Page 211 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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VI: Skew Lattices in Rings
6.1 Quadratic skew lattices in rings
Recall that a quadratic skew lattice in a ring R is any multiplicative band in R that is also
closed under the circle operation: x○y = x + y – xy. Letting • denote multiplication, by Theorem
2.1.7 such a band S satisfies the following absorption identities that guarantee that (S, ○, •) is
indeed a skew lattice:
x • (x ○ y) = x = (y ○ x) • x.
x ○ (x•y) = x = (y•x) ○ x.
In particular both (S, ○) and (S, •) are regular bands. We typically identify ○ as the join ∨ and •
as the meet ∧. Joins are generally higher than their constituent elements and this is certainly the
case here. In particular, in matrix rings
rank(e • f) ≤ rank(e), rank(f) ≤ rank(e ○ f).
By Theorem 2.1.9 every maximal right [left] regular multiplicative band S in a ring is closed
under ○, making (S, ○, •) is a maximal right-handed quadratic skew lattice in the ring. Our first
result is the dual of this theorem.
Theorem 6.1.1. Let R be a ring and let S be a subset of R forming a left regular band in
R under the circle operation. If S is a maximal such ○-band in S, then S is also closed under
multiplication and forms a maximal right-handed quadratic skew lattice in S.
Proof. First suppose that R has an identity 1. Then γ(x) = 1 – x induces a bijection on E(R) such
that γ(xy) = γ(x)○γ(y) and γ(x○y) = γ(x)γ(y) regardless of the outcomes also being idempotent.
Hence, if S is a maximal left regular band in R under ○, γ[S] must be a maximal left regular
multiplicative band in R and thus along with ○ forms a maximal left-handed skew lattice in R.
Clearly S = γγ[S] is indeed a maximal right-handed skew lattice in R.
If R does not have an identity, then it can be embedded in a ring Rʹ with identity 1. In Rʹ
we extend S to a maximal left regular ○-band Sʹ that forms a maximal right-handed skew lattice
in Rʹ. Hence S itself must generate a right-handed skew lattice in R. But given maximal status of
S in R, it is this skew lattice. £
Our main emphasis in this section is with classes of quadratic skew lattices. To begin,
recall that for any e ∈E(R), its R-set Re = e + eR(1 – e) is the maximal right-zero semigroup in R
containing e. Since x○y = yx on Re, Re is a maximal right-rectangular skew lattice in R. Recall
also that if e > f in E(R), then Re ∪ Rf forms a maximal right-primitive skew lattice in R. This
has an immediate generalization.
209
6.1 Quadratic skew lattices in rings
Recall that a quadratic skew lattice in a ring R is any multiplicative band in R that is also
closed under the circle operation: x○y = x + y – xy. Letting • denote multiplication, by Theorem
2.1.7 such a band S satisfies the following absorption identities that guarantee that (S, ○, •) is
indeed a skew lattice:
x • (x ○ y) = x = (y ○ x) • x.
x ○ (x•y) = x = (y•x) ○ x.
In particular both (S, ○) and (S, •) are regular bands. We typically identify ○ as the join ∨ and •
as the meet ∧. Joins are generally higher than their constituent elements and this is certainly the
case here. In particular, in matrix rings
rank(e • f) ≤ rank(e), rank(f) ≤ rank(e ○ f).
By Theorem 2.1.9 every maximal right [left] regular multiplicative band S in a ring is closed
under ○, making (S, ○, •) is a maximal right-handed quadratic skew lattice in the ring. Our first
result is the dual of this theorem.
Theorem 6.1.1. Let R be a ring and let S be a subset of R forming a left regular band in
R under the circle operation. If S is a maximal such ○-band in S, then S is also closed under
multiplication and forms a maximal right-handed quadratic skew lattice in S.
Proof. First suppose that R has an identity 1. Then γ(x) = 1 – x induces a bijection on E(R) such
that γ(xy) = γ(x)○γ(y) and γ(x○y) = γ(x)γ(y) regardless of the outcomes also being idempotent.
Hence, if S is a maximal left regular band in R under ○, γ[S] must be a maximal left regular
multiplicative band in R and thus along with ○ forms a maximal left-handed skew lattice in R.
Clearly S = γγ[S] is indeed a maximal right-handed skew lattice in R.
If R does not have an identity, then it can be embedded in a ring Rʹ with identity 1. In Rʹ
we extend S to a maximal left regular ○-band Sʹ that forms a maximal right-handed skew lattice
in Rʹ. Hence S itself must generate a right-handed skew lattice in R. But given maximal status of
S in R, it is this skew lattice. £
Our main emphasis in this section is with classes of quadratic skew lattices. To begin,
recall that for any e ∈E(R), its R-set Re = e + eR(1 – e) is the maximal right-zero semigroup in R
containing e. Since x○y = yx on Re, Re is a maximal right-rectangular skew lattice in R. Recall
also that if e > f in E(R), then Re ∪ Rf forms a maximal right-primitive skew lattice in R. This
has an immediate generalization.
209
