Page 119 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 119
III: Quasilattices, Paralattices and their Congruences
Proof. Suppose first that p is prime with p = 2n + 1 and let θ be a congruence with x θ y where x
≠ y. As with the previous theorem, things may be reduced to the case where x = ni and y = nj in
the middle row of the ∨-array and the ascending diagonal of the ∧-array. If F(X, Y) = n0 ∨ (X ∧
Y), then F(ni, nj) = n (i + j)/2 in the same θ-class as ni = n0 ∨ ni. (Here (i + j)/2 is calculated in
&p.) Since n0, n1, … , n p–1 generates the algebra, simplicity follows if we can show that from
ni and nj one can F-generate the entire nth ∨-row. This is equivalent to showing that from any
two i ≠ j in &p, all of &p is generated via the function f(x, y) = (x + y)/2. Let S be the set of all
numbers in &p thus generated. If 0 ∈ S, then S is closed under addition and thus is a nontrivial
subgroup of &p which forces S = &p. Indeed f(0, (x + y)/2) = (x + y)/4, f(0, (x + y)/4) = (x + y)/8,
etc. Hence all (x + y)/2n lie in S. Since some power of 2 equals 1 in &p, we get x + y ∈ S so that
S is as claimed. Otherwise, suppose 0 ∉ S. From f(x + k, y + k) = f(x, y) + k, the general case can
be shifted to the 0-case, so that no matter what pair i, j is given, the f-generated set is all of &p.
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). The argument that α is a congruence is identical to that in
the case of de la Loubère’s rule.
A number of further examples of magic squares that induce simple antilattices are given
in Leech [2005b].
Historical remarks
Nearly all results in the first five sections appeared in a 2002 paper by Gratiela Laslo and
Jonathan Leech that studied congruences on noncommutative lattices. The paper was written
while Laslo was working her dissertation at the University of Cluj-Napoca; several results are
from that dissertation. The material on isotopy is of more recent vintage, and was developed
from remarks in an email from Michael Kinyon. It appears here for the first time. The final
section on recreational mathematics and antilattices appeared Leech’s 2005 paper.
References
G. Laslo and J. Leech,
Green’s equivalences on noncommutative lattices, Acti Sci. Math (Szeged) 68 (2002),
501-533.
J, Leech,
Magic squares, finite planes and simple quasilattices, Ars Combinatoria 71 (2005), 75 -
96
117
Proof. Suppose first that p is prime with p = 2n + 1 and let θ be a congruence with x θ y where x
≠ y. As with the previous theorem, things may be reduced to the case where x = ni and y = nj in
the middle row of the ∨-array and the ascending diagonal of the ∧-array. If F(X, Y) = n0 ∨ (X ∧
Y), then F(ni, nj) = n (i + j)/2 in the same θ-class as ni = n0 ∨ ni. (Here (i + j)/2 is calculated in
&p.) Since n0, n1, … , n p–1 generates the algebra, simplicity follows if we can show that from
ni and nj one can F-generate the entire nth ∨-row. This is equivalent to showing that from any
two i ≠ j in &p, all of &p is generated via the function f(x, y) = (x + y)/2. Let S be the set of all
numbers in &p thus generated. If 0 ∈ S, then S is closed under addition and thus is a nontrivial
subgroup of &p which forces S = &p. Indeed f(0, (x + y)/2) = (x + y)/4, f(0, (x + y)/4) = (x + y)/8,
etc. Hence all (x + y)/2n lie in S. Since some power of 2 equals 1 in &p, we get x + y ∈ S so that
S is as claimed. Otherwise, suppose 0 ∉ S. From f(x + k, y + k) = f(x, y) + k, the general case can
be shifted to the 0-case, so that no matter what pair i, j is given, the f-generated set is all of &p.
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). The argument that α is a congruence is identical to that in
the case of de la Loubère’s rule.
A number of further examples of magic squares that induce simple antilattices are given
in Leech [2005b].
Historical remarks
Nearly all results in the first five sections appeared in a 2002 paper by Gratiela Laslo and
Jonathan Leech that studied congruences on noncommutative lattices. The paper was written
while Laslo was working her dissertation at the University of Cluj-Napoca; several results are
from that dissertation. The material on isotopy is of more recent vintage, and was developed
from remarks in an email from Michael Kinyon. It appears here for the first time. The final
section on recreational mathematics and antilattices appeared Leech’s 2005 paper.
References
G. Laslo and J. Leech,
Green’s equivalences on noncommutative lattices, Acti Sci. Math (Szeged) 68 (2002),
501-533.
J, Leech,
Magic squares, finite planes and simple quasilattices, Ars Combinatoria 71 (2005), 75 -
96
117
