Page 130 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 130
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Proposition 4.2.3. Let θ be a congruence on a skew Boolean algebra A viewed as an
orthosum ∑1r Di0 of primitive subalgebras Di0 , the Di being the atomic D-classes. Then:
1) If d θ 0 for d ∈Di, then Di ⊆ [0]θ, the congruence class of 0.
2) If d1 θ d2 with d1 ∈Di, d2 ∈ Dj, but Di ≠ Dj, then Di ∪ Dj ⊆ [0]θ.
3) Thus if some Di ⊆ [0]θ, then upon re-indexing, D1∪ … ∪Dk ⊆ [0]θ, with the remaining
Di
refined by θ-classes and
( )∑ ∑rDi0θ≅ r Di0 θi
1 k+1
where θ i = θ | Di0 × Di0 and Di0 θi is primitive for each i ≥ k + 1.
∑Thus, given S = r Di0 and a homomorphism of skew Boolean algebras f : S → T:
1
4) f[S] is an orthosum with summands f ⎡⎣Di0 ⎦⎤ , each of which is either primitive or else
just {0T}. In the former case, f[ Di ] is at least atomic in f[S].
5) f ⎡⎣ Di0 ⎤⎦ ∩ f ⎡ D0j ⎤ ≠ {0T} implies i = j so that Di0 = D0j .
⎣ ⎦
Finally, in the purely primitive case:
6) Given left-[right]handed primitive algebras D10 and D02 , a non-0 homomorphism
from D10 to D 0 is any map sending 0 to 0, and elements in D1 to elements in D2.
2
7) In general, all non-0 homomorphisms f : D10 → D20 are obtained as follows:
(a) Send 0 to 0.
(b) Pick a ∈D1 and b ∈D2 and any map λ: La → Lb and any map ρ: Ra → Rb.
(c) Finally set f(0) = 0 and for all x ∈La and y ∈Ra, set f(x∧y) = λ(x) ∧ ρ(y).
Proof. (1) should be clear. (2) For d1 = d1∧ d1 is θ-related to d1∧ d2 = 0 and likewise d2 θ 0.
The conclusion now follows from (1). (3) – (5) should also be clear. (6) and (7) recall some of
our remarks in the final part of Section 1.3. £
As a consequence, we have the following particularly crisp result:
128
Proposition 4.2.3. Let θ be a congruence on a skew Boolean algebra A viewed as an
orthosum ∑1r Di0 of primitive subalgebras Di0 , the Di being the atomic D-classes. Then:
1) If d θ 0 for d ∈Di, then Di ⊆ [0]θ, the congruence class of 0.
2) If d1 θ d2 with d1 ∈Di, d2 ∈ Dj, but Di ≠ Dj, then Di ∪ Dj ⊆ [0]θ.
3) Thus if some Di ⊆ [0]θ, then upon re-indexing, D1∪ … ∪Dk ⊆ [0]θ, with the remaining
Di
refined by θ-classes and
( )∑ ∑rDi0θ≅ r Di0 θi
1 k+1
where θ i = θ | Di0 × Di0 and Di0 θi is primitive for each i ≥ k + 1.
∑Thus, given S = r Di0 and a homomorphism of skew Boolean algebras f : S → T:
1
4) f[S] is an orthosum with summands f ⎡⎣Di0 ⎦⎤ , each of which is either primitive or else
just {0T}. In the former case, f[ Di ] is at least atomic in f[S].
5) f ⎡⎣ Di0 ⎤⎦ ∩ f ⎡ D0j ⎤ ≠ {0T} implies i = j so that Di0 = D0j .
⎣ ⎦
Finally, in the purely primitive case:
6) Given left-[right]handed primitive algebras D10 and D02 , a non-0 homomorphism
from D10 to D 0 is any map sending 0 to 0, and elements in D1 to elements in D2.
2
7) In general, all non-0 homomorphisms f : D10 → D20 are obtained as follows:
(a) Send 0 to 0.
(b) Pick a ∈D1 and b ∈D2 and any map λ: La → Lb and any map ρ: Ra → Rb.
(c) Finally set f(0) = 0 and for all x ∈La and y ∈Ra, set f(x∧y) = λ(x) ∧ ρ(y).
Proof. (1) should be clear. (2) For d1 = d1∧ d1 is θ-related to d1∧ d2 = 0 and likewise d2 θ 0.
The conclusion now follows from (1). (3) – (5) should also be clear. (6) and (7) recall some of
our remarks in the final part of Section 1.3. £
As a consequence, we have the following particularly crisp result:
128
