Page 111 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 111
III: Quasilattices, Paralattices and their Congruences
⎡8 1 6 ⎤ ⎡1 2 3 ⎤
⎢3 7⎥ ⎢4 6⎥ .
(∧) ⎣⎢4 5 ⎥ (∨) ⎢⎣7 5 ⎥
9 ⎦ 8 ⎦
2 9
Square arrays also come from finite planes, that is, vector spaces of dimension 2 over
finite fields. For instance, given the field &5, the plane &5 × &5 can be represented as the 5×5
array of ordered pairs on the left below, but with parentheses deleted. Alternatively, one could
view these pairs as base 5 representations of integers in base 10. Thus 3,2 represents 325 = 1710.
The planar array could thus be encoded using the numbers in the right array.
⎡0, 0 1, 0 2, 0 3, 0 4, 0⎤ ⎡0 1 2 3 4 ⎤
⎢ ⎥
⎢ 0,1 1,1 2,1 3,1 4,1 ⎥ ⎢ 5 6 78 9 ⎥
⎢⎢0, 2 1, 2 ⎥
2, 2 3, 2 4, 2 ⎥ ⎢10 11 12 13 14 ⎥
⎢⎢15 ⎥
⎢0, 3 1, 3 2, 3 3, 3 4, 3 ⎥ 16 17 18 19 ⎥
⎣⎢0, 4 1, 4 2, 4 3, 4 4, 4 ⎥⎦
⎢⎣20 21 22 23 24⎥⎦
In either case, this plane has 25 points and 30 lines, the latter arranged in six classes of five
parallel lines each. The rows of the array consist of all lines of slope 0, the columns consist of all
lines of undefined slope, the main diagonal plus the four broken descending diagonals yield all
five lines of slope 1, and the counter-diagonal plus all four broken ascending counter-diagonals
yield all five lines of slope 4. In all, between the rows, columns, diagonals and counter-
diagonals, 20 out of 30 lines are accounted for, with only lines of slopes 2 or 3 left out.
Alternatively, &5 × &5 can be represented by storing the five lines of slope 1 in the five
rows and the five lines of slope 4 in the five columns in the left array below.
⎡0, 0 1,1 2, 2 3, 3 4, 4⎤ ⎡ 0 6 12 18 24⎤
⎢2, 3 3, 4 4, 0 0,1 1, 2⎥ ⎢13 19 20 1 7 ⎥
⎢⎢4,1 ⎥ ⎢⎢21 ⎥
0, 2 1, 3 2, 4 3, 0 ⎥ 2 8 14 15 ⎥
⎢1, 4 2, 0 3,1 4, 2 0, 3 ⎥ ⎢ 9 10 16 22 3⎥
⎣⎢3, 2 4, 3 0, 4 1, 0 2,1⎥⎦ ⎢⎣17 23 4 5 11 ⎦⎥
In the right array, not only do all rows, columns, the main diagonal and the counter-diagonal sum
to 60, but so do all broken diagonals and counter-diagonals. This makes the right array a
pandiagonal magic square. Returning to the left array, the (broken) diagonals and counter-
diagonals are precisely the lines of slope 3 and 2 respectively. Indeed, the line arrangement of the
left array forces the right array to be pandiagonal. That finite planes can yield pandiagonal
squares is well known. Together, the two representations of this plane in integer format describe
an antilattice induced from a magic square. As will be shown below, both antilattices derived
from the two magic squares thus far encountered are simple. Is this true in general?
109
⎡8 1 6 ⎤ ⎡1 2 3 ⎤
⎢3 7⎥ ⎢4 6⎥ .
(∧) ⎣⎢4 5 ⎥ (∨) ⎢⎣7 5 ⎥
9 ⎦ 8 ⎦
2 9
Square arrays also come from finite planes, that is, vector spaces of dimension 2 over
finite fields. For instance, given the field &5, the plane &5 × &5 can be represented as the 5×5
array of ordered pairs on the left below, but with parentheses deleted. Alternatively, one could
view these pairs as base 5 representations of integers in base 10. Thus 3,2 represents 325 = 1710.
The planar array could thus be encoded using the numbers in the right array.
⎡0, 0 1, 0 2, 0 3, 0 4, 0⎤ ⎡0 1 2 3 4 ⎤
⎢ ⎥
⎢ 0,1 1,1 2,1 3,1 4,1 ⎥ ⎢ 5 6 78 9 ⎥
⎢⎢0, 2 1, 2 ⎥
2, 2 3, 2 4, 2 ⎥ ⎢10 11 12 13 14 ⎥
⎢⎢15 ⎥
⎢0, 3 1, 3 2, 3 3, 3 4, 3 ⎥ 16 17 18 19 ⎥
⎣⎢0, 4 1, 4 2, 4 3, 4 4, 4 ⎥⎦
⎢⎣20 21 22 23 24⎥⎦
In either case, this plane has 25 points and 30 lines, the latter arranged in six classes of five
parallel lines each. The rows of the array consist of all lines of slope 0, the columns consist of all
lines of undefined slope, the main diagonal plus the four broken descending diagonals yield all
five lines of slope 1, and the counter-diagonal plus all four broken ascending counter-diagonals
yield all five lines of slope 4. In all, between the rows, columns, diagonals and counter-
diagonals, 20 out of 30 lines are accounted for, with only lines of slopes 2 or 3 left out.
Alternatively, &5 × &5 can be represented by storing the five lines of slope 1 in the five
rows and the five lines of slope 4 in the five columns in the left array below.
⎡0, 0 1,1 2, 2 3, 3 4, 4⎤ ⎡ 0 6 12 18 24⎤
⎢2, 3 3, 4 4, 0 0,1 1, 2⎥ ⎢13 19 20 1 7 ⎥
⎢⎢4,1 ⎥ ⎢⎢21 ⎥
0, 2 1, 3 2, 4 3, 0 ⎥ 2 8 14 15 ⎥
⎢1, 4 2, 0 3,1 4, 2 0, 3 ⎥ ⎢ 9 10 16 22 3⎥
⎣⎢3, 2 4, 3 0, 4 1, 0 2,1⎥⎦ ⎢⎣17 23 4 5 11 ⎦⎥
In the right array, not only do all rows, columns, the main diagonal and the counter-diagonal sum
to 60, but so do all broken diagonals and counter-diagonals. This makes the right array a
pandiagonal magic square. Returning to the left array, the (broken) diagonals and counter-
diagonals are precisely the lines of slope 3 and 2 respectively. Indeed, the line arrangement of the
left array forces the right array to be pandiagonal. That finite planes can yield pandiagonal
squares is well known. Together, the two representations of this plane in integer format describe
an antilattice induced from a magic square. As will be shown below, both antilattices derived
from the two magic squares thus far encountered are simple. Is this true in general?
109
