Page 124 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 124
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Proof. Let S be a normal, symmetric skew lattice with 0 such that S/D is a generalized Boolean
lattice. Then S is distributive by Theorem 2.3.2. Since each ⎡e⎤ is a Boolean lattice, \ is
implicitly determined by (ii). The converse is clear. £
Examples 4.1.1. The following skew lattices are examples of skew Boolean algebras:
(a) Maximal normal bands in rings form skew Boolean algebras upon setting
e∧f = ef , e∨f = e∇f and e\f = e – efe. (See Cvetko-Vah and Leech, [2011] and
[2012].)
(b) Any partial function set P(A, B) with ∨ and ∧ defined as before and with 0 = ∅
and f \ g = f⎮F \ G where F and G are the supports in A for f and g respectively
(c) More generally, any ring of partial functions provided the underlying lattice of
subsets of A is a generalized Boolean lattice and contains the empty set.
(d) Given a rectangular skew lattice D, a primitive skew Boolean algebra is formed
by D0 upon setting x \ y = x if y = 0, but 0 otherwise. £
Theorem 4.1.3. Skew Boolean algebras satisfy:
(iii) e \ f = e \ (e∧f∧e).
(iv) e \ (f ∨ g) = (e \ f) ∧ (e \ g) and e \ (f ∧ g) = (e \ f) ∨ (e \ g).
(v) e \ (e \ f) = e ∧ f ∧ e.
Proof. These identities are immediate consequences of each inner ideal ⎡e⎤ being a Boolean
lattice. £
Given a skew lattice S, an ideal of S is a subset I of S such that given x, y in I, and z in S
x∨y, z∧x and x∧z are in I. Given any element a in a skew lattice S, the principal ideal of a is the
set (a) = {x ∈ S⎮x ≺ a}. Clearly, x ∈(a) if and only if x ≺ b for all b ∈Da. If S has a zero element
0, the annihilator of a is the set ann(a) = {x ∈ S⎮x ∧ a = 0}. Due to Theorem 2.1.2, x ∈ ann(a) if
and only if x ∧ b = 0 = b ∧ x for all b ∈Da. If the skew lattice is distributive, then ann(a) is easily
seen to be an ideal. Clearly (a) and ann(a) can also be parameterized by the relevant D-class
A = Da as (A) and ann(A) respectively since due to Theorem 2.1.2, any b in Da induces the same
pair of sets,. When S is a skew Boolean algebra, the situation can be sharpened to give a
decomposition of primary importance in understanding these algebras.
Theorem 4.1.4. Given a D-class A of a skew Boolean algebra S, then:
(i) Both (A) and ann(A) are ideals of S.
(ii) All elements of (A) commute with all elements of ann(A).
(iii) In particular, for all u ∈(A) and all v ∈ ann(A), u∧v = 0 = v∧u.
(iv) The map µ: (A) × ann(A) → S defined by µ(e1, e2) = e1 ∨ e2 is an isomorphism.
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Proof. Let S be a normal, symmetric skew lattice with 0 such that S/D is a generalized Boolean
lattice. Then S is distributive by Theorem 2.3.2. Since each ⎡e⎤ is a Boolean lattice, \ is
implicitly determined by (ii). The converse is clear. £
Examples 4.1.1. The following skew lattices are examples of skew Boolean algebras:
(a) Maximal normal bands in rings form skew Boolean algebras upon setting
e∧f = ef , e∨f = e∇f and e\f = e – efe. (See Cvetko-Vah and Leech, [2011] and
[2012].)
(b) Any partial function set P(A, B) with ∨ and ∧ defined as before and with 0 = ∅
and f \ g = f⎮F \ G where F and G are the supports in A for f and g respectively
(c) More generally, any ring of partial functions provided the underlying lattice of
subsets of A is a generalized Boolean lattice and contains the empty set.
(d) Given a rectangular skew lattice D, a primitive skew Boolean algebra is formed
by D0 upon setting x \ y = x if y = 0, but 0 otherwise. £
Theorem 4.1.3. Skew Boolean algebras satisfy:
(iii) e \ f = e \ (e∧f∧e).
(iv) e \ (f ∨ g) = (e \ f) ∧ (e \ g) and e \ (f ∧ g) = (e \ f) ∨ (e \ g).
(v) e \ (e \ f) = e ∧ f ∧ e.
Proof. These identities are immediate consequences of each inner ideal ⎡e⎤ being a Boolean
lattice. £
Given a skew lattice S, an ideal of S is a subset I of S such that given x, y in I, and z in S
x∨y, z∧x and x∧z are in I. Given any element a in a skew lattice S, the principal ideal of a is the
set (a) = {x ∈ S⎮x ≺ a}. Clearly, x ∈(a) if and only if x ≺ b for all b ∈Da. If S has a zero element
0, the annihilator of a is the set ann(a) = {x ∈ S⎮x ∧ a = 0}. Due to Theorem 2.1.2, x ∈ ann(a) if
and only if x ∧ b = 0 = b ∧ x for all b ∈Da. If the skew lattice is distributive, then ann(a) is easily
seen to be an ideal. Clearly (a) and ann(a) can also be parameterized by the relevant D-class
A = Da as (A) and ann(A) respectively since due to Theorem 2.1.2, any b in Da induces the same
pair of sets,. When S is a skew Boolean algebra, the situation can be sharpened to give a
decomposition of primary importance in understanding these algebras.
Theorem 4.1.4. Given a D-class A of a skew Boolean algebra S, then:
(i) Both (A) and ann(A) are ideals of S.
(ii) All elements of (A) commute with all elements of ann(A).
(iii) In particular, for all u ∈(A) and all v ∈ ann(A), u∧v = 0 = v∧u.
(iv) The map µ: (A) × ann(A) → S defined by µ(e1, e2) = e1 ∨ e2 is an isomorphism.
122
