Page 116 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
where πR and πC are outer/inner partitions splitting rows[columns] 1 & 4 against rows[columns]
2 & 3 in the standard array.
Examples 3.6.2. Consider the following magic squares in the Ollerenshaw-Bondi listing:
⎡1 7 12 14⎤ ⎡1 4 15 14 ⎤ ⎡1 16 6 11⎤
⎢10 5⎥ ⎢13 2⎥ ⎢13 7⎥
(1) ⎢15 16 3 4⎥ (9) ⎢12 16 3 7⎥ (25) ⎢12 4 10 2⎥ .
9 6 9 6 5 15
⎣⎢8 2 13 11⎥⎦ ⎢⎣8 5 10 11⎥⎦ ⎢⎣8 9 3 14 ⎥⎦
Square (1) induces a simple antilattice because 1 - 4 lie in distinct rows and columns
(denying πR) and 1, 5, 9, 13 lie in distinct rows and columns (denying πC). By contrast both πR
and πC work for (9), while πC works, but not πR, for square (25). Thus both (9) and (25) are
nonsimple. (Caveat. In the Ollerenshaw-Bondi list, the arrays actually store 0 – 15, instead of 1
– 16, and do so in base 4 notation.)
A survey of the 220 leading squares in the Ollerenshaw-Bondi list yields, upon applying
the test of Theorem 3.6.8, the following statistic:
Corollary 3.6.9. Of the 880 magic squares storing 1 – 16, 416 cases yield simple
antilattices and 464 yield nonsimple antilattices, giving a breakdown of 47.27% to 52.73%.
Proof of Theorem 3.6.8. (All arrays in this proof are identified to within row and column
permutations.) To begin, all possible cartesian partitions of a 4×4 square with distinct elements
can only have indices among the following: 1, 2, 3, 4. 6, 8, 9, 12, 16. Thus if A is nonsimple, the
index |µ| of its maximal proper congruence µ must lie among 2, 3, 4, 6, 8, 9, 12.
If |µ| = 2, then any cartesian partition of the standard array is one of four cases: one row
and three rows, or one column and three columns, or two rows and two rows, or two columns and
two columns. The first two cases are impossible when A is included, as no row or column in the
standard array has the magic sum of 34. In the final cases, the sum of each pair of rows or
columns must be 2 × 34 = 68. This occurs only for {row 1 ∪ row 4⎮row 2 ∪ row 3} or
{column 1 ∪ column 4⎮column 2 ∪ column 3}, just as stated.
|µ| = 3 is impossible in the antilattice context since that would mean a row or column in
the standard array would sum to 34 (because it would appear as a row or column in A), which is
impossible.
|µ| = 4 is possible. But in this case, the quotient algebra A/µ would have order 4 and thus
be nonsimple by Proposition 3.2.2. Hence µ was not really maximal after all.
|µ| = 6 is also possible with the cartesian partition of the standard array having either
⎡[1 × 2] [1 × 2]⎤ ⎡[1 × 1] [1 × 3]⎤
⎢⎢⎢⎣[[12 22]]⎥⎥⎦⎥ ⎢⎢⎣⎢[[12 33]]⎦⎥⎥⎥
template × 2] [1 × or template × 1] [1 × or a transpose. In all these cases, the
× 2] [2 × × 1] [2 ×
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where πR and πC are outer/inner partitions splitting rows[columns] 1 & 4 against rows[columns]
2 & 3 in the standard array.
Examples 3.6.2. Consider the following magic squares in the Ollerenshaw-Bondi listing:
⎡1 7 12 14⎤ ⎡1 4 15 14 ⎤ ⎡1 16 6 11⎤
⎢10 5⎥ ⎢13 2⎥ ⎢13 7⎥
(1) ⎢15 16 3 4⎥ (9) ⎢12 16 3 7⎥ (25) ⎢12 4 10 2⎥ .
9 6 9 6 5 15
⎣⎢8 2 13 11⎥⎦ ⎢⎣8 5 10 11⎥⎦ ⎢⎣8 9 3 14 ⎥⎦
Square (1) induces a simple antilattice because 1 - 4 lie in distinct rows and columns
(denying πR) and 1, 5, 9, 13 lie in distinct rows and columns (denying πC). By contrast both πR
and πC work for (9), while πC works, but not πR, for square (25). Thus both (9) and (25) are
nonsimple. (Caveat. In the Ollerenshaw-Bondi list, the arrays actually store 0 – 15, instead of 1
– 16, and do so in base 4 notation.)
A survey of the 220 leading squares in the Ollerenshaw-Bondi list yields, upon applying
the test of Theorem 3.6.8, the following statistic:
Corollary 3.6.9. Of the 880 magic squares storing 1 – 16, 416 cases yield simple
antilattices and 464 yield nonsimple antilattices, giving a breakdown of 47.27% to 52.73%.
Proof of Theorem 3.6.8. (All arrays in this proof are identified to within row and column
permutations.) To begin, all possible cartesian partitions of a 4×4 square with distinct elements
can only have indices among the following: 1, 2, 3, 4. 6, 8, 9, 12, 16. Thus if A is nonsimple, the
index |µ| of its maximal proper congruence µ must lie among 2, 3, 4, 6, 8, 9, 12.
If |µ| = 2, then any cartesian partition of the standard array is one of four cases: one row
and three rows, or one column and three columns, or two rows and two rows, or two columns and
two columns. The first two cases are impossible when A is included, as no row or column in the
standard array has the magic sum of 34. In the final cases, the sum of each pair of rows or
columns must be 2 × 34 = 68. This occurs only for {row 1 ∪ row 4⎮row 2 ∪ row 3} or
{column 1 ∪ column 4⎮column 2 ∪ column 3}, just as stated.
|µ| = 3 is impossible in the antilattice context since that would mean a row or column in
the standard array would sum to 34 (because it would appear as a row or column in A), which is
impossible.
|µ| = 4 is possible. But in this case, the quotient algebra A/µ would have order 4 and thus
be nonsimple by Proposition 3.2.2. Hence µ was not really maximal after all.
|µ| = 6 is also possible with the cartesian partition of the standard array having either
⎡[1 × 2] [1 × 2]⎤ ⎡[1 × 1] [1 × 3]⎤
⎢⎢⎢⎣[[12 22]]⎥⎥⎦⎥ ⎢⎢⎣⎢[[12 33]]⎦⎥⎥⎥
template × 2] [1 × or template × 1] [1 × or a transpose. In all these cases, the
× 2] [2 × × 1] [2 ×
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