Page 30 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
consider the cell-maps ϕ: A → Ab0A ⊆ B and ψ: B → Bc0B ⊆ C defined by ϕ(a) = ab0a and
ψ(b) = bc0b for b0 ∈ B and c0 ∈ C. Then for all a ∈ A, ψoϕ(a) = ab0ac0ab0a = = ab0c0b0a by
Corollary 1.2.8. Thus ψoϕ: A → C is a cell-map form A to the cell of b0c0b0 in C. Finally, given
c ∈ C let ζ: A → C be cell-map defined by ζ(a) = aca. By CCC again, b0 ∈ B exists such that
b0 ≥ c. By Lemma 1.2.8 again, for all a ∈ A, aca = ab0cb0a = ab0acab0a. Thus, if
ϕ: A → Ab0A ⊆ B and ψ: B → BcB ⊆ C are the cell-maps ϕ(a) = ab0a and ψ(b) = bcb, then ψoϕ
is precisely ζ: A → C. £
The above need not hold in all regular bands, as is seen in LReg{a, b, c}. The cell-map
from {a} to the bottom D-class in LReg{a, b, c} sending a to abc does not factor through the
intermediate class {ac, ca}. Likewise the cell-map sending a to acb does not factor though the
class {ab, ba}.
Proposition. 1.2.18. The class of all bands satisfying the class covering condition are
closed under products and homomorphic images.
Proof. Closure under products is clear. Suppose that band S satisfies the CCC and that f: S → T
is a homomorphism. Let a ≻ b in f[S] with a = f(x) and b = f(y) for x, y in S. Then x ≻ yxy in S
with f(yxy) = bab = b in T. By CCC, xʹ ∈ Dx exists such that xʹ ≥ yxy. Clearly aʹ = f(xʹ) ≥ b and
aʹ ∈ Da. £
Thus bands satisfying the class covering condition do not form a variety. This condition
holds, however, for both band reducts (S, ∧) and (S, ∨) of any skew lattice (S, ∨, ∧). And skew
lattices do form a variety.
1.3 Noncommutative lattices – initial observations
In general, a noncommutative lattice is an algebra (N: ∨, ∧) where both ∨ and ∧ are
associative, idempotent binary operations satisfying a specified set of absorption identities. We
continue to call ∨ the join and ∧ the meet. The adjective “noncommutative” is used here in the
inclusive sense of “not-necessarily-commutative”. Thus lattices will play an important role in the
general study of noncommutative lattices. Thus far, nearly all types of noncommutative lattices
that have been studied assume absorption identities from among the following:
B1: a ∧ (a ∨ b) = a. C1: a ∨ (a ∧ b) = a.
B2: (b ∨ a) ∧ a = a. C2: (b ∧ a) ∨ a = a.
B3: a ∧ (b ∨ a) = a. C3: a ∨ (b ∧ a) = a.
B4: (a ∨ b) ∧ a = a. C4: (a ∧ b) ∨ a = a.
If ∨ and ∧ are commutative, then clearly the B’s merge together as do the C’s. In general we
require that ∨ and ∧ satisfy at least a pair of identities that reduce to B1 and C1 when ∨ and ∧ are
commutative. We consider several classes of such algebras.
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consider the cell-maps ϕ: A → Ab0A ⊆ B and ψ: B → Bc0B ⊆ C defined by ϕ(a) = ab0a and
ψ(b) = bc0b for b0 ∈ B and c0 ∈ C. Then for all a ∈ A, ψoϕ(a) = ab0ac0ab0a = = ab0c0b0a by
Corollary 1.2.8. Thus ψoϕ: A → C is a cell-map form A to the cell of b0c0b0 in C. Finally, given
c ∈ C let ζ: A → C be cell-map defined by ζ(a) = aca. By CCC again, b0 ∈ B exists such that
b0 ≥ c. By Lemma 1.2.8 again, for all a ∈ A, aca = ab0cb0a = ab0acab0a. Thus, if
ϕ: A → Ab0A ⊆ B and ψ: B → BcB ⊆ C are the cell-maps ϕ(a) = ab0a and ψ(b) = bcb, then ψoϕ
is precisely ζ: A → C. £
The above need not hold in all regular bands, as is seen in LReg{a, b, c}. The cell-map
from {a} to the bottom D-class in LReg{a, b, c} sending a to abc does not factor through the
intermediate class {ac, ca}. Likewise the cell-map sending a to acb does not factor though the
class {ab, ba}.
Proposition. 1.2.18. The class of all bands satisfying the class covering condition are
closed under products and homomorphic images.
Proof. Closure under products is clear. Suppose that band S satisfies the CCC and that f: S → T
is a homomorphism. Let a ≻ b in f[S] with a = f(x) and b = f(y) for x, y in S. Then x ≻ yxy in S
with f(yxy) = bab = b in T. By CCC, xʹ ∈ Dx exists such that xʹ ≥ yxy. Clearly aʹ = f(xʹ) ≥ b and
aʹ ∈ Da. £
Thus bands satisfying the class covering condition do not form a variety. This condition
holds, however, for both band reducts (S, ∧) and (S, ∨) of any skew lattice (S, ∨, ∧). And skew
lattices do form a variety.
1.3 Noncommutative lattices – initial observations
In general, a noncommutative lattice is an algebra (N: ∨, ∧) where both ∨ and ∧ are
associative, idempotent binary operations satisfying a specified set of absorption identities. We
continue to call ∨ the join and ∧ the meet. The adjective “noncommutative” is used here in the
inclusive sense of “not-necessarily-commutative”. Thus lattices will play an important role in the
general study of noncommutative lattices. Thus far, nearly all types of noncommutative lattices
that have been studied assume absorption identities from among the following:
B1: a ∧ (a ∨ b) = a. C1: a ∨ (a ∧ b) = a.
B2: (b ∨ a) ∧ a = a. C2: (b ∧ a) ∨ a = a.
B3: a ∧ (b ∨ a) = a. C3: a ∨ (b ∧ a) = a.
B4: (a ∨ b) ∧ a = a. C4: (a ∧ b) ∨ a = a.
If ∨ and ∧ are commutative, then clearly the B’s merge together as do the C’s. In general we
require that ∨ and ∧ satisfy at least a pair of identities that reduce to B1 and C1 when ∨ and ∧ are
commutative. We consider several classes of such algebras.
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