Page 32 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 32
athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
abc bcf
(∨) d e f (∧) a e i
ghi d gh
The result of either x∨y or x∧y is the element lying in the row of x and the column of y in the
relevant array. Thus a∨f = c while a∧f = i. In this example, none of the Bi or Ci are satisfied for i
≤ 4. In particular, we have neither a skew lattice nor a skew* lattice.
If both arrays coincide, then x ∨ y = x ∧ y and N is a skew* lattice. If they are transposes
(so that L = R* and R = L*), then x ∨ y = y ∧ x and N is a skew lattice. In both special cases only
the ∧-array is needed. (In the transpose case, one has x(λ, ρ) ∧ x(µ, σ) = x(µ, ρ).) In general, both
arrays are needed and all we can assert is that N is a refined quasi-lattice.
We return to the first eight absorption identities. Initially, each pair (Bi, Ci) is a dual pair
in that either is obtained from the other by switching ∨ with ∧. But other forms of duality exist.
Lemma 1.3.1. Given (N, ∨, ∧) where both ∨ and ∧ are associative and idempotent:
B1 - a ∧ (a ∨ b) = a - asserts that for x, y ∈ N, x (∨)≻R y implies x L≺ (∧) y.
B2 - (b ∨ a) ∧ a = a - asserts that for x, y ∈ N, x (∨)≻L y implies x R≺ (∧) y.
B3 - a ∧ (b ∨ a) = a - asserts that for x, y ∈ N, x (∨)≻L y implies x L≺ (∧) y.
B4 - (a ∨ b) ∧ a = a - asserts that for x, y ∈ N, x (∨)≻R y implies x R≺ (∧) y.
C1 - a ∨ (a ∧ b) = a - asserts that for x, y ∈ N, x (∧)≻R y implies x L≺ (∨) y.
C2 - (b ∧ a) ∨ a = a - asserts that for x, y ∈ N, x (∧)≻L y implies x R≺ (∨) y.
C3 - a ∨ (b ∧ a) = a - asserts that for x, y ∈ N, x (∧)≻L y implies x L≺ (∨) y.
C4 - (a ∧ b) ∨ a = a - asserts that for x, y ∈ N, x (∧)≻R y implies x R≺ (∨) y.
Thus, we have the following pairs inducing converse implications:
B1 and C2, B2 and C1, B3 and C3 and B4 and C4. £
Theorem 1.3.2. (Laslo [1997]) An algebra (N, ∨, ∧) where ∨ and ∧ are associative,
idempotent binary operations is a lattice if and only if B1, B2, C3 and C4 all hold or the
complementary set B3, B4, C1 and C2 all hold. (Thus absorption can imply commutativity.)
Proof. B1, B2, C3 and C4 yield x (∨)≻R y ⇒ x L≺ (∧) y ⇒ x (∨)≻L y ⇒ x R≺ (∧) y ⇒ x (∨)≻R y.
Thus, for each operation, both ≻R and ≻L reduce to ≥ and both natural partial orders dualize each
other and we have a lattice. The case for B3, B4, C1 and C2 is similar. The converse is clear. £
Similarly we obtain:
30
abc bcf
(∨) d e f (∧) a e i
ghi d gh
The result of either x∨y or x∧y is the element lying in the row of x and the column of y in the
relevant array. Thus a∨f = c while a∧f = i. In this example, none of the Bi or Ci are satisfied for i
≤ 4. In particular, we have neither a skew lattice nor a skew* lattice.
If both arrays coincide, then x ∨ y = x ∧ y and N is a skew* lattice. If they are transposes
(so that L = R* and R = L*), then x ∨ y = y ∧ x and N is a skew lattice. In both special cases only
the ∧-array is needed. (In the transpose case, one has x(λ, ρ) ∧ x(µ, σ) = x(µ, ρ).) In general, both
arrays are needed and all we can assert is that N is a refined quasi-lattice.
We return to the first eight absorption identities. Initially, each pair (Bi, Ci) is a dual pair
in that either is obtained from the other by switching ∨ with ∧. But other forms of duality exist.
Lemma 1.3.1. Given (N, ∨, ∧) where both ∨ and ∧ are associative and idempotent:
B1 - a ∧ (a ∨ b) = a - asserts that for x, y ∈ N, x (∨)≻R y implies x L≺ (∧) y.
B2 - (b ∨ a) ∧ a = a - asserts that for x, y ∈ N, x (∨)≻L y implies x R≺ (∧) y.
B3 - a ∧ (b ∨ a) = a - asserts that for x, y ∈ N, x (∨)≻L y implies x L≺ (∧) y.
B4 - (a ∨ b) ∧ a = a - asserts that for x, y ∈ N, x (∨)≻R y implies x R≺ (∧) y.
C1 - a ∨ (a ∧ b) = a - asserts that for x, y ∈ N, x (∧)≻R y implies x L≺ (∨) y.
C2 - (b ∧ a) ∨ a = a - asserts that for x, y ∈ N, x (∧)≻L y implies x R≺ (∨) y.
C3 - a ∨ (b ∧ a) = a - asserts that for x, y ∈ N, x (∧)≻L y implies x L≺ (∨) y.
C4 - (a ∧ b) ∨ a = a - asserts that for x, y ∈ N, x (∧)≻R y implies x R≺ (∨) y.
Thus, we have the following pairs inducing converse implications:
B1 and C2, B2 and C1, B3 and C3 and B4 and C4. £
Theorem 1.3.2. (Laslo [1997]) An algebra (N, ∨, ∧) where ∨ and ∧ are associative,
idempotent binary operations is a lattice if and only if B1, B2, C3 and C4 all hold or the
complementary set B3, B4, C1 and C2 all hold. (Thus absorption can imply commutativity.)
Proof. B1, B2, C3 and C4 yield x (∨)≻R y ⇒ x L≺ (∧) y ⇒ x (∨)≻L y ⇒ x R≺ (∧) y ⇒ x (∨)≻R y.
Thus, for each operation, both ≻R and ≻L reduce to ≥ and both natural partial orders dualize each
other and we have a lattice. The case for B3, B4, C1 and C2 is similar. The converse is clear. £
Similarly we obtain:
30
