Page 13 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 13
PRELIMINARIES
Noncommutative variants of lattices have been studied for over sixty-five years. The first
person, to our knowledge, to engage in their extended study was Pascual Jordan who published
numerous articles over a span of seventeen years. Since then papers on this subject have been
written by various authors from a variety of perspectives.
Why study noncommutative lattices? One reason comes from an interest in axioms.
Clearly many important algebraic structures are characterized by axioms expressed as algebraic
identities. In particular, lattices are defined as algebras (L; ∨, ∧) where ∨ and ∧ are binary
operations on a set L satisfying the following pairs of associative, absorption and commutative
identities.
a ∧ (b ∧ c) = (a ∧ b) ∧ c. a ∨ (b ∨ c) = (a ∨ b) ∨ c.
a ∧ (a ∨ b) = a. a ∨ (a ∨ b) = a.
a ∧ b = b ∧ a. a ∨ b = b ∨ a.
An initial result in lattice theory is that the idempotent identities (a ∧ a = a = a ∨ a)
follow from the two absorption identities above without recourse to either the associative or
commutative identities.
If one deletes both commutative identities, then the four remaining identities are satisfied
by genuinely noncommutative structures. (Consider, e.g., any set A of size greater than 1.
Define ∨ and ∧ on A by setting a ∨ b = a = a ∧ b.) On the other hand, if one deletes the
commutative laws, and combines instead the associative identities with a modified and expanded
set of absorption identities
a ∧ (a ∨ b) = a, (b ∨ a) ∧ a = a,
a ∨ (b ∧ a) = a, (a ∧ b) ∨ a = a,
then these identities also characterize lattices. (See Theorem 1.3.2 below.) The point is
that axiomatic studies of lattices opened the door to considering noncommutative variants of
lattices. Indeed, during the period when Jordan studied noncommutative lattices, others were
studying axiomatic issues of lattices with an awareness of noncommutative possibilities. Thus an
interest in axioms combined with a curiosity about possible noncommutative variations of lattices
virtually insured that such variants would appear and then studied to some degree.
11
Noncommutative variants of lattices have been studied for over sixty-five years. The first
person, to our knowledge, to engage in their extended study was Pascual Jordan who published
numerous articles over a span of seventeen years. Since then papers on this subject have been
written by various authors from a variety of perspectives.
Why study noncommutative lattices? One reason comes from an interest in axioms.
Clearly many important algebraic structures are characterized by axioms expressed as algebraic
identities. In particular, lattices are defined as algebras (L; ∨, ∧) where ∨ and ∧ are binary
operations on a set L satisfying the following pairs of associative, absorption and commutative
identities.
a ∧ (b ∧ c) = (a ∧ b) ∧ c. a ∨ (b ∨ c) = (a ∨ b) ∨ c.
a ∧ (a ∨ b) = a. a ∨ (a ∨ b) = a.
a ∧ b = b ∧ a. a ∨ b = b ∨ a.
An initial result in lattice theory is that the idempotent identities (a ∧ a = a = a ∨ a)
follow from the two absorption identities above without recourse to either the associative or
commutative identities.
If one deletes both commutative identities, then the four remaining identities are satisfied
by genuinely noncommutative structures. (Consider, e.g., any set A of size greater than 1.
Define ∨ and ∧ on A by setting a ∨ b = a = a ∧ b.) On the other hand, if one deletes the
commutative laws, and combines instead the associative identities with a modified and expanded
set of absorption identities
a ∧ (a ∨ b) = a, (b ∨ a) ∧ a = a,
a ∨ (b ∧ a) = a, (a ∧ b) ∨ a = a,
then these identities also characterize lattices. (See Theorem 1.3.2 below.) The point is
that axiomatic studies of lattices opened the door to considering noncommutative variants of
lattices. Indeed, during the period when Jordan studied noncommutative lattices, others were
studying axiomatic issues of lattices with an awareness of noncommutative possibilities. Thus an
interest in axioms combined with a curiosity about possible noncommutative variations of lattices
virtually insured that such variants would appear and then studied to some degree.
11
