Page 115 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 115
III: Quasilattices, Paralattices and their Congruences
Clearly a = α – β – γ and b = α – β. What is the next smallest element? If it is α – γ, then the
ascending sequence
α–β–γ < α–β < α–γ < α–β+γ < α < α+β–γ < α+γ < α+β < α+β+γ
must occur. In this case we have the displayed array. Otherwise, we must have:
α–β–γ < α–β < α–β+γ < α–γ < α < α+γ < α+β–γ < α+β < α+β+γ
yielding an array complementary to A.
This theorem has the following consequence:
Corollary 3.6.6. Given the arrays A and Aʹ of the prior theorem, the induced antilattice
A is simple if and only if A and Aʹ are complementary. Otherwise, Con(A) is a 3-element chain.
In general, all antilattices induced from 3 × 3 magic squares are congruence distributive.
Proof. In the complementary case, any pair of distinct elements generates A, which thus is
simple. Otherwise, a single nontrivial, proper congruence is given by [a, b, c, g, h, i⎮d, e, f].
The 4 × 4 case
We next consider antilattices induced from 4 × 4 magic squares storing 1 - 16. While just
one 3×3 magic square stores 1 - 9 (with eight dihedral variations), 880 essentially distinct magic
squares store 1 to 16. A list of all 880 squares was given by Bernard Frénicle de Bessy in a
posthumous 1693 publication. A mathematical analysis was given in the 1983 paper of Dame
Kathleen Ollerenshaw and Sir Hermann Bondi [5]. Thanks to the following observation, these
880 cases decompose into 220 classes of 4.
Lemma 3.6.7. Given a 4 × 4 magic square A, let squares B, C and D be induced from A
by simultaneous row and column permutations determined by (2 3), (1 2)(3 4) and (1 3 4 2)
respectively. Then A – D are all magic squares, but none are dihedrally equivalent. Moreover all
four squares induce the same antilattice.
Thus one can get by checking the leading array in each row of four squares in the
Ollerenshaw-Bondi list. Among these, the nonsimple cases are easily spotted, thanks to a
theorem about semimagic squares (all rows and columns add up to 34). In its statement, the
index of a congruence µ counts its number of congruence classes.
Theorem 3.6.8. If a semi-magic square A storing 1 – 16 induces a nonsimple antilattice
A, then A has a maximal congruence µ of index 2 whose corresponding congruence class
partition is either
πR = {1 – 4, 13 – 16⎮5 – 12}
or
πC = {1, 4, 5, 8, 9, 12, 13, 16⎮2, 3, 6, 7, 10, 11, 14, 15}
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Clearly a = α – β – γ and b = α – β. What is the next smallest element? If it is α – γ, then the
ascending sequence
α–β–γ < α–β < α–γ < α–β+γ < α < α+β–γ < α+γ < α+β < α+β+γ
must occur. In this case we have the displayed array. Otherwise, we must have:
α–β–γ < α–β < α–β+γ < α–γ < α < α+γ < α+β–γ < α+β < α+β+γ
yielding an array complementary to A.
This theorem has the following consequence:
Corollary 3.6.6. Given the arrays A and Aʹ of the prior theorem, the induced antilattice
A is simple if and only if A and Aʹ are complementary. Otherwise, Con(A) is a 3-element chain.
In general, all antilattices induced from 3 × 3 magic squares are congruence distributive.
Proof. In the complementary case, any pair of distinct elements generates A, which thus is
simple. Otherwise, a single nontrivial, proper congruence is given by [a, b, c, g, h, i⎮d, e, f].
The 4 × 4 case
We next consider antilattices induced from 4 × 4 magic squares storing 1 - 16. While just
one 3×3 magic square stores 1 - 9 (with eight dihedral variations), 880 essentially distinct magic
squares store 1 to 16. A list of all 880 squares was given by Bernard Frénicle de Bessy in a
posthumous 1693 publication. A mathematical analysis was given in the 1983 paper of Dame
Kathleen Ollerenshaw and Sir Hermann Bondi [5]. Thanks to the following observation, these
880 cases decompose into 220 classes of 4.
Lemma 3.6.7. Given a 4 × 4 magic square A, let squares B, C and D be induced from A
by simultaneous row and column permutations determined by (2 3), (1 2)(3 4) and (1 3 4 2)
respectively. Then A – D are all magic squares, but none are dihedrally equivalent. Moreover all
four squares induce the same antilattice.
Thus one can get by checking the leading array in each row of four squares in the
Ollerenshaw-Bondi list. Among these, the nonsimple cases are easily spotted, thanks to a
theorem about semimagic squares (all rows and columns add up to 34). In its statement, the
index of a congruence µ counts its number of congruence classes.
Theorem 3.6.8. If a semi-magic square A storing 1 – 16 induces a nonsimple antilattice
A, then A has a maximal congruence µ of index 2 whose corresponding congruence class
partition is either
πR = {1 – 4, 13 – 16⎮5 – 12}
or
πC = {1, 4, 5, 8, 9, 12, 13, 16⎮2, 3, 6, 7, 10, 11, 14, 15}
113
