Page 165 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 165
V: FURTHER TOPICS IN SKEW LATTICES
Skew lattices were introduced in Section 1.3 and then studied in Chapter 2. Two main
sources of initial examples were encountered:
i) Partial function algebras P(A, B) that form noncommutative variants of Boolean
algebras defined on power sets P(A). (Example 2.6.1.)
ii) Sets of idempotents in rings that are closed under multiplication and the (often
called) circle operation: e ¡ f = e + f – ef. (Theorem 2.1.7.)
Both types of examples satisfy the primary conditions for an algebraic system (S; ∨, ∧) to be a
skew lattice: ∨ and ∧ are associative idempotent binary operations on S satisfying the absorption
identities x∧(x ∨ y) = x = (y ∨ x)∧x and x∨(x ∧ y) = x = (y ∧ x)∨x that guarantee the basic
dualities, x∧y = y iff x∨y = x and x∧y = x iff x∨y = y. These examples are also a source of
optional conditions that any skew lattice might satisfy.
In both cases commutation was symmetric in that x∧y = y∧x iff x∨y = y∨x, thus making
instances of commutation unambiguous. Also in both cases the skew lattices were distributive in
that both x∧(y ∨ z)∧x = (x∧y∧x) ∨ (x∧z∧x) and x∨(y ∧ z)∨x = (x∨y∨x) ∧ (x∨z∨x) hold. For partial
function algebras a stronger pair of identities hold: x∧(y ∨ z) = (x∧y) ∨ (x∧z) and
(y ∨ z)∧x = (y∧x) ∨ (z∧x). In Section 2.3 we saw that a chain of implications holds for all skew
lattices:
∀x, y, z ∈ S, x∧(y ∨ z) = (x∧y) ∨ (x∧z) & (y ∨ z)∧x = (y∧x) ∨ (z∧x)
⇒ ∀x, y, z, w ∈ S, x∧(y ∨ z)∧w = (x∧y∧w) ∨ (x∧z∧w)
⇒ ∀x, y, z ∈ S, x∧(y ∨ z)∧x = (x∧y∧x) ∨ (x∧z∧x)
& x∨(y ∧ z)∨x = (x∨y∨x) ∧ (x∨z∨x).
In general these implications are strict, but if the skew lattice is symmetric the converse of the
first implication holds; likewise if the skew lattice is normal in that x∧y∧z∧w = x∧z∧y∧w, the
converse of the second implication holds. We also observed that x∧(y ∨ z)∧x = (x∧y∧x) ∨ (x∧z∧x)
and its dual are not equivalent, but given the assumption of symmetry we will prove they are.
In addition, both types of algebras are categorical in that nonempty composites ψ¡ϕ of
successive coset bijections from D-classes to say lower D-classes are also coset bijections.
Indeed partial function algebras (and normal skew lattices in general) are strictly categorical in
that all composites of coset bijections between successive D-classes are also nonempty.
In this chapter we study these properties in greater detail.
163
Skew lattices were introduced in Section 1.3 and then studied in Chapter 2. Two main
sources of initial examples were encountered:
i) Partial function algebras P(A, B) that form noncommutative variants of Boolean
algebras defined on power sets P(A). (Example 2.6.1.)
ii) Sets of idempotents in rings that are closed under multiplication and the (often
called) circle operation: e ¡ f = e + f – ef. (Theorem 2.1.7.)
Both types of examples satisfy the primary conditions for an algebraic system (S; ∨, ∧) to be a
skew lattice: ∨ and ∧ are associative idempotent binary operations on S satisfying the absorption
identities x∧(x ∨ y) = x = (y ∨ x)∧x and x∨(x ∧ y) = x = (y ∧ x)∨x that guarantee the basic
dualities, x∧y = y iff x∨y = x and x∧y = x iff x∨y = y. These examples are also a source of
optional conditions that any skew lattice might satisfy.
In both cases commutation was symmetric in that x∧y = y∧x iff x∨y = y∨x, thus making
instances of commutation unambiguous. Also in both cases the skew lattices were distributive in
that both x∧(y ∨ z)∧x = (x∧y∧x) ∨ (x∧z∧x) and x∨(y ∧ z)∨x = (x∨y∨x) ∧ (x∨z∨x) hold. For partial
function algebras a stronger pair of identities hold: x∧(y ∨ z) = (x∧y) ∨ (x∧z) and
(y ∨ z)∧x = (y∧x) ∨ (z∧x). In Section 2.3 we saw that a chain of implications holds for all skew
lattices:
∀x, y, z ∈ S, x∧(y ∨ z) = (x∧y) ∨ (x∧z) & (y ∨ z)∧x = (y∧x) ∨ (z∧x)
⇒ ∀x, y, z, w ∈ S, x∧(y ∨ z)∧w = (x∧y∧w) ∨ (x∧z∧w)
⇒ ∀x, y, z ∈ S, x∧(y ∨ z)∧x = (x∧y∧x) ∨ (x∧z∧x)
& x∨(y ∧ z)∨x = (x∨y∨x) ∧ (x∨z∨x).
In general these implications are strict, but if the skew lattice is symmetric the converse of the
first implication holds; likewise if the skew lattice is normal in that x∧y∧z∧w = x∧z∧y∧w, the
converse of the second implication holds. We also observed that x∧(y ∨ z)∧x = (x∧y∧x) ∨ (x∧z∧x)
and its dual are not equivalent, but given the assumption of symmetry we will prove they are.
In addition, both types of algebras are categorical in that nonempty composites ψ¡ϕ of
successive coset bijections from D-classes to say lower D-classes are also coset bijections.
Indeed partial function algebras (and normal skew lattices in general) are strictly categorical in
that all composites of coset bijections between successive D-classes are also nonempty.
In this chapter we study these properties in greater detail.
163
