Page 166 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 166
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
In Section 5.1 we consider symmetric skew lattices where commutation is unambiguous
as well as variations of this condition. Symmetry is first bisected into lower symmetry where
commutation shifts “downward” (x∨y = y∨x ⇒ x∧y = y∧x) and its dual, upper symmetry, where
commutation shifts “upward” (x∧y = y∧x ⇒ x∨y = y∨x). Characterizing identities for each are
given in Theorem 5.1.1. Both types of partial symmetry are characterized in Theorem 5.1.2 by a
pair of forbidden subalgebras (one right-handed and one left-handed). The absence of all four as
subalgebras characterizes symmetric skew lattices.
In Section 5.2 we prove the equivalence of x∧(y ∨ z)∧x = (x∧y∧x) ∨ (x∧z∧x) and its dual
x∨(y ∧ z)∨x = (x∨y∨x) ∧ (x∨z∨x) in the presence of symmetry. In particular, Theorem 5.2.3
states that given lower symmetry, x∧(y ∨ z)∧x = (x∧y∧x) ∨ (x∧z∧x) implies
x∨(y ∧ z)∨x = (x∨y∨x) ∧ (x∨z∨x),
and dually, in the presence of upper symmetry, the converse implication holds. These results are
due to Matthew Spinks, who had obtained in a very lengthy computer-generated proof, but
eventually reduced the proof to one of a moderate size that was then “humanized” by Karin
Cvetko-Vah. (All relevant references are given in the sections.)
For lattices, not only is x∧(y ∨ z) = (x∧y) ∨ (x∧z) equivalent to x∨(y ∧ z) = (x∨y) ∧ (x∨z),
both identities in turn are equivalent to a lattice not having copies of M3 or N5 as subalgebras.
Another equivalent condition for lattices is this form of cancellation:
x∨z = y∨z & x∧z = y∧z ⇒ x = y.
For skew lattices, however, these conditions are mutually nonequivalent in the absence of other
qualifying assumptions. Either a distributive identity or the cancellative implication will by itself
rule out M3 or N5 occurring as subalgebras, but not conversely. A skew lattice is called
cancellative if both the above implication (cancelling on the right) and its dual,
x∨y = x∨z & x∧y = x∧z ⇒ y = z (cancelling on the left),
hold. Cancellative skew lattices are always strongly symmetric. Since every skew chain is
cancellative (Proposition 5.3.2), but not necessarily distributive (Example 5.3.3), being
cancellative does not imply being distributive. Conversely the four forbidden algebras of Section
5.1 are distributive, but not cancellative. Cancellative skew lattices, and their close variants, are
studied in Section 5.3. They all form varieties that are characterized by small sets of forbidden
subalgebras (Theorem 5.3.8). Their coset structure also produces some interesting counting
features. (See Theorems 5.3.10 and 5.3.11.) Both partial function algebras and skew lattices of
idempotents in rings are cancellative.
Like being symmetric, being categorical is desirable in a skew lattice as it makes cosets
and their bijections well-behaved globally. Indeed, all distributive skew lattices are categorical
(by Theorem 5.4.2). This condition is studied more closely in Section 5.4. Categorical skew
lattices form a proper subvariety of skew lattices and in Theorem 5.4.4 we give a countable
family of forbidden subalgebras that characterize this subvariety. Strictly categorical skew
164
In Section 5.1 we consider symmetric skew lattices where commutation is unambiguous
as well as variations of this condition. Symmetry is first bisected into lower symmetry where
commutation shifts “downward” (x∨y = y∨x ⇒ x∧y = y∧x) and its dual, upper symmetry, where
commutation shifts “upward” (x∧y = y∧x ⇒ x∨y = y∨x). Characterizing identities for each are
given in Theorem 5.1.1. Both types of partial symmetry are characterized in Theorem 5.1.2 by a
pair of forbidden subalgebras (one right-handed and one left-handed). The absence of all four as
subalgebras characterizes symmetric skew lattices.
In Section 5.2 we prove the equivalence of x∧(y ∨ z)∧x = (x∧y∧x) ∨ (x∧z∧x) and its dual
x∨(y ∧ z)∨x = (x∨y∨x) ∧ (x∨z∨x) in the presence of symmetry. In particular, Theorem 5.2.3
states that given lower symmetry, x∧(y ∨ z)∧x = (x∧y∧x) ∨ (x∧z∧x) implies
x∨(y ∧ z)∨x = (x∨y∨x) ∧ (x∨z∨x),
and dually, in the presence of upper symmetry, the converse implication holds. These results are
due to Matthew Spinks, who had obtained in a very lengthy computer-generated proof, but
eventually reduced the proof to one of a moderate size that was then “humanized” by Karin
Cvetko-Vah. (All relevant references are given in the sections.)
For lattices, not only is x∧(y ∨ z) = (x∧y) ∨ (x∧z) equivalent to x∨(y ∧ z) = (x∨y) ∧ (x∨z),
both identities in turn are equivalent to a lattice not having copies of M3 or N5 as subalgebras.
Another equivalent condition for lattices is this form of cancellation:
x∨z = y∨z & x∧z = y∧z ⇒ x = y.
For skew lattices, however, these conditions are mutually nonequivalent in the absence of other
qualifying assumptions. Either a distributive identity or the cancellative implication will by itself
rule out M3 or N5 occurring as subalgebras, but not conversely. A skew lattice is called
cancellative if both the above implication (cancelling on the right) and its dual,
x∨y = x∨z & x∧y = x∧z ⇒ y = z (cancelling on the left),
hold. Cancellative skew lattices are always strongly symmetric. Since every skew chain is
cancellative (Proposition 5.3.2), but not necessarily distributive (Example 5.3.3), being
cancellative does not imply being distributive. Conversely the four forbidden algebras of Section
5.1 are distributive, but not cancellative. Cancellative skew lattices, and their close variants, are
studied in Section 5.3. They all form varieties that are characterized by small sets of forbidden
subalgebras (Theorem 5.3.8). Their coset structure also produces some interesting counting
features. (See Theorems 5.3.10 and 5.3.11.) Both partial function algebras and skew lattices of
idempotents in rings are cancellative.
Like being symmetric, being categorical is desirable in a skew lattice as it makes cosets
and their bijections well-behaved globally. Indeed, all distributive skew lattices are categorical
(by Theorem 5.4.2). This condition is studied more closely in Section 5.4. Categorical skew
lattices form a proper subvariety of skew lattices and in Theorem 5.4.4 we give a countable
family of forbidden subalgebras that characterize this subvariety. Strictly categorical skew
164
