Page 110 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 110
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Pastijn’s theorem also shows that the lattice of all varieties of fully distributive
quasilattices is Boolean of order 32. The larger lattice of varieties is also Boolean, but of order
210. (Pastijn works with identities, aba + a + aba = a and (a+b+a)a(a+b+a) = a, the ID-semiring
equivalents of B5 and C5.)
Returning to refined quasilattices from the last section, one may ask about the effect of
distributivity of (S, ∨, ∧) on its isotopic left-handed skew lattice variant (S, ∨*, ∧*) where
x∨*y = y∨x∨y and x∧*y = x∧y∧x,
Propositon 3.5.4. If a refined quasilattice (S, ∨, ∧) is distributive, then its isotopic left-
handed skew lattice (S, ∨*, ∧*) is also distributive.
Proof. Given that x∨*y = y∨x∨y = y∨*x∨*y and x∧*y = x∧y∧x = x∧*y∧*x, suppose that (S, ∨, ∧)
is distributive. Then
x ∧* (y ∨* z) ∧* x = x∧(y ∨* z)∧x = x∧(z ∨ y ∨ z)∧x = (x∧z∧x) ∨ (x∧y∧x) ∨ (x∧z∧x)
and
(x∧*y∧*x) ∨* (x∧*z∧*x) = (x∧y∧x) ∨* (x∧z∧x) = (x∧z∧x) ∨ (x∧y∧x) ∨ (x∧z∧x),
so that (S, ∨*, ∧*) satisfies D7 and likewise its dual D8. £
Conversely, if (S, ∨*, ∧*) is distributive, then (S, ∨, ∧) is easily seen to satisfy the weaker
identity x∧(z∨y∨z)∧x = (x∧z∧x)∨(x∧y∧x)∨(x∧z∧x) and its dual. If in addition (S, ∨, ∧) is already
known to be a skew lattice, then one can show it must be distributive.
3.6 Deriving simple antilattices from magic squares
Recall that a magic square is a square array of distinct numbers where all rows, columns
and the two diagonals have a common magic sum. A classic instance is the Lo-Shu with a magic
sum of 15:
816
3 5 7.
492
Given a magic square, its derived antilattice arises by letting the given square be the ∧-array and
letting the ∨-array be the square array storing the same numbers entered in their natural ordering.
Thus in the case of the Lo-Shu one gets:
108
Pastijn’s theorem also shows that the lattice of all varieties of fully distributive
quasilattices is Boolean of order 32. The larger lattice of varieties is also Boolean, but of order
210. (Pastijn works with identities, aba + a + aba = a and (a+b+a)a(a+b+a) = a, the ID-semiring
equivalents of B5 and C5.)
Returning to refined quasilattices from the last section, one may ask about the effect of
distributivity of (S, ∨, ∧) on its isotopic left-handed skew lattice variant (S, ∨*, ∧*) where
x∨*y = y∨x∨y and x∧*y = x∧y∧x,
Propositon 3.5.4. If a refined quasilattice (S, ∨, ∧) is distributive, then its isotopic left-
handed skew lattice (S, ∨*, ∧*) is also distributive.
Proof. Given that x∨*y = y∨x∨y = y∨*x∨*y and x∧*y = x∧y∧x = x∧*y∧*x, suppose that (S, ∨, ∧)
is distributive. Then
x ∧* (y ∨* z) ∧* x = x∧(y ∨* z)∧x = x∧(z ∨ y ∨ z)∧x = (x∧z∧x) ∨ (x∧y∧x) ∨ (x∧z∧x)
and
(x∧*y∧*x) ∨* (x∧*z∧*x) = (x∧y∧x) ∨* (x∧z∧x) = (x∧z∧x) ∨ (x∧y∧x) ∨ (x∧z∧x),
so that (S, ∨*, ∧*) satisfies D7 and likewise its dual D8. £
Conversely, if (S, ∨*, ∧*) is distributive, then (S, ∨, ∧) is easily seen to satisfy the weaker
identity x∧(z∨y∨z)∧x = (x∧z∧x)∨(x∧y∧x)∨(x∧z∧x) and its dual. If in addition (S, ∨, ∧) is already
known to be a skew lattice, then one can show it must be distributive.
3.6 Deriving simple antilattices from magic squares
Recall that a magic square is a square array of distinct numbers where all rows, columns
and the two diagonals have a common magic sum. A classic instance is the Lo-Shu with a magic
sum of 15:
816
3 5 7.
492
Given a magic square, its derived antilattice arises by letting the given square be the ∧-array and
letting the ∨-array be the square array storing the same numbers entered in their natural ordering.
Thus in the case of the Lo-Shu one gets:
108
