Page 181 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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V: Further Topics in Skew Lattices
Being left-handed, both aʹ > aʹ∧b and b∨cʹ > cʹ are clear. Trivially cʹ∧(aʹ∧b) = cʹ and
(b∨cʹ)∨aʹ = aʹ. On the other hand, (aʹ∧b)∧cʹ = (aʹ∧b)∧(aʹ∧c) = aʹ∧(b∧c) = aʹ∧c = cʹ also so that
aʹ∧b > cʹ. In dual fashion aʹ > b∨cʹ. Notice that both
aʹ ∧ b ≠ b ∨ cʹ = aʹ ∧ (b ∨ cʹ) = aʹ ∧ [b ∨ (aʹ ∧ b ∧ cʹ)].
and
b ∨ cʹ ≠ aʹ ∧ b = (aʹ ∧ b) ∨ cʹ = [(aʹ ∨ b ∨ cʹ) ∧ b] ∨ cʹ.
The lemma follows by contraposition. £
Comment: The converse (in the left-handed case) has been shown using Prover9. These
identities do not imply left-handedness. As a consequence of this lemma we have:
Theorem 5.4.2. Any skew lattice satisfying either distributive identity (5.2.1) or (5.2.2)
is categorical. In particular, all distributive skew lattices are categorical.
Proof. To begin, if a left-handed skew lattice satisfies (5.2.1), then (5.4.1) follows:
x ∧ [y ∨ (x ∧ y ∧ z)] = (x∧y) ∨ [x ∧ (x ∧ y ∧ z)] = (x∧y) ∨ (x ∧ y ∧ z) = x∧y.
Likewise a left-handed skew lattice satisfying (5.2.2) must satisfy (5.4.2). Dually, a right-handed
skew lattice satisfying either (5.2.1) or (5.2.2) are categorical. In general, if a skew lattice S
satisfies either distributive identity, then so do its factors S/R and S/L. Hence each factor is
categorical and thus so is S, since it is isomorphic to a subalgebra of S/R × S/L. £
Forbidden subalgebras
Given a comparable D-classes, A > B, if a, aʹ ∈ A lie in a common B-coset, we denote
this by a –B aʹ; likewise b –A bʹ in B if b, bʹ lie in a common A-coset. Of interest here are skew
chains A > B > C, since a skew lattice is categorical if and only if all its skew chains are thus.
Two elements b and bʹ in the middle class B are AC-connected if a finite sequence
b = b0, b1, b2, … , bn = bʹ exists such that bi –A bi+1 or bi –C bi+1 for all i ≤ n – 1. Clearly this
defines an equivalence relation on B. An AC-component of B (or just component when the
context is clear) is an equivalence class for this relation, that is, a maximally AC-connected
subset of B. (Connectedness is actually a congruence on the rectangular algebra B, making the
components subalgebras of B. Indeed it is the join-equivalence of the congruence partitions
given by the A-cosets and by the B-cosets.) In the examples below, B is connected. Given a
component B1 in the middle class B, a sub-skew chain is given by A > B1 > C. Indeed if A1 is a
B-coset in A and C1 is a B-coset in C, then A1 > B1 > C1 is also a sub-skew chain; moreover, a
skew chain is categorical if and only if all such sub-skew chains are categorical.
Our classification of forbidden subalgebras rests on the next lemma and its right-handed
dual.
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Being left-handed, both aʹ > aʹ∧b and b∨cʹ > cʹ are clear. Trivially cʹ∧(aʹ∧b) = cʹ and
(b∨cʹ)∨aʹ = aʹ. On the other hand, (aʹ∧b)∧cʹ = (aʹ∧b)∧(aʹ∧c) = aʹ∧(b∧c) = aʹ∧c = cʹ also so that
aʹ∧b > cʹ. In dual fashion aʹ > b∨cʹ. Notice that both
aʹ ∧ b ≠ b ∨ cʹ = aʹ ∧ (b ∨ cʹ) = aʹ ∧ [b ∨ (aʹ ∧ b ∧ cʹ)].
and
b ∨ cʹ ≠ aʹ ∧ b = (aʹ ∧ b) ∨ cʹ = [(aʹ ∨ b ∨ cʹ) ∧ b] ∨ cʹ.
The lemma follows by contraposition. £
Comment: The converse (in the left-handed case) has been shown using Prover9. These
identities do not imply left-handedness. As a consequence of this lemma we have:
Theorem 5.4.2. Any skew lattice satisfying either distributive identity (5.2.1) or (5.2.2)
is categorical. In particular, all distributive skew lattices are categorical.
Proof. To begin, if a left-handed skew lattice satisfies (5.2.1), then (5.4.1) follows:
x ∧ [y ∨ (x ∧ y ∧ z)] = (x∧y) ∨ [x ∧ (x ∧ y ∧ z)] = (x∧y) ∨ (x ∧ y ∧ z) = x∧y.
Likewise a left-handed skew lattice satisfying (5.2.2) must satisfy (5.4.2). Dually, a right-handed
skew lattice satisfying either (5.2.1) or (5.2.2) are categorical. In general, if a skew lattice S
satisfies either distributive identity, then so do its factors S/R and S/L. Hence each factor is
categorical and thus so is S, since it is isomorphic to a subalgebra of S/R × S/L. £
Forbidden subalgebras
Given a comparable D-classes, A > B, if a, aʹ ∈ A lie in a common B-coset, we denote
this by a –B aʹ; likewise b –A bʹ in B if b, bʹ lie in a common A-coset. Of interest here are skew
chains A > B > C, since a skew lattice is categorical if and only if all its skew chains are thus.
Two elements b and bʹ in the middle class B are AC-connected if a finite sequence
b = b0, b1, b2, … , bn = bʹ exists such that bi –A bi+1 or bi –C bi+1 for all i ≤ n – 1. Clearly this
defines an equivalence relation on B. An AC-component of B (or just component when the
context is clear) is an equivalence class for this relation, that is, a maximally AC-connected
subset of B. (Connectedness is actually a congruence on the rectangular algebra B, making the
components subalgebras of B. Indeed it is the join-equivalence of the congruence partitions
given by the A-cosets and by the B-cosets.) In the examples below, B is connected. Given a
component B1 in the middle class B, a sub-skew chain is given by A > B1 > C. Indeed if A1 is a
B-coset in A and C1 is a B-coset in C, then A1 > B1 > C1 is also a sub-skew chain; moreover, a
skew chain is categorical if and only if all such sub-skew chains are categorical.
Our classification of forbidden subalgebras rests on the next lemma and its right-handed
dual.
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