Page 60 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 60
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Applying the identity e∇f∇e = e + f – fef (replace g by e in the above calculation of e∇f∇g and
then simplify) we get
(e∇f∇e) (e∇g∇e) = (e + f – fef)(e + g – geg)
= e + eg – eg + fe + fg – fgeg – fe – feg + fefgeg
= e + fg – feg = e + fg – fgefg = e∇(fg)∇e.
Thus S is also distributive. £
The above theorem has several consequences of significance for skew lattices.
Theorem 2.3.7. If R is a ring for which E(R) is closed under multiplication, then E(R) is
a normal skew lattice under ∇ and •. In particular, E(R) is closed under multiplication if R
satisfies the identity, abcd = acbd. E(R) is a Boolean lattice when R has an identity, 1.
Proof. We prove the last statement first. So let R be a ring with identity 1 for which E(R) is
closed under multiplication. Then for all e ∈ E(R), 1 – e ∈ E(R) also with e(1 – e) = 0. Since
E(R) is a band, for all f ∈ E(R), ef(1 – e) = 0 also (The Clifford-McLean Theorem). Thus ef = efe
holds for all e, f ∈E(R). Likewise, from (1 – e)fe = 0 we get fe = efe. Thus E(R) is commutative
under multiplication, forcing (E(R); ○, •, 1, 0) to be a Boolean lattice.
Suppose next that R is any ring for which E(R) is a multiplicative band. Given e ∈ E(R),
eE(R)e = E(eRe) is a sub-band that is necessarily commutative since e is the identity of eRe.
Thus E(R) is a normal band. Moreover, for any e, f ∈ E(R),
(e∇f)2 = (e + f + fe – efe – fef)2 = e + (fe + f – fef) + fe – efe – fe = (e∇f)
where each term in the third expression is a reduction of x(e∇f) where x is one of the terms in
e∇f. Thus E(R) is closed under ∇, and being normal forms a skew lattice under ∇ and •.
Finally, if R satisfies the identity abcd = acbd, then (ef)2 = efef = eeff = ef for any pair of
idempotents e and f. Thus E(R) is indeed a multiplicative band and the theorem follows. £
Corollary 2.3.8. A normal band in a ring generates a normal skew lattice under ∇ and
•. A maximal normal band in a ring R thus forms a normal skew lattice under ∇ and •.
Proof. For such a band B, the subring S generated from B satisfy xyzw = xzyw. Thus B ⊆ E(S)
which is a normal band forming a skew lattice under ∇ and •. By maximality, B = E(S) . £
Example 2.3.3. If R is the semigroup ring A[B] with A a commutative ring and B a
normal band, then E(R) is a normal skew lattice. £
Example 2.3.4. [Karin Cvetko-Vah] Consider the band of all (n+2)×(n+2) matrices of
the form:
58
Applying the identity e∇f∇e = e + f – fef (replace g by e in the above calculation of e∇f∇g and
then simplify) we get
(e∇f∇e) (e∇g∇e) = (e + f – fef)(e + g – geg)
= e + eg – eg + fe + fg – fgeg – fe – feg + fefgeg
= e + fg – feg = e + fg – fgefg = e∇(fg)∇e.
Thus S is also distributive. £
The above theorem has several consequences of significance for skew lattices.
Theorem 2.3.7. If R is a ring for which E(R) is closed under multiplication, then E(R) is
a normal skew lattice under ∇ and •. In particular, E(R) is closed under multiplication if R
satisfies the identity, abcd = acbd. E(R) is a Boolean lattice when R has an identity, 1.
Proof. We prove the last statement first. So let R be a ring with identity 1 for which E(R) is
closed under multiplication. Then for all e ∈ E(R), 1 – e ∈ E(R) also with e(1 – e) = 0. Since
E(R) is a band, for all f ∈ E(R), ef(1 – e) = 0 also (The Clifford-McLean Theorem). Thus ef = efe
holds for all e, f ∈E(R). Likewise, from (1 – e)fe = 0 we get fe = efe. Thus E(R) is commutative
under multiplication, forcing (E(R); ○, •, 1, 0) to be a Boolean lattice.
Suppose next that R is any ring for which E(R) is a multiplicative band. Given e ∈ E(R),
eE(R)e = E(eRe) is a sub-band that is necessarily commutative since e is the identity of eRe.
Thus E(R) is a normal band. Moreover, for any e, f ∈ E(R),
(e∇f)2 = (e + f + fe – efe – fef)2 = e + (fe + f – fef) + fe – efe – fe = (e∇f)
where each term in the third expression is a reduction of x(e∇f) where x is one of the terms in
e∇f. Thus E(R) is closed under ∇, and being normal forms a skew lattice under ∇ and •.
Finally, if R satisfies the identity abcd = acbd, then (ef)2 = efef = eeff = ef for any pair of
idempotents e and f. Thus E(R) is indeed a multiplicative band and the theorem follows. £
Corollary 2.3.8. A normal band in a ring generates a normal skew lattice under ∇ and
•. A maximal normal band in a ring R thus forms a normal skew lattice under ∇ and •.
Proof. For such a band B, the subring S generated from B satisfy xyzw = xzyw. Thus B ⊆ E(S)
which is a normal band forming a skew lattice under ∇ and •. By maximality, B = E(S) . £
Example 2.3.3. If R is the semigroup ring A[B] with A a commutative ring and B a
normal band, then E(R) is a normal skew lattice. £
Example 2.3.4. [Karin Cvetko-Vah] Consider the band of all (n+2)×(n+2) matrices of
the form:
58
