Page 43 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 43
SKEW LATTICES
We take a closer look at skew lattices, i.e., at algebras (S; ∨, ∧) with associative binary
operations ∨ and ∧ satisfying the absorption identities:
B1: a ∧ (a ∨ b) = a. C1: a ∨ (a ∧ b) = a.
B2: (b ∨ a) ∧ a = a. C2: (b ∧ a) ∨ a = a.
By Theorem 1.3.4, both ∨ and ∧ are idempotent: x ∨ x = x = x ∧ x. Equivalently, skew lattices
are characterized as the double bands (S; ∨, ∧) satisfying the dualities:
a ∨ b = a if and only if a ∧ b = b.
a ∨ b = b if and only if a ∧ b = a.
In this chapter a basic theory for these algebras is developed. For every statement about skew
lattices, clearly a parallel statement holds for skew* lattices. Thus, from this point on, the latter
will not be mentioned until more general structures are considered in Chapter 3.
We begin in Section 1 with some fundamental results about skew lattices. Of particular
importance are two core structural results for skew lattices: analogues of the Clifford-McLean
Theorem and the Kimura Factorization Theorem (Theorems 2.1.2 and 2.1.5), given originally for
bands and regular bands respectively (Theorem 1.2.6 and what follows). We also initiate our
study of skew lattices of idempotents in rings, a source of both examples and conceptual
motivation, in Theorems 2.1.7 and 2.1.9. (In this case, ∧ and ∨ are given first as e∧f = ef and
e∨f = e +f – ef.) Due to the latter theorem, a band can be embedded into some skew lattice as a
sub-band of its ∧-reduct (or of its ∨-reduct) if and only if the band itself is regular (Theorem
2.1.10).
In Section 2 we consider the role of commutativity in skew lattices. The center of a skew
lattice S, Z(S) = {e ∈ S⎪e∨x = x∨e and e∧x = x∧e for all x ∈ S} is characterized in Theorem 2.2.2
as the sublattice of all elements that form singleton D-classes. We look at the important property
of symmetry (a∨b = b∨a iff a∧b = b∧a) and some of its consequences. These include Theorem
2.2.10 that asserts that any symmetric skew lattice with a countable maximal lattice image has a
lattice section (that is, a sublattice meeting each D-class of S at a single point). Example 2.2.2
41
We take a closer look at skew lattices, i.e., at algebras (S; ∨, ∧) with associative binary
operations ∨ and ∧ satisfying the absorption identities:
B1: a ∧ (a ∨ b) = a. C1: a ∨ (a ∧ b) = a.
B2: (b ∨ a) ∧ a = a. C2: (b ∧ a) ∨ a = a.
By Theorem 1.3.4, both ∨ and ∧ are idempotent: x ∨ x = x = x ∧ x. Equivalently, skew lattices
are characterized as the double bands (S; ∨, ∧) satisfying the dualities:
a ∨ b = a if and only if a ∧ b = b.
a ∨ b = b if and only if a ∧ b = a.
In this chapter a basic theory for these algebras is developed. For every statement about skew
lattices, clearly a parallel statement holds for skew* lattices. Thus, from this point on, the latter
will not be mentioned until more general structures are considered in Chapter 3.
We begin in Section 1 with some fundamental results about skew lattices. Of particular
importance are two core structural results for skew lattices: analogues of the Clifford-McLean
Theorem and the Kimura Factorization Theorem (Theorems 2.1.2 and 2.1.5), given originally for
bands and regular bands respectively (Theorem 1.2.6 and what follows). We also initiate our
study of skew lattices of idempotents in rings, a source of both examples and conceptual
motivation, in Theorems 2.1.7 and 2.1.9. (In this case, ∧ and ∨ are given first as e∧f = ef and
e∨f = e +f – ef.) Due to the latter theorem, a band can be embedded into some skew lattice as a
sub-band of its ∧-reduct (or of its ∨-reduct) if and only if the band itself is regular (Theorem
2.1.10).
In Section 2 we consider the role of commutativity in skew lattices. The center of a skew
lattice S, Z(S) = {e ∈ S⎪e∨x = x∨e and e∧x = x∧e for all x ∈ S} is characterized in Theorem 2.2.2
as the sublattice of all elements that form singleton D-classes. We look at the important property
of symmetry (a∨b = b∨a iff a∧b = b∧a) and some of its consequences. These include Theorem
2.2.10 that asserts that any symmetric skew lattice with a countable maximal lattice image has a
lattice section (that is, a sublattice meeting each D-class of S at a single point). Example 2.2.2
41
