Page 113 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 113
III: Quasilattices, Paralattices and their Congruences
immediately generates its dual pair in the opposite array. In this sense, these two arrays are
complementary 3×3 arrays, so that any pair of elements lying in a common row or column in
either array generates the entire antilattice which thus is simple.
What can be said in general about congruences on an antilattice?
Given a rectangular array A, a cartesian partition of A is a partition P that is induced in
cartesian fashion from a partition of the rows and a partition of the columns of A. For example,
one cartesian partition of
⎡a b c d e⎤
⎢f g h i j ⎥
⎢⎣k ⎥
l m n o ⎦
is given by: ⎡⎡a b c⎤ ⎡d e⎤⎤
⎢⎢⎢⎣ f g h ⎦⎥ ⎢⎣i j ⎦⎥⎥⎥ .
⎣⎢[k l m][n o] ⎥⎦
In all, 260 = 52 × 5 cartesian partitions of this array are given by the fifty-two partitions of {a, b,
c, d, e} and the seven partitions of {a, f, k}.
Given such a partition P, an equivalence P# on A is given by a P# b if a and b lie in the
same P-class. Such an equivalence is called a cartesian equivalence on A. In the case of
rectangular bands, the congruences on an array that are consistent with a single rectangular band
operation (using just one of ∨ or ∧) are precisely its cartesian equivalences. Thus:
Proposition 3.6.3. Given an antilattice A, its congruences arise from pairs of cartesian
partitions of its two arrays sharing the same equivalence classes.
The general 3×3 case.
The Lo-Shu is one of infinitely many possible 3×3 magic squares that can arise if we
agree to store integers besides 1 - 9. Others include the following two squares:
⎡71 89 17 ⎤ ⎡252 171 363⎤
⎢5 59 113⎥ ⎢373 262 151⎥
⎢⎣101 29 ⎥ ⎣⎢161 353 272⎦⎥
47 ⎦
The magic square on the left consists entirely of primes, with a magic sum of 177, the least
possible such sum for any magic square of primes. The magic square on the right consists of 3-
digit palindromes. Like the Lo Shu case, both examples induce simple antilattices. Is this true
for all 3 × 3 magic squares? To answer this, we begin with the following result.
111
immediately generates its dual pair in the opposite array. In this sense, these two arrays are
complementary 3×3 arrays, so that any pair of elements lying in a common row or column in
either array generates the entire antilattice which thus is simple.
What can be said in general about congruences on an antilattice?
Given a rectangular array A, a cartesian partition of A is a partition P that is induced in
cartesian fashion from a partition of the rows and a partition of the columns of A. For example,
one cartesian partition of
⎡a b c d e⎤
⎢f g h i j ⎥
⎢⎣k ⎥
l m n o ⎦
is given by: ⎡⎡a b c⎤ ⎡d e⎤⎤
⎢⎢⎢⎣ f g h ⎦⎥ ⎢⎣i j ⎦⎥⎥⎥ .
⎣⎢[k l m][n o] ⎥⎦
In all, 260 = 52 × 5 cartesian partitions of this array are given by the fifty-two partitions of {a, b,
c, d, e} and the seven partitions of {a, f, k}.
Given such a partition P, an equivalence P# on A is given by a P# b if a and b lie in the
same P-class. Such an equivalence is called a cartesian equivalence on A. In the case of
rectangular bands, the congruences on an array that are consistent with a single rectangular band
operation (using just one of ∨ or ∧) are precisely its cartesian equivalences. Thus:
Proposition 3.6.3. Given an antilattice A, its congruences arise from pairs of cartesian
partitions of its two arrays sharing the same equivalence classes.
The general 3×3 case.
The Lo-Shu is one of infinitely many possible 3×3 magic squares that can arise if we
agree to store integers besides 1 - 9. Others include the following two squares:
⎡71 89 17 ⎤ ⎡252 171 363⎤
⎢5 59 113⎥ ⎢373 262 151⎥
⎢⎣101 29 ⎥ ⎣⎢161 353 272⎦⎥
47 ⎦
The magic square on the left consists entirely of primes, with a magic sum of 177, the least
possible such sum for any magic square of primes. The magic square on the right consists of 3-
digit palindromes. Like the Lo Shu case, both examples induce simple antilattices. Is this true
for all 3 × 3 magic squares? To answer this, we begin with the following result.
111
