Page 168 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 168
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
In the final seventh section we present some combinatorial results about skew chains and
skew diamonds. These counting theorems are a continuation of some results in Section 2.4, in
particular Theorem 2.4.10 and its corollaries. Whereas the former are over 20 years old, the
results in Section 7 were published in the past few years and are due primarily to Cvetko-Vah,
Leech and Pita de Costa. The section concludes with some consequences for cancellative skew
lattices. This is followed by a few remarks of a bibliographic nature.
5.1 Symmetric skew lattices
Recall from Theorem 2.2.4 that symmetric skew lattices are characterized as skew lattices
by the following identities with both sides respectively equaling x∧y∧x and x∨y∨x:
x∧y∧(x∨y∨x) = (x∨y∨x)∧y∧x (5.1.1)
and
x∨y∨(x∧y∧x) = (x∧y∧x)∨y∨x. (5.1.2)
Matthew Spinks in [Spinks 1998] observed that these identities can be trimmed to
x∧y∧(x∨y) = (y∨x)∧y∧x, (5.1.3)
and (5.1.4)
x∨y∨(x∧y) = (y∧x)∨y∨x
since x∧y∧x ≺ both x∧y and x∨y, so that x∨y∨(x∧y) = x∨y∨(x∧y∧x)∨(x∧y) = x∨y∨(x∧y∧x) due to
x∧y∧x R x∧y holding in any skew lattice. The three other terms are handled similarly.
Symmetry is parsed as follows. A skew lattice is lower symmetric if x∨y = y∨x implies
x∧y = y∧x. Dually, a skew lattice is upper symmetric if x∧y = y∧x implies x∨y = y∨x.
Theorem 5.1.1. Lower symmetry is characterized by identity (5.1.3). Dually, upper
symmetry is characterized by identity (5.1.4). Hence both classes of skew lattices form varieties.
A skew lattice S is thus upper [lower] symmetric if and only if both S/L and S/R are.
Proof. Indeed, if x∨y = y∨x, then (5.1.3) plus absorption gives x∧y = y∧x so that lower symmetry
holds. Conversely, since x ≥ x∨y∨x, occurrences of x and (x∨y∨x)∧y∧(x∨y∨x) must ∨-commute.
Hence if a skew lattice is lower symmetric, x and (x∨y∨x)∧y∧(x∨y∨x) also ∧-commute so that
(5.1.1) and hence (5.1.3) follows. Similar remarks hold for (5.1.4) and upper symmetry. £
Consider the following pair of Hasse diagrams, each determining a right-handed skew
diamond and its left-handed dual. (The dotted lines denote the natural partial order ≥.)
166
In the final seventh section we present some combinatorial results about skew chains and
skew diamonds. These counting theorems are a continuation of some results in Section 2.4, in
particular Theorem 2.4.10 and its corollaries. Whereas the former are over 20 years old, the
results in Section 7 were published in the past few years and are due primarily to Cvetko-Vah,
Leech and Pita de Costa. The section concludes with some consequences for cancellative skew
lattices. This is followed by a few remarks of a bibliographic nature.
5.1 Symmetric skew lattices
Recall from Theorem 2.2.4 that symmetric skew lattices are characterized as skew lattices
by the following identities with both sides respectively equaling x∧y∧x and x∨y∨x:
x∧y∧(x∨y∨x) = (x∨y∨x)∧y∧x (5.1.1)
and
x∨y∨(x∧y∧x) = (x∧y∧x)∨y∨x. (5.1.2)
Matthew Spinks in [Spinks 1998] observed that these identities can be trimmed to
x∧y∧(x∨y) = (y∨x)∧y∧x, (5.1.3)
and (5.1.4)
x∨y∨(x∧y) = (y∧x)∨y∨x
since x∧y∧x ≺ both x∧y and x∨y, so that x∨y∨(x∧y) = x∨y∨(x∧y∧x)∨(x∧y) = x∨y∨(x∧y∧x) due to
x∧y∧x R x∧y holding in any skew lattice. The three other terms are handled similarly.
Symmetry is parsed as follows. A skew lattice is lower symmetric if x∨y = y∨x implies
x∧y = y∧x. Dually, a skew lattice is upper symmetric if x∧y = y∧x implies x∨y = y∨x.
Theorem 5.1.1. Lower symmetry is characterized by identity (5.1.3). Dually, upper
symmetry is characterized by identity (5.1.4). Hence both classes of skew lattices form varieties.
A skew lattice S is thus upper [lower] symmetric if and only if both S/L and S/R are.
Proof. Indeed, if x∨y = y∨x, then (5.1.3) plus absorption gives x∧y = y∧x so that lower symmetry
holds. Conversely, since x ≥ x∨y∨x, occurrences of x and (x∨y∨x)∧y∧(x∨y∨x) must ∨-commute.
Hence if a skew lattice is lower symmetric, x and (x∨y∨x)∧y∧(x∨y∨x) also ∧-commute so that
(5.1.1) and hence (5.1.3) follows. Similar remarks hold for (5.1.4) and upper symmetry. £
Consider the following pair of Hasse diagrams, each determining a right-handed skew
diamond and its left-handed dual. (The dotted lines denote the natural partial order ≥.)
166
