Page 159 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 159
IV: Skew Boolean Algebras
Proof. If a > 0 in B, then (a, a) D (a, 0) but (a, a) ≠ (a, 0) in ω(B). So let a nontrivial
congruenec θ ⊆ D be given and suppose that (a, aʹ) θ (a, aʺ) with a ≥ aʹ, aʺ in B, but aʹ ≠ aʺ.
Applying ∧ (b, b) we get (a∧b, aʹ∧b) θ (a∧b, aʺ∧b) for all b ∈ B. Since aʹ ≠ aʺ, one of these does
not equal aʹ ∧ aʺ in B, say aʹ. Setting b = aʹ we get (aʹ, aʹ) θ (aʹ, aʺ∧ aʹ) with aʹ > aʺ∧ aʹ. Thus
we may assume that (a, a) θ (a, aʹ) with a > aʹ in B from the outset. Next, setting b = a \ aʹ, we
get that a ≥ b > 0. Then (a∧b, a∧b) θ (a∧b, aʹ∧b), that is, (b, b) θ (b, 0). We show that the
θ-class of (b, b) is in fact its entire D-class. Since b > 0, this forces ω(B)/θ to have a singleton
D-class other than {0}, making its center nontrivial. Applying ∨ (c, c) to (b, b) θ (b, 0) for any
c ≤ b, we get
(b, b) ∨ (c, c) = (b∨c, (b \ c) ∨ c) = (b, b) and (b, 0) ∨ (c, c) = (b∨c, (0 \ c) ∨ c) = (b, c).
Thus (b, b) θ (b, c) for all c ≤ b, so that the θ-class of (b, b) is precisely its D-class in ω(B). £
Minimal skew Boolean covers can be created without the ω-construction. The latter is an
elegant way of achieving this since (1) it gives us minimal covers and (2) it provides a systematic
way of carrying this out.
While ω(B)/θ has a nontrivial center for every nontrivial congruence θ ⊆ D, a second
class of congruences exists for which the quotient algebra always has a trivial center, namely the
class of ∩-congruences β that also preserve intersections in that a β b implies a∩c β b∩c, in
which case S/β also has finite intersections. As we have seen, such congruences arise from ideals
of S. Given an ideal I, an ∩-congruence βI is defined by: a βI b if (a \ (a ∩ b)) ∨ (b \ (a ∩ b)) ∈ I.
Conversely, every ∩-congruence β determines an ideal Iβ given by the congruence class β[0] of 0.
The two assignment processes are reciprocal. (See Lemma 4.4.7 and Theorem 4.4.8.)
Lemma 4.5.13. Given any ∩-congruence β on ω(B), its image ω(B)/β is isomorphic to
an omega algebra and thus has trivial center. Put otherwise ∩-homomorphic images of omega
algebras are copies of omega algebras.
Proof. Let β be derived from ideal I. In turn let I0 ≅ I/D be the ideal of B consisting of all left-
coordinates of elements in I so that I = ω(I0). If β0 is the congruence on B induced from I0, then
the map f: ω(B) → ω(B/β0) defined by f[(a, aʹ)] = (β0[a], β0[aʹ]) is a ∩-homomorphism for which
f–1{(0, 0)} is precisely I. Thus β is the congruence induced by f and ω(B)/β ≅ ω(B/β0). £
We can now decompose the congruence lattice of ω(B) as a subdirect product of (i) the
interval [Δ, D] of all congruences (inclusively) between the trivial congruence Δ and D and (ii)
the lattice Con∩(ω(B)) of all ∩-congruences on ω(B). Indeed Theorem 4.4.9 gives us:
Theorem 4.5.14. The full lattice of congruences Con(ω(B)) is a subdirect product of the
interval [Δ, D] and Con∩(ω(B)) under the embedding Con(ω(B)) → [Δ, D] × Con∩(ω(B)) given
by θ → (θ∩D, βθ[0]). For all θ ∈[Δ, D] except Δ, the image ω(B)/θ has nontrivial center, while
for all β ∈ Con∩(ω(B)) the image ω(B)/β has a trivial center.
