Page 104 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 104
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Corollary 3.4.9. Let (S, ∨, ∧) be a refined quasilattice with a ∈ S. The following are
equivalent:
1. For all b ∈ S, a∨b = b∨a.
2. For all b ∈ S, a∧b = b∧a.
3. The D-class of a is the singleton, {a}.
Thus the union of all singleton D-classes is the center of S. £
These results, in contrast with that of Theorem 3.1.7, show the effect of a coherent
natural partial order on the structure of a quasilattice. It turns out that (even irregular) refined
quasilattices and skew lattices are closely connected in a very precise way. To see this, we
borrow a term arising in a number of other contexts both mathematical and scientific.
An isotopy of quasi-lattices (Q, ∨, ∧) and (Qʹ, ∨ʹ, ∧ʹ) with respective natural pre-orders ≻
and ≻ʹ is a bijection f: Q → Qʹ such that x ≻ y in Q iff f(x) ≻ʹ f(y) in Qʹ.
An isotopy of paralattices (P, ∨, ∧) and (Pʹ, ∨ʹ, ∧ʹ) with natural partial orders ≥ and ≥ʹ is
a bijection f: P → Pʹ such that x ≥ y in P iff f(x) ≥ʹ f(y) in Pʹ.
Finally, an isotopy of refined quasi-lattices (S, ∨, ∧) and (Sʹ, ∨ʹ, ∧ʹ) is any bijection from
S to Sʹ that is simultaneously an isotopy of quasi-lattices and an isotopy of paralattices.
In each case, all isomorphisms are isotopies. Also, in each of these cases, given an
algebra (S, ∨, ∧), upon defining ∨• by x∨•y = y∨x and ∧• by x∧•y = y∧x, all three derived algebras
(S, ∨•, ∧), (S, ∨, ∧•) and (S, ∨•, ∧•) are all isotopes of (S, ∨, ∧) under the identity map on S. Indeed
x∨•y∨•x = x∨y∨x and x∧•y∧•x = x∧y∧x on S, in the case of quasilattices, thus guaranteeing that ≤ is
the same for all four algebras. Likewise x∨•y = y∨•x iff x∨y = y∨x on S with both outcomes
agreeing, and x∧•y = y∧•x iff x∧y = y∧x on S with both outcomes agreeing in the case of
paralattices, thus guaranteeing that ≤ is the same for all four algebras. Note that antilattices are
isotopic precisely if the have the same size.
In general we have the following assertions, the first of which is near-obvious:
Proposition 3.4.10. Let quasilattices (Q, ∨, ∧) and (Qʹ, ∨ʹ, ∧ʹ) be given. Then any
isotopy f: Q → Qʹ induces a unique isomorphism f*: Q/D → Qʹ/D making the following diagram
commute, where δ: Q → Q/D and δʹ Qʹ → Qʹ/D denote the canonical epimorphisms.
Q ⎯⎯f → Q′
δ↓ ⎯⎯f *→ ↓ δ′
Q /D
Q′ / D
102
Corollary 3.4.9. Let (S, ∨, ∧) be a refined quasilattice with a ∈ S. The following are
equivalent:
1. For all b ∈ S, a∨b = b∨a.
2. For all b ∈ S, a∧b = b∧a.
3. The D-class of a is the singleton, {a}.
Thus the union of all singleton D-classes is the center of S. £
These results, in contrast with that of Theorem 3.1.7, show the effect of a coherent
natural partial order on the structure of a quasilattice. It turns out that (even irregular) refined
quasilattices and skew lattices are closely connected in a very precise way. To see this, we
borrow a term arising in a number of other contexts both mathematical and scientific.
An isotopy of quasi-lattices (Q, ∨, ∧) and (Qʹ, ∨ʹ, ∧ʹ) with respective natural pre-orders ≻
and ≻ʹ is a bijection f: Q → Qʹ such that x ≻ y in Q iff f(x) ≻ʹ f(y) in Qʹ.
An isotopy of paralattices (P, ∨, ∧) and (Pʹ, ∨ʹ, ∧ʹ) with natural partial orders ≥ and ≥ʹ is
a bijection f: P → Pʹ such that x ≥ y in P iff f(x) ≥ʹ f(y) in Pʹ.
Finally, an isotopy of refined quasi-lattices (S, ∨, ∧) and (Sʹ, ∨ʹ, ∧ʹ) is any bijection from
S to Sʹ that is simultaneously an isotopy of quasi-lattices and an isotopy of paralattices.
In each case, all isomorphisms are isotopies. Also, in each of these cases, given an
algebra (S, ∨, ∧), upon defining ∨• by x∨•y = y∨x and ∧• by x∧•y = y∧x, all three derived algebras
(S, ∨•, ∧), (S, ∨, ∧•) and (S, ∨•, ∧•) are all isotopes of (S, ∨, ∧) under the identity map on S. Indeed
x∨•y∨•x = x∨y∨x and x∧•y∧•x = x∧y∧x on S, in the case of quasilattices, thus guaranteeing that ≤ is
the same for all four algebras. Likewise x∨•y = y∨•x iff x∨y = y∨x on S with both outcomes
agreeing, and x∧•y = y∧•x iff x∧y = y∧x on S with both outcomes agreeing in the case of
paralattices, thus guaranteeing that ≤ is the same for all four algebras. Note that antilattices are
isotopic precisely if the have the same size.
In general we have the following assertions, the first of which is near-obvious:
Proposition 3.4.10. Let quasilattices (Q, ∨, ∧) and (Qʹ, ∨ʹ, ∧ʹ) be given. Then any
isotopy f: Q → Qʹ induces a unique isomorphism f*: Q/D → Qʹ/D making the following diagram
commute, where δ: Q → Q/D and δʹ Qʹ → Qʹ/D denote the canonical epimorphisms.
Q ⎯⎯f → Q′
δ↓ ⎯⎯f *→ ↓ δ′
Q /D
Q′ / D
102
