Page 112 - 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
To begin, given antilattice A and any pair a, b ∈A, recall that the principal congruence
θ(a,b) is the smallest congruence on A relating a and b. Clearly: θ(a,b) = ∩{θ ∈ Con(A)⎮a θ b}.
In particular, θ(a,b) = Δ precisely when a = b. Clearly:
Lemma 3.6.1. An algebra A of any type is simple if and only iff θ(a,b) = ∇ for all a ≠ b in
its underlying set .
For antilattices, this obvious criterion can be simplified. Consider an antilattice A deter-
mined by a pair of rectangular arrays. Let R0 and C0 represent a row and a column of, say, the ∨-
array of A. (Which array is not important. But R0 and C0 must come from the same array.)
Lemma 3.6.2. (Simplicity Criterion for Antilattices) Given an antilattice A determined
by a pair of rectangular arrays, let R0 and C0 denote respectively a row and a column of the
∨-array. Then A is a simple algebra iff θ(a,b) = ∇ for all a ≠ b in R0 and all a ≠ b in C0. In
particular, any given θ(a,b) must equal ∇ if A is generated from {a, b} using both ∨ and ∧.
Proof. The condition is clearly necessary. To see sufficiency, suppose that the condition holds
for row R0 and column C0 intersecting at element c in the ∨-array. Given a ≠ b in A, both c∨a
and c∨b lie in R0, while a∨c and b∨c lie in C0. Since a ≠ b, either c∨a ≠ c∨b in R0 or a∨c ≠ b∨c
in C0. Say c∨a ≠ c∨b, so that θ(c∨a,c∨b) = ∇. But since c∨a θ(a,b)c∨b, θ(c∨a,c∨b) refines θ(a,b) so
that θ(a,b) = ∇ also. Thus θ(a,b) = ∇ for all a ≠ b ∈ A and A is simple. Since ∨ and ∧ are
idempotent, the subalgebra 〈a, b〉 generated from {a, b} lies in the θ(a,b)-class of a. The final
statement follows.
In the case of a square antilattice determined from a pair of n×n arrays, this theorem says
that the number of principal congruences needing to be checked can be reduced from (n4 – n2)/2
to just n2 – n. Although the check to see that θ(a,b) = ∇ for a ≠ b in either R0×R0 or C0×C0 can be
initially tedious, as the check continues some random recursion enters the process. Thus, if say
θ(a,b) has been shown to equal ∇ and a θ(c, d) b is encountered in the check of θ(c, d), then one can
immediately conclude that θ(c, d) = ∇ also holds.
⎡1 2 3 ⎤ ⎡8 1 6 ⎤
Example 3.6.1. The Lo-Shu antilattice (∨) ⎢⎢⎣74 ⎥ ⎣⎢⎢34 ⎥
5 6 ⎥ (∧) 5 7 ⎥ is simple.
8 9 ⎦ 9 2 ⎦
To begin, take {1, 2}. From the ∧-array, it is clear that 6, 9 ∈〈1, 2〉, the subalgebra generated
from {1, 2}. But {1, 2, 6, 9} clearly generates the ∨-array and thus the algebra. Hence θ(1,2) = ∇.
Similar remarks hold for any other pair a ≠ b in any row or column of either array.
These remarks deserve a more precise analysis. Given distinct elements a and b in a
common row (column) of a 3×3 array, the elements c and d lying in neither the row (column) or
the two columns (rows) of a and b is called the dual pair. The relationship is symmetrical. Thus
{1, 2} and {6, 9} form dual pairs in the ∨-array above, but not in the ∧-array. Any pair of dual
pairs in a 3×3 array generates the entire array under the ambient idempotent operation. Given
two distinct elements in a common row or column of one of the above Lo-Shu arrays, this pair
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To begin, given antilattice A and any pair a, b ∈A, recall that the principal congruence
θ(a,b) is the smallest congruence on A relating a and b. Clearly: θ(a,b) = ∩{θ ∈ Con(A)⎮a θ b}.
In particular, θ(a,b) = Δ precisely when a = b. Clearly:
Lemma 3.6.1. An algebra A of any type is simple if and only iff θ(a,b) = ∇ for all a ≠ b in
its underlying set .
For antilattices, this obvious criterion can be simplified. Consider an antilattice A deter-
mined by a pair of rectangular arrays. Let R0 and C0 represent a row and a column of, say, the ∨-
array of A. (Which array is not important. But R0 and C0 must come from the same array.)
Lemma 3.6.2. (Simplicity Criterion for Antilattices) Given an antilattice A determined
by a pair of rectangular arrays, let R0 and C0 denote respectively a row and a column of the
∨-array. Then A is a simple algebra iff θ(a,b) = ∇ for all a ≠ b in R0 and all a ≠ b in C0. In
particular, any given θ(a,b) must equal ∇ if A is generated from {a, b} using both ∨ and ∧.
Proof. The condition is clearly necessary. To see sufficiency, suppose that the condition holds
for row R0 and column C0 intersecting at element c in the ∨-array. Given a ≠ b in A, both c∨a
and c∨b lie in R0, while a∨c and b∨c lie in C0. Since a ≠ b, either c∨a ≠ c∨b in R0 or a∨c ≠ b∨c
in C0. Say c∨a ≠ c∨b, so that θ(c∨a,c∨b) = ∇. But since c∨a θ(a,b)c∨b, θ(c∨a,c∨b) refines θ(a,b) so
that θ(a,b) = ∇ also. Thus θ(a,b) = ∇ for all a ≠ b ∈ A and A is simple. Since ∨ and ∧ are
idempotent, the subalgebra 〈a, b〉 generated from {a, b} lies in the θ(a,b)-class of a. The final
statement follows.
In the case of a square antilattice determined from a pair of n×n arrays, this theorem says
that the number of principal congruences needing to be checked can be reduced from (n4 – n2)/2
to just n2 – n. Although the check to see that θ(a,b) = ∇ for a ≠ b in either R0×R0 or C0×C0 can be
initially tedious, as the check continues some random recursion enters the process. Thus, if say
θ(a,b) has been shown to equal ∇ and a θ(c, d) b is encountered in the check of θ(c, d), then one can
immediately conclude that θ(c, d) = ∇ also holds.
⎡1 2 3 ⎤ ⎡8 1 6 ⎤
Example 3.6.1. The Lo-Shu antilattice (∨) ⎢⎢⎣74 ⎥ ⎣⎢⎢34 ⎥
5 6 ⎥ (∧) 5 7 ⎥ is simple.
8 9 ⎦ 9 2 ⎦
To begin, take {1, 2}. From the ∧-array, it is clear that 6, 9 ∈〈1, 2〉, the subalgebra generated
from {1, 2}. But {1, 2, 6, 9} clearly generates the ∨-array and thus the algebra. Hence θ(1,2) = ∇.
Similar remarks hold for any other pair a ≠ b in any row or column of either array.
These remarks deserve a more precise analysis. Given distinct elements a and b in a
common row (column) of a 3×3 array, the elements c and d lying in neither the row (column) or
the two columns (rows) of a and b is called the dual pair. The relationship is symmetrical. Thus
{1, 2} and {6, 9} form dual pairs in the ∨-array above, but not in the ∧-array. Any pair of dual
pairs in a 3×3 array generates the entire array under the ambient idempotent operation. Given
two distinct elements in a common row or column of one of the above Lo-Shu arrays, this pair
110