157
Proof. If a > 0 in B, then (a, a) D (a, 0) but (a, a) ≠ (a, 0) in ω(B). So let a nontrivial
congruenec θ ⊆ D be given and suppose that (a, aʹ) θ (a, aʺ) with a ≥ aʹ, aʺ in B, but aʹ ≠ aʺ.
Applying ∧ (b, b) we get (a∧b, aʹ∧b) θ (a∧b, aʺ∧b) for all b ∈ B. Since aʹ ≠ aʺ, one of these does
not equal aʹ ∧ aʺ in B, say aʹ. Setting b = aʹ we get (aʹ, aʹ) θ (aʹ, aʺ∧ aʹ) with aʹ > aʺ∧ aʹ. Thus
we may assume that (a, a) θ (a, aʹ) with a > aʹ in B from the outset. Next, setting b = a \ aʹ, we
get that a ≥ b > 0. Then (a∧b, a∧b) θ (a∧b, aʹ∧b), that is, (b, b) θ (b, 0). We show that the
θ-class of (b, b) is in fact its entire D-class. Since b > 0, this forces ω(B)/θ to have a singleton
D-class other than {0}, making its center nontrivial. Applying ∨ (c, c) to (b, b) θ (b, 0) for any
c ≤ b, we get
(b, b) ∨ (c, c) = (b∨c, (b \ c) ∨ c) = (b, b) and (b, 0) ∨ (c, c) = (b∨c, (0 \ c) ∨ c) = (b, c).
Thus (b, b) θ (b, c) for all c ≤ b, so that the θ-class of (b, b) is precisely its D-class in ω(B). £
Minimal skew Boolean covers can be created without the ω-construction. The latter is an
elegant way of achieving this since (1) it gives us minimal covers and (2) it provides a systematic
way of carrying this out.
While ω(B)/θ has a nontrivial center for every nontrivial congruence θ ⊆ D, a second
class of congruences exists for which the quotient algebra always has a trivial center, namely the
class of ∩-congruences β that also preserve intersections in that a β b implies a∩c β b∩c, in
which case S/β also has finite intersections. As we have seen, such congruences arise from ideals
of S. Given an ideal I, an ∩-congruence βI is defined by: a βI b if (a \ (a ∩ b)) ∨ (b \ (a ∩ b)) ∈ I.
Conversely, every ∩-congruence β determines an ideal Iβ given by the congruence class β[0] of 0.
The two assignment processes are reciprocal. (See Lemma 4.4.7 and Theorem 4.4.8.)
Lemma 4.5.13. Given any ∩-congruence β on ω(B), its image ω(B)/β is isomorphic to
an omega algebra and thus has trivial center. Put otherwise ∩-homomorphic images of omega
algebras are copies of omega algebras.
Proof. Let β be derived from ideal I. In turn let I0 ≅ I/D be the ideal of B consisting of all left-
coordinates of elements in I so that I = ω(I0). If β0 is the congruence on B induced from I0, then
the map f: ω(B) → ω(B/β0) defined by f[(a, aʹ)] = (β0[a], β0[aʹ]) is a ∩-homomorphism for which
f–1{(0, 0)} is precisely I. Thus β is the congruence induced by f and ω(B)/β ≅ ω(B/β0). £
We can now decompose the congruence lattice of ω(B) as a subdirect product of (i) the
interval [Δ, D] of all congruences (inclusively) between the trivial congruence Δ and D and (ii)
the lattice Con∩(ω(B)) of all ∩-congruences on ω(B). Indeed Theorem 4.4.9 gives us:
Theorem 4.5.14. The full lattice of congruences Con(ω(B)) is a subdirect product of the
interval [Δ, D] and Con∩(ω(B)) under the embedding Con(ω(B)) → [Δ, D] × Con∩(ω(B)) given
by θ → (θ∩D, βθ[0]). For all θ ∈[Δ, D] except Δ, the image ω(B)/θ has nontrivial center, while
for all β ∈ Con∩(ω(B)) the image ω(B)/β has a trivial center.
157
