Page 114 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 114
athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Lemma 3.6.4. Given 3×3 arrays, A and Aʹ, each storing the same 9 distinct elements,
the following assertions are equivalent:
1. A and Aʹ form a complimentary pair of 3×3 arrays.
2. If two distinct elements are either row-related or column-related in either array, they
are unrelated in either sense in the other array.
3. The rows [columns] in A either all become (extended) diagonals in Aʹ or all become
extended counterdiagonals in Aʹ; similar remarks hold in passing from Aʹ to A.
Proof. Clearly (1) implies (2). For the converse, observe that the status of (2) is unchanged if
either array undergoes row or columns interchanged! Thus, we assume (2) in the case where
⎡a b c ⎤
⎢⎣⎢dg ⎥
elements a and b lie in a common row of A, as in e f ⎥ . Assertion (2) implies that Aʹ is of
h i ⎦
⎡a f x⎤ ⎡a i x⎤ ⎡a f h ⎤ ⎡a i e ⎤
form ⎢i b y ⎥ or ⎢ f b y ⎥ . ⎢i d ⎥ or ⎢ f b g⎥ its
⎣⎢u v w⎥⎦ ⎣⎢u ⎥ Applying (2) further, Aʹ must be either ⎢⎣e b ⎥ ⎢⎣h ⎥
⎦ g ⎦ ⎦
v w c d c
trnaspose. In either case we have a complementary pair of arrays. Similarly, assuming a and b
lie in the same column of A, (2) forces Aʹ to be a complementary array. Likewise, if a and b are
row-[column-] related in Aʹ, then (2) forces A to be a complementary to Aʹ. Thus, (1) and (2) are
equivalent. Clearly (3) implies (1) and (2). Given the latter, every row/column in either array
must be an (extended) [counter]diagonal in the other array. This can only happen if (3) holds.
We are ready to state our main result about antilattices induced from 3×3 magic squares.
⎡a b c ⎤
Theorem 3.6.5. Given a 3×3 array A = ⎢⎢d ⎥
e f ⎥ consisting of distinct positive
⎢⎣g h i ⎥⎦
integers in their natural (increasing) order and a second 3 × 3 magic square Aʹ storing the same
⎡b i c⎤
⎢ ⎥
integers, then either A and Aʹ are complementary or Aʹ is a dihedral variation of ⎢ f e d ⎥ .
⎢⎣g a h ⎦⎥
⎡1 3 4 ⎤ ⎡3 11 4⎤
⎢5 6 7⎥ ⎢⎣⎢87 ⎥
(An instance of the latter is the pair A = ⎣⎢8 ⎥ and Aʹ = 6 5 ⎥ .)
⎦ 1 9 ⎦
9 11
Proof. Using a dihedral replacement of Aʹ if need be, distinct β > γ > 0 and α > β + γ exist such
that:
⎡1 1 1⎤ ⎡−1 1 0 ⎤ ⎡ 0 1 − 1⎤ ⎡ α − β α + β + γ α − γ ⎤
Aʹ = α ⎢⎣⎢11 11⎦⎥⎥ ⎢ 11⎥⎥⎦ ⎢ 01⎥⎦⎥ ⎢⎢⎣αα++βγ− γ ⎥
1 + β ⎢ 1 0− + γ ⎢ −1 0 = α α − β +γ ⎥ .
1 ⎣ 0 −1 ⎣ 1− 1 α −β−γ α+β ⎦
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Lemma 3.6.4. Given 3×3 arrays, A and Aʹ, each storing the same 9 distinct elements,
the following assertions are equivalent:
1. A and Aʹ form a complimentary pair of 3×3 arrays.
2. If two distinct elements are either row-related or column-related in either array, they
are unrelated in either sense in the other array.
3. The rows [columns] in A either all become (extended) diagonals in Aʹ or all become
extended counterdiagonals in Aʹ; similar remarks hold in passing from Aʹ to A.
Proof. Clearly (1) implies (2). For the converse, observe that the status of (2) is unchanged if
either array undergoes row or columns interchanged! Thus, we assume (2) in the case where
⎡a b c ⎤
⎢⎣⎢dg ⎥
elements a and b lie in a common row of A, as in e f ⎥ . Assertion (2) implies that Aʹ is of
h i ⎦
⎡a f x⎤ ⎡a i x⎤ ⎡a f h ⎤ ⎡a i e ⎤
form ⎢i b y ⎥ or ⎢ f b y ⎥ . ⎢i d ⎥ or ⎢ f b g⎥ its
⎣⎢u v w⎥⎦ ⎣⎢u ⎥ Applying (2) further, Aʹ must be either ⎢⎣e b ⎥ ⎢⎣h ⎥
⎦ g ⎦ ⎦
v w c d c
trnaspose. In either case we have a complementary pair of arrays. Similarly, assuming a and b
lie in the same column of A, (2) forces Aʹ to be a complementary array. Likewise, if a and b are
row-[column-] related in Aʹ, then (2) forces A to be a complementary to Aʹ. Thus, (1) and (2) are
equivalent. Clearly (3) implies (1) and (2). Given the latter, every row/column in either array
must be an (extended) [counter]diagonal in the other array. This can only happen if (3) holds.
We are ready to state our main result about antilattices induced from 3×3 magic squares.
⎡a b c ⎤
Theorem 3.6.5. Given a 3×3 array A = ⎢⎢d ⎥
e f ⎥ consisting of distinct positive
⎢⎣g h i ⎥⎦
integers in their natural (increasing) order and a second 3 × 3 magic square Aʹ storing the same
⎡b i c⎤
⎢ ⎥
integers, then either A and Aʹ are complementary or Aʹ is a dihedral variation of ⎢ f e d ⎥ .
⎢⎣g a h ⎦⎥
⎡1 3 4 ⎤ ⎡3 11 4⎤
⎢5 6 7⎥ ⎢⎣⎢87 ⎥
(An instance of the latter is the pair A = ⎣⎢8 ⎥ and Aʹ = 6 5 ⎥ .)
⎦ 1 9 ⎦
9 11
Proof. Using a dihedral replacement of Aʹ if need be, distinct β > γ > 0 and α > β + γ exist such
that:
⎡1 1 1⎤ ⎡−1 1 0 ⎤ ⎡ 0 1 − 1⎤ ⎡ α − β α + β + γ α − γ ⎤
Aʹ = α ⎢⎣⎢11 11⎦⎥⎥ ⎢ 11⎥⎥⎦ ⎢ 01⎥⎦⎥ ⎢⎢⎣αα++βγ− γ ⎥
1 + β ⎢ 1 0− + γ ⎢ −1 0 = α α − β +γ ⎥ .
1 ⎣ 0 −1 ⎣ 1− 1 α −β−γ α+β ⎦
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