Page 209 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 209
VI: SKEW LATTICES IN RINGS
From the initial research into skew lattices in the 1980s, skew lattices of idempotents in
rings and their particular examples have provided fundamental ideas about the subject. The
absorption identities came from observing that nonempty sets of idempotents in a ring that were
closed under both multiplication and the circle operation (x ○ y = x + y – xy) satisfied them. The
significance of the distributive identities a∧(b ∨ c)∧a = (a∧b∧a) ∨ (a∧c∧a) and its dual was due
to the fact they hold for all skew lattices in rings, whereas say a∧(b ∨ c) = (a∧b) ∨ (a∧c) need not
hold. Occurrences of being symmetric, cancellative or categorical were observed first in the ring
context. Maximal left-regular (right-regular) multiplicative bands turned out to be maximal left-
handed (right-handed) skew lattices under • and ○; and maximal normal bands formed skew
Boolean algebras in their host rings (with ○ often replaced by ∇). In this chapter we look more
closely at skew lattices (of idempotents) in rings.
In Section 6.1 left- and right-handed skew lattice extensions of a lattice of idempotents S0
in a ring are introduced. These are the uniquely largest left-handed and right-handed skew
lattices containing S0 as a lattice section. More generally, quadratic skew lattices (with join ○)
are studied. Theorems about the center Z(S) of a quadratic skew lattice S and decompositions of
S related to its center are given. Attention is given to what occurs in matrix rings over fields.
Section 6.2 looks at ∇-bands, that is, multiplicative bands of idempotents that are closed
also under the cubic join ∇ where x∇y = (x ○ y)2 = x + y + yx – xyx – yxy. Even when ∇ is not
associative, ∇-bands share many properties of quadratic skew lattices. (Theorem 6.2.2.) ∇ is
thus a nocommutative join. When it is associative, the ∇-band is a cubic skew lattice that is
necessarily distributive and cancellative, and hence categorical and symmetric. (Quadratic skew
lattices are seen as being trivially cubic.) We consider various criteria given by Cvetko-Vah and
Leech [2007] for the ∇-operation in ∇-bands to be associative. These include the following: the
L and R congruences relative to multiplication are also ∇-congruences; ∇ is associative on every
primitive subalgebra of comparable D-classes A > B in the band. In particular, normal ∇-bands
where multiplication is normal, are seen (again) to be normal skew lattices. For many ∇-bands S
where ∇ is not associative, a closely related associative join ∨ exists making (S; ∨, •) a skew
lattice. (See Theorems 6.2.12 and 6.2.13 and the preceding discussion.) This holds for ∇-bands
in finite dimensional algebras over fields (in the ring-theoretic sense of “algebra”).
While maximal left [right] regular bands and normal bands in a ring form skew lattices
under • and ∇, maximal regular bands in rings need not form skew lattices or even ∇-bands.
Equivalently, a regular band in a ring need not generate a ∇-band under • and ∇. In Section 6.3
we consider ∇-inductive conditions. These are conditions which guarantee that a regular band
satisfying them will generate at least a ∇-band in the host ring. (See Theorems 6.3.1 and 6.3.2.)
Being normal or being left [right] regular are cases of ∇-inductive conditions. We conclude with
Cvetko-Vah’s nice theorem (6.3.5) stating that a regular band in a ring with totally ordered D-
207
From the initial research into skew lattices in the 1980s, skew lattices of idempotents in
rings and their particular examples have provided fundamental ideas about the subject. The
absorption identities came from observing that nonempty sets of idempotents in a ring that were
closed under both multiplication and the circle operation (x ○ y = x + y – xy) satisfied them. The
significance of the distributive identities a∧(b ∨ c)∧a = (a∧b∧a) ∨ (a∧c∧a) and its dual was due
to the fact they hold for all skew lattices in rings, whereas say a∧(b ∨ c) = (a∧b) ∨ (a∧c) need not
hold. Occurrences of being symmetric, cancellative or categorical were observed first in the ring
context. Maximal left-regular (right-regular) multiplicative bands turned out to be maximal left-
handed (right-handed) skew lattices under • and ○; and maximal normal bands formed skew
Boolean algebras in their host rings (with ○ often replaced by ∇). In this chapter we look more
closely at skew lattices (of idempotents) in rings.
In Section 6.1 left- and right-handed skew lattice extensions of a lattice of idempotents S0
in a ring are introduced. These are the uniquely largest left-handed and right-handed skew
lattices containing S0 as a lattice section. More generally, quadratic skew lattices (with join ○)
are studied. Theorems about the center Z(S) of a quadratic skew lattice S and decompositions of
S related to its center are given. Attention is given to what occurs in matrix rings over fields.
Section 6.2 looks at ∇-bands, that is, multiplicative bands of idempotents that are closed
also under the cubic join ∇ where x∇y = (x ○ y)2 = x + y + yx – xyx – yxy. Even when ∇ is not
associative, ∇-bands share many properties of quadratic skew lattices. (Theorem 6.2.2.) ∇ is
thus a nocommutative join. When it is associative, the ∇-band is a cubic skew lattice that is
necessarily distributive and cancellative, and hence categorical and symmetric. (Quadratic skew
lattices are seen as being trivially cubic.) We consider various criteria given by Cvetko-Vah and
Leech [2007] for the ∇-operation in ∇-bands to be associative. These include the following: the
L and R congruences relative to multiplication are also ∇-congruences; ∇ is associative on every
primitive subalgebra of comparable D-classes A > B in the band. In particular, normal ∇-bands
where multiplication is normal, are seen (again) to be normal skew lattices. For many ∇-bands S
where ∇ is not associative, a closely related associative join ∨ exists making (S; ∨, •) a skew
lattice. (See Theorems 6.2.12 and 6.2.13 and the preceding discussion.) This holds for ∇-bands
in finite dimensional algebras over fields (in the ring-theoretic sense of “algebra”).
While maximal left [right] regular bands and normal bands in a ring form skew lattices
under • and ∇, maximal regular bands in rings need not form skew lattices or even ∇-bands.
Equivalently, a regular band in a ring need not generate a ∇-band under • and ∇. In Section 6.3
we consider ∇-inductive conditions. These are conditions which guarantee that a regular band
satisfying them will generate at least a ∇-band in the host ring. (See Theorems 6.3.1 and 6.3.2.)
Being normal or being left [right] regular are cases of ∇-inductive conditions. We conclude with
Cvetko-Vah’s nice theorem (6.3.5) stating that a regular band in a ring with totally ordered D-
207
