Page 118 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 118
athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Proof. Suppose that x θ y where x ≠ y. By Lemma 3.6.2, we may assume that x and y either lie in
a common row or in a common column of the ∨-array. If x and y are in the same column, then
their meets yield θ-related elements u and v in distinct columns of the ∨-array. From
{u, v, u∨v, v∨u} we gain a pair of distinct θ-related elements lying in a common row of the
∨-array.
Thus at the outset we may assume that x θ y with x and y distinct elements in a common
row of the ∨-array. If the order of the magic square is p = 2n + 1 (and the order of the algebra is
p2), then n0 ∨ x and n0 ∨ y must be distinct θ-related elements in the middle row of the ∨-array,
say ni and nj. But ni and nj are also lie the main ascending diagonal of the ∧-array and from them
we can generate via f(X, Y) = ni ∨ (X ∧ Y), all n, i±mk where k = j – i. If p is prime, the main
ascending diagonal in the ∧-array must lie in a common θ-class. Since this diagonal generates the
entire algebra, θ = ∇.
If p is composite, say p = ab with 1 < a, b < p, then define an equivalence α by ij α kl if
both i ≡ k (mod a) and j ≡ l (mod a). That α is a ∨-congruence is clear. In the case of ∧, observe
that in the ∧-array any horizontal or vertical displacement of a positions from any starting
position yields an α-related element. Conversely, any pair of α-related elements are connected
by a sequence of such displacements. Thus given x α y and u α v, the ∧-columns of x∧u and y∧v,
being the ∧-columns of u and v, differ in their position by a multiple of a. Likewise the ∧-rows of
x ∧ u and y ∧ v, being the ∧-rows of x and y, differ in their position by a multiple of a. It follows
that x ∧ u α y ∧ v so that α is a ∧–congruence also. Clearly α is neither Δ or ∇.
A variation of the above rule had been given previously by Claude Gaspar Bachet de
Méziriac (who in 1621 published the famous edition of Diophantus’ Arithmetica).
Bachet de Méziriac’s Rule. Place 00 directly above the middle position of an n×n array.
In ascending (broken) diagonal fashion place in order, the remaining 01 through 0, n–1. Next,
place 10 two rows directly above 0, n–1. In ascending (broken) diagonal fashion place, in order,
11 through 1, n–1. Next, place 20 two rows directly above 1, n–1. Repeat the process until an
entire n×n array is filled to produce a magic square storing 0 through n2–1 in base n.
Theorem 3.6.11. The magic squares of odd order given by Bachet de Méziriac’s rule
induce simple antilattices precisely when the order is prime.
⎡00 01 02 03 04 05 06⎤ ⎡63 20 54 11 45 02 36⎤
⎢⎢⎢1200 ⎥ ⎢⎢⎢5226 ⎥
11 12 13 14 15 16 ⎥ 53 10 44 01 35 62 ⎥
21 22 23 24 25 26 ⎥ 16 43 00 34 61 25 ⎥
⎢30 31 32 33 34 35 36⎥ ⎢ 15 42 06 33 60 24 51 ⎥
⎢⎢40 ⎥ ⎢⎢41 ⎥
41 42 43 44 45 46 ⎥ 05 32 66 23 50 14 ⎥
⎢50 51 52 53 54 55 56 ⎥ ⎢04 31 65 22 56 13 40 ⎥
⎣⎢60 61 62 63 64 65 66 ⎥⎦ ⎢⎣30 64 21 55 12 46 03 ⎦⎥
(Standard array for 0 – 48, base 7) (de Méziriac array for 0 – 48. base 7)
116
Proof. Suppose that x θ y where x ≠ y. By Lemma 3.6.2, we may assume that x and y either lie in
a common row or in a common column of the ∨-array. If x and y are in the same column, then
their meets yield θ-related elements u and v in distinct columns of the ∨-array. From
{u, v, u∨v, v∨u} we gain a pair of distinct θ-related elements lying in a common row of the
∨-array.
Thus at the outset we may assume that x θ y with x and y distinct elements in a common
row of the ∨-array. If the order of the magic square is p = 2n + 1 (and the order of the algebra is
p2), then n0 ∨ x and n0 ∨ y must be distinct θ-related elements in the middle row of the ∨-array,
say ni and nj. But ni and nj are also lie the main ascending diagonal of the ∧-array and from them
we can generate via f(X, Y) = ni ∨ (X ∧ Y), all n, i±mk where k = j – i. If p is prime, the main
ascending diagonal in the ∧-array must lie in a common θ-class. Since this diagonal generates the
entire algebra, θ = ∇.
If p is composite, say p = ab with 1 < a, b < p, then define an equivalence α by ij α kl if
both i ≡ k (mod a) and j ≡ l (mod a). That α is a ∨-congruence is clear. In the case of ∧, observe
that in the ∧-array any horizontal or vertical displacement of a positions from any starting
position yields an α-related element. Conversely, any pair of α-related elements are connected
by a sequence of such displacements. Thus given x α y and u α v, the ∧-columns of x∧u and y∧v,
being the ∧-columns of u and v, differ in their position by a multiple of a. Likewise the ∧-rows of
x ∧ u and y ∧ v, being the ∧-rows of x and y, differ in their position by a multiple of a. It follows
that x ∧ u α y ∧ v so that α is a ∧–congruence also. Clearly α is neither Δ or ∇.
A variation of the above rule had been given previously by Claude Gaspar Bachet de
Méziriac (who in 1621 published the famous edition of Diophantus’ Arithmetica).
Bachet de Méziriac’s Rule. Place 00 directly above the middle position of an n×n array.
In ascending (broken) diagonal fashion place in order, the remaining 01 through 0, n–1. Next,
place 10 two rows directly above 0, n–1. In ascending (broken) diagonal fashion place, in order,
11 through 1, n–1. Next, place 20 two rows directly above 1, n–1. Repeat the process until an
entire n×n array is filled to produce a magic square storing 0 through n2–1 in base n.
Theorem 3.6.11. The magic squares of odd order given by Bachet de Méziriac’s rule
induce simple antilattices precisely when the order is prime.
⎡00 01 02 03 04 05 06⎤ ⎡63 20 54 11 45 02 36⎤
⎢⎢⎢1200 ⎥ ⎢⎢⎢5226 ⎥
11 12 13 14 15 16 ⎥ 53 10 44 01 35 62 ⎥
21 22 23 24 25 26 ⎥ 16 43 00 34 61 25 ⎥
⎢30 31 32 33 34 35 36⎥ ⎢ 15 42 06 33 60 24 51 ⎥
⎢⎢40 ⎥ ⎢⎢41 ⎥
41 42 43 44 45 46 ⎥ 05 32 66 23 50 14 ⎥
⎢50 51 52 53 54 55 56 ⎥ ⎢04 31 65 22 56 13 40 ⎥
⎣⎢60 61 62 63 64 65 66 ⎥⎦ ⎢⎣30 64 21 55 12 46 03 ⎦⎥
(Standard array for 0 – 48, base 7) (de Méziriac array for 0 – 48. base 7)
116
