Page 117 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 117
III: Quasilattices, Paralattices and their Congruences
two bottom cells would still be adjacent in the cartesian partition of the magic square, thus
inducing a congruence of index two.
Similarly for the remaining indices of 8, 9 and 12, congruences with these indices must
be refined by a properly larger congruence of index 2, thus returning us to the |µ| = 2 case.
Examples 3.6.3. Pandiagonal magic squares first appear in the 4×4 case. In considering
pandiagonal magic squares in general, two such squares of the same dimension are equivalent if
either is obtained from the other using a combination of dihedral operations, along with cyclic
permutations of the rows and/or the columns. Any given pandiagonal square storing 1 through 16
is thus one of 8 × 4 × 4 = 128 equivalent pandiagonal magic squares. In the 4×4 case, 48
dihedrally distinct pandiagonal squares exist, all being equivalent to exactly one of the following
three pandiagonal magic squares:
⎡1 8 10 15⎤ ⎢⎡1 8 11 14 ⎤ ⎡ 1 14 4 15⎤
⎥
⎢⎢12 13 ⎥ ⎢12 13 2 7 ⎥ ⎢ 10⎥⎥ .
3 6 ⎥ ⎢⎥ ⎢ 8 11 5
⎢7 2 16 9 ⎥ ⎢6 3 16 9 ⎥ ⎢13 2 16 3 ⎥
⎢⎣14 11 5 4 ⎥⎦ ⎢⎣12 7 9 6 ⎦⎥
⎢⎥
⎣15 10 5 4⎦
The first two pandiagonal magic arrays induce simple antilattices; but not the third array.
Two classic constructions
In 1693, Simon de la Loubère gave the following rule for constructing magic squares for
any odd order n. Again we will work with base n ordered pairs, often in abbreviated notation.
De la Loubère’s Rule. Place 00 in the middle of the first row. In ascending (broken)
diagonal fashion place in order the remaining 01 through 0, n–1. Beneath 0, n–1 place 10 and
again in ascending diagonal fashion place 11 through 1, n–1. Beneath 1, n–1 place 20, and
repeat the process until an entire n×n array is filled. The resulting array is a magic square of
odd order n storing 0 through n2–1 in base n.
The array to the right is the n = 5 case in base 5 notation storing 0 – 24.
⎡00 01 02 03 04 ⎤ ⎡31 43 00 12 24⎤
⎢10 11 12 13 14 ⎥ ⎢42 04 11 23 30 ⎥
(∨) ⎢⎢20 ⎥ (∧) ⎢⎢03 10 22 34 41⎥⎥
21 22 23 24 ⎥ ⎢14 21 33 40 02⎥
⎢⎣20 32 44 01 13 ⎥⎦
⎢30 31 32 33 34 ⎥
⎢⎣40 41 42 43 44 ⎥⎦
Theorem 3.6.10. The magic squares of odd order n given by De la Loubère’s rule along
with their corresponding standard array yield simple antilattices precisely when n is prime.
115
two bottom cells would still be adjacent in the cartesian partition of the magic square, thus
inducing a congruence of index two.
Similarly for the remaining indices of 8, 9 and 12, congruences with these indices must
be refined by a properly larger congruence of index 2, thus returning us to the |µ| = 2 case.
Examples 3.6.3. Pandiagonal magic squares first appear in the 4×4 case. In considering
pandiagonal magic squares in general, two such squares of the same dimension are equivalent if
either is obtained from the other using a combination of dihedral operations, along with cyclic
permutations of the rows and/or the columns. Any given pandiagonal square storing 1 through 16
is thus one of 8 × 4 × 4 = 128 equivalent pandiagonal magic squares. In the 4×4 case, 48
dihedrally distinct pandiagonal squares exist, all being equivalent to exactly one of the following
three pandiagonal magic squares:
⎡1 8 10 15⎤ ⎢⎡1 8 11 14 ⎤ ⎡ 1 14 4 15⎤
⎥
⎢⎢12 13 ⎥ ⎢12 13 2 7 ⎥ ⎢ 10⎥⎥ .
3 6 ⎥ ⎢⎥ ⎢ 8 11 5
⎢7 2 16 9 ⎥ ⎢6 3 16 9 ⎥ ⎢13 2 16 3 ⎥
⎢⎣14 11 5 4 ⎥⎦ ⎢⎣12 7 9 6 ⎦⎥
⎢⎥
⎣15 10 5 4⎦
The first two pandiagonal magic arrays induce simple antilattices; but not the third array.
Two classic constructions
In 1693, Simon de la Loubère gave the following rule for constructing magic squares for
any odd order n. Again we will work with base n ordered pairs, often in abbreviated notation.
De la Loubère’s Rule. Place 00 in the middle of the first row. In ascending (broken)
diagonal fashion place in order the remaining 01 through 0, n–1. Beneath 0, n–1 place 10 and
again in ascending diagonal fashion place 11 through 1, n–1. Beneath 1, n–1 place 20, and
repeat the process until an entire n×n array is filled. The resulting array is a magic square of
odd order n storing 0 through n2–1 in base n.
The array to the right is the n = 5 case in base 5 notation storing 0 – 24.
⎡00 01 02 03 04 ⎤ ⎡31 43 00 12 24⎤
⎢10 11 12 13 14 ⎥ ⎢42 04 11 23 30 ⎥
(∨) ⎢⎢20 ⎥ (∧) ⎢⎢03 10 22 34 41⎥⎥
21 22 23 24 ⎥ ⎢14 21 33 40 02⎥
⎢⎣20 32 44 01 13 ⎥⎦
⎢30 31 32 33 34 ⎥
⎢⎣40 41 42 43 44 ⎥⎦
Theorem 3.6.10. The magic squares of odd order n given by De la Loubère’s rule along
with their corresponding standard array yield simple antilattices precisely when n is prime.
115
