Page 142 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 142
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
In particular, a partial function algebra P(A, B) is completely reducible and thus must
have arbitrary intersections. Indeed, upon viewing partial functions from A to B as subsets of A ×
B, intersections in our sense are just intersections of subsets: f∩g = f ∩ g.
In particular, a skew Boolean algebra S for which S/D is finite has arbitrary intersections.
Example 4.4.4. Let R be a C-ring: an associative ring such that every for each x ∈R a
central idempotent C(x) exists such that xC(x) = x and C(x) is the least central idempotent with
respect to this property. Cornish [1980] showed that a left-handed skew Boolean algebra can be
defined from any C-ring upon setting: x∧y = xC(y), x∨y = x + y – xC(y) and x \ y = x – xC(y).
The intersection of a pair of elements is given by x∩y = [1 – C(x – y)]x.
Example 4.4.5. Kudryavtseva and Leech [2016] have shown that free skew Boolean
algebras have finite intersections. This is discussed later in this section.
Example 4.4.6. Given any maximal normal band S in a ring R, we have seen that S
forms a skew Boolean algebra upon setting e∧f = ef , e∨f = e∇f and e f = e – efe. They have
intersections if the ring is semisimple and Artinian, or in particular, is a matrix ring over a field.
Example 4.4.7. (Counterexample) Let * be the set of natural numbers and let S denote
the subset of P(*, {0, 1}) consisting of all partial functions having domains that are either finite
or cofinite (in that the complement in * is finite). It is easily verified that S is closed under the
skew Boolean operations, ∨ and ∧ and \. Thus S is a skew Boolean algebra. Clearly f∩g does not
exist for the partial functions f, g ∈ S, both with full domain *, but with
f(n) = 0 and g(n) = ⎧0 if n is even .
⎨⎩1 if n is odd
Recall that an ideal is a nonempty subset I of a skew lattice S such that for all x, y ∈I and
all z ∈S, x∨y, x∧z and z∧x all lie in I. While arbitrary sub-skew Boolean algebras of skew
Boolean ∩-algebras need not have finite intersections, ideals of skew Boolean ∩-algebras do.
Extending Theorem 4.1.4, we have:
Theorem 4.4.5. Given a principal ideal 〈a〉 in a skew Boolean algebra S and its
associated annihilator ideal ann(a), S has finite intersections if and only if both 〈a〉 and ann(a)
have finite intersections, in which case the map µ: 〈a〉 × ann(a) → S defined by µ(x, y) = x∨y is an
isomorphism of skew Boolean ∩-algebras. In particular, given x, y in 〈a〉 and u, v in ann(a),
(x ∨ u) ∩ (y ∨ v) = (x ∩ y) ∨ (u ∩ v). £
As a consequence we have:
140
In particular, a partial function algebra P(A, B) is completely reducible and thus must
have arbitrary intersections. Indeed, upon viewing partial functions from A to B as subsets of A ×
B, intersections in our sense are just intersections of subsets: f∩g = f ∩ g.
In particular, a skew Boolean algebra S for which S/D is finite has arbitrary intersections.
Example 4.4.4. Let R be a C-ring: an associative ring such that every for each x ∈R a
central idempotent C(x) exists such that xC(x) = x and C(x) is the least central idempotent with
respect to this property. Cornish [1980] showed that a left-handed skew Boolean algebra can be
defined from any C-ring upon setting: x∧y = xC(y), x∨y = x + y – xC(y) and x \ y = x – xC(y).
The intersection of a pair of elements is given by x∩y = [1 – C(x – y)]x.
Example 4.4.5. Kudryavtseva and Leech [2016] have shown that free skew Boolean
algebras have finite intersections. This is discussed later in this section.
Example 4.4.6. Given any maximal normal band S in a ring R, we have seen that S
forms a skew Boolean algebra upon setting e∧f = ef , e∨f = e∇f and e f = e – efe. They have
intersections if the ring is semisimple and Artinian, or in particular, is a matrix ring over a field.
Example 4.4.7. (Counterexample) Let * be the set of natural numbers and let S denote
the subset of P(*, {0, 1}) consisting of all partial functions having domains that are either finite
or cofinite (in that the complement in * is finite). It is easily verified that S is closed under the
skew Boolean operations, ∨ and ∧ and \. Thus S is a skew Boolean algebra. Clearly f∩g does not
exist for the partial functions f, g ∈ S, both with full domain *, but with
f(n) = 0 and g(n) = ⎧0 if n is even .
⎨⎩1 if n is odd
Recall that an ideal is a nonempty subset I of a skew lattice S such that for all x, y ∈I and
all z ∈S, x∨y, x∧z and z∧x all lie in I. While arbitrary sub-skew Boolean algebras of skew
Boolean ∩-algebras need not have finite intersections, ideals of skew Boolean ∩-algebras do.
Extending Theorem 4.1.4, we have:
Theorem 4.4.5. Given a principal ideal 〈a〉 in a skew Boolean algebra S and its
associated annihilator ideal ann(a), S has finite intersections if and only if both 〈a〉 and ann(a)
have finite intersections, in which case the map µ: 〈a〉 × ann(a) → S defined by µ(x, y) = x∨y is an
isomorphism of skew Boolean ∩-algebras. In particular, given x, y in 〈a〉 and u, v in ann(a),
(x ∨ u) ∩ (y ∨ v) = (x ∩ y) ∨ (u ∩ v). £
As a consequence we have:
140
