Page 97 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 97
III: Quasilattices, Paralattices and their Congruences
Proposition 3.2.2. No 4-element antiilattice is simple.
Proof. Any such antilattice A falls into one of three cases. First case: A is flat. In this case
Con(A) equals Equ(A) and has order greater than 2. Second case: either (A, ∨) or (A, ∧) is flat,
but not both. Then both L and R for the nontrivial operation are congruences that are distinct
from Δ and ∇. Third case: neither (A, ∨) nor (A, ∧) is flat. Then the elementary combinatorics of
2×2 squares forces one of L(∨) or R(∨) to equal one of L(∧) or R(∧), with the equated equivalence
being a nontrivial congruence. £
Thus simple antilattices of finite order > 2 must have composite order greater than 5.
We may now state:
Theorem 3.2.3. Simple antilattices exist for all composite orders > 5. In all such cases, none
of R(∨), L(∨), R(∧) or L(∧) are congruences.
Proof. Given a set A of order mn ≥ 6 with n ≥ m ≥ 2, store its elements in each of the following
m×n arrays, so that a∨b is in the row of a and the column of b of the array on the left and a∧b is
similarly obtained from the array on the right. (First array indices are used in the second array to
show how the elements have been rearranged.)
a11 a12 a13 ... a1n
a21 a22 a23 ... a2n
(∨) a31 a32 a33 ... a3n
am1 am2 am3 ... amn
a11 a12 a1,n−m+1 a1,n−m+2 a1, n−2 a1,n−1 a21
a22 a23 a2,n−m+2 a2,n−m+3 a2, n−1 a31 a32
(∧) a33 a34 a3,n−m+3 a3,n−m+4 a41 a42 a43
am, m am, m+1 am, n am−1, n a3, n a2, n a1, n
Let θ be a congruence on A and let distinct a, b in A exist such that aθb. We show that
θ = ∇. To begin, let a11θa21 be given. Then (a11∨a1i)θ(a21∨a1i) for i = 1, 2, …, n yields a
sequence of congruent pairs a1iθa2i in the first two rows of the ∨-array. Upon taking ∧-products
of these θ-related pairs, all elements in the first two rows of the ∧-array are seen to be θ-related.
In particular, a21θa31. This leads first to a sequence of relations a2iθa3i in the ∨-array and then to
all elements in the first three rows of the ∧-array being θ-related. The process continues until the
entire ∧-array, and hence A, is absorbed into a single θ-class, showing that θ = ∇.
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Proposition 3.2.2. No 4-element antiilattice is simple.
Proof. Any such antilattice A falls into one of three cases. First case: A is flat. In this case
Con(A) equals Equ(A) and has order greater than 2. Second case: either (A, ∨) or (A, ∧) is flat,
but not both. Then both L and R for the nontrivial operation are congruences that are distinct
from Δ and ∇. Third case: neither (A, ∨) nor (A, ∧) is flat. Then the elementary combinatorics of
2×2 squares forces one of L(∨) or R(∨) to equal one of L(∧) or R(∧), with the equated equivalence
being a nontrivial congruence. £
Thus simple antilattices of finite order > 2 must have composite order greater than 5.
We may now state:
Theorem 3.2.3. Simple antilattices exist for all composite orders > 5. In all such cases, none
of R(∨), L(∨), R(∧) or L(∧) are congruences.
Proof. Given a set A of order mn ≥ 6 with n ≥ m ≥ 2, store its elements in each of the following
m×n arrays, so that a∨b is in the row of a and the column of b of the array on the left and a∧b is
similarly obtained from the array on the right. (First array indices are used in the second array to
show how the elements have been rearranged.)
a11 a12 a13 ... a1n
a21 a22 a23 ... a2n
(∨) a31 a32 a33 ... a3n
am1 am2 am3 ... amn
a11 a12 a1,n−m+1 a1,n−m+2 a1, n−2 a1,n−1 a21
a22 a23 a2,n−m+2 a2,n−m+3 a2, n−1 a31 a32
(∧) a33 a34 a3,n−m+3 a3,n−m+4 a41 a42 a43
am, m am, m+1 am, n am−1, n a3, n a2, n a1, n
Let θ be a congruence on A and let distinct a, b in A exist such that aθb. We show that
θ = ∇. To begin, let a11θa21 be given. Then (a11∨a1i)θ(a21∨a1i) for i = 1, 2, …, n yields a
sequence of congruent pairs a1iθa2i in the first two rows of the ∨-array. Upon taking ∧-products
of these θ-related pairs, all elements in the first two rows of the ∧-array are seen to be θ-related.
In particular, a21θa31. This leads first to a sequence of relations a2iθa3i in the ∨-array and then to
all elements in the first three rows of the ∧-array being θ-related. The process continues until the
entire ∧-array, and hence A, is absorbed into a single θ-class, showing that θ = ∇.
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