Page 236 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 236
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Theorem 6.4.2 If a ring R is idempotent-closed, then:
i) E(R) is a normal band under multiplication.
ii) E(R) is also closed under ∇ which is idempotent and associative on E(R).
iii) (E(R); ∇, •) is a strongly distributive (distributive, normal and symmetric)
skew lattice.
iv) Upon setting e \ f = e – efe, (E(R); ∇, •, \, 0) is a skew Boolean algebra.
Proof. (i) Given e ∈ E(R), the principal subring eRe has an identity e and thus eE(R)e = E(eRe)
is commutative by the above proposition. Hence E(R) satisfies (c) and is normal. (ii) and (iii)
follow from Theorems 2.3.6 and 2.3.7. Finally, let e\f as e – efe in R. As such, e\f is the
complement of e∧f∧e in E(eRe) and satisfies the characterizing identities in Section 4.1:
e = (e∧f∧e) ∨ (e \ f) and (e∧f∧e) ∧ (e \ f) = 0. £
Recall that E(R) is partially ordered by e ≥ f if ef = f = fe with e > f denoting e ≥ f ≠ e.
Recall also that an idempotent e > 0 is primitive if no idempotent f exists such that e > f > 0. E(R)
is 0-primitive if all of its non-0 elements are primitive. In this case, M(R) denotes E(R)\{0}.
Recall that a band is rectangular if efe = e, or equivalently, efg = eg holds. A 0-rectangular band
M0 consist of a rectangular band M and a distinct element 0 ∉M, so that x0 = 0 = 0x. Abstractly
viewed, every primitive skew Boolean algebra has operations induced from a 0-rectangular band
M0: given x, y ∈ M, 0∧x = 0 = x∧0, 0∨x = x = x∨0, x∧y = xy = y∨x, x\0 = x and 0\x = 0 = x\y.
In this simple manner, 0-rectangular bands are in 1-1 correspondence to the class of primitive
skew Boolean algebras.
Theorem 6.4.2 If E(R) is 0-primitive and closed under multiplication, then under
multiplication, E(R) is a 0-rectangular band with e∇f = fe on M(R).
Proof. Given the assumptions on E(R), let e ≠ f in M(R). Then e ≥ efe ≥ 0 in E(R) so that efe is
either 0 or e. If efe = 0, then so are fef, ef and fe (since, e.g., ef = efef = 0). It follows that e + f is
an idempotent greater than either e or f and hence not primitive. Thus efe = e for all e, f in M(R)
making M(R) a rectangular band with e∇f = e + f + fe – efe – fef reducing to fe. £
Idempotent-dominated rings
How does E(R) being multiplicatively closed affect the behavior of R? To answer this
one is pressed to find reasonable assumptions connecting E(R) to all of R in order to obtain
consequences for all of R. To do so we begin with the following lemma.
Theorem 6.4.3 Given a ring R, set Γ(R) ={x⎪ex = x = xf, for some e, f ∈ E(R)}. Then:
i) Γ(R) is a multiplicatively closed set containing E(R) that also has negatives:
x ∈ Γ(R) implies −x ∈ Γ(R).
ii) When E(R) is idempotent-closed, Γ(R) = ∪{eRe⎪e ∈ E(R)}.
iii) In general, the set Q(R) of all finite sums of elements in Γ(R) is the smallest ideal
in R containing E(R).
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Theorem 6.4.2 If a ring R is idempotent-closed, then:
i) E(R) is a normal band under multiplication.
ii) E(R) is also closed under ∇ which is idempotent and associative on E(R).
iii) (E(R); ∇, •) is a strongly distributive (distributive, normal and symmetric)
skew lattice.
iv) Upon setting e \ f = e – efe, (E(R); ∇, •, \, 0) is a skew Boolean algebra.
Proof. (i) Given e ∈ E(R), the principal subring eRe has an identity e and thus eE(R)e = E(eRe)
is commutative by the above proposition. Hence E(R) satisfies (c) and is normal. (ii) and (iii)
follow from Theorems 2.3.6 and 2.3.7. Finally, let e\f as e – efe in R. As such, e\f is the
complement of e∧f∧e in E(eRe) and satisfies the characterizing identities in Section 4.1:
e = (e∧f∧e) ∨ (e \ f) and (e∧f∧e) ∧ (e \ f) = 0. £
Recall that E(R) is partially ordered by e ≥ f if ef = f = fe with e > f denoting e ≥ f ≠ e.
Recall also that an idempotent e > 0 is primitive if no idempotent f exists such that e > f > 0. E(R)
is 0-primitive if all of its non-0 elements are primitive. In this case, M(R) denotes E(R)\{0}.
Recall that a band is rectangular if efe = e, or equivalently, efg = eg holds. A 0-rectangular band
M0 consist of a rectangular band M and a distinct element 0 ∉M, so that x0 = 0 = 0x. Abstractly
viewed, every primitive skew Boolean algebra has operations induced from a 0-rectangular band
M0: given x, y ∈ M, 0∧x = 0 = x∧0, 0∨x = x = x∨0, x∧y = xy = y∨x, x\0 = x and 0\x = 0 = x\y.
In this simple manner, 0-rectangular bands are in 1-1 correspondence to the class of primitive
skew Boolean algebras.
Theorem 6.4.2 If E(R) is 0-primitive and closed under multiplication, then under
multiplication, E(R) is a 0-rectangular band with e∇f = fe on M(R).
Proof. Given the assumptions on E(R), let e ≠ f in M(R). Then e ≥ efe ≥ 0 in E(R) so that efe is
either 0 or e. If efe = 0, then so are fef, ef and fe (since, e.g., ef = efef = 0). It follows that e + f is
an idempotent greater than either e or f and hence not primitive. Thus efe = e for all e, f in M(R)
making M(R) a rectangular band with e∇f = e + f + fe – efe – fef reducing to fe. £
Idempotent-dominated rings
How does E(R) being multiplicatively closed affect the behavior of R? To answer this
one is pressed to find reasonable assumptions connecting E(R) to all of R in order to obtain
consequences for all of R. To do so we begin with the following lemma.
Theorem 6.4.3 Given a ring R, set Γ(R) ={x⎪ex = x = xf, for some e, f ∈ E(R)}. Then:
i) Γ(R) is a multiplicatively closed set containing E(R) that also has negatives:
x ∈ Γ(R) implies −x ∈ Γ(R).
ii) When E(R) is idempotent-closed, Γ(R) = ∪{eRe⎪e ∈ E(R)}.
iii) In general, the set Q(R) of all finite sums of elements in Γ(R) is the smallest ideal
in R containing E(R).
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