lattice consisting of two rectangular algebras A > B,
where b∧a∧b = b or dually a∨b∨a = a, for all a ∈ A and b ∈ B. We are especially interested in
the case when B = {0}. We denote such an algebra by A0. Suppose we are given two such
algebras, A0 and B0. Then the following assertions follow from the behavior of 0 and the results
above
Proposition 1.3.19. Given primitive skew lattices A0 and B0, Hom(A0, B0) consists of (i)
all constant maps from A0 to B0 and (ii) all maps f from A0 to B0 for which f(0A) = (0B) and the
restriction f|A is a homomorphism from A to B. (Thus the nontrivial part of f is governed by the
conclusions of the preceding results.)
Corollary 1.3.20. Given a primitive skew lattice A0, the nontrivial congruences on A0
(where not all elements are related), are equivalences for which {0} forms a single equivalence
class and their restriction to A is a congruence on the subalgebra A.
38
