Page 232 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 232
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
6.3 The question of s-closure
While every s-band in a ring is regular as a band, not every regular band in a ring need
generate a s-band in that ring. Thus far no simple necessary and sufficient criteria for a regular
band in a ring to generate a s-band are known, though cases exist where successful generation is
guaranteed. (By contrast, every regular band S is isomorphic to a regular band Sʹ in some ring
such that Sʹ generates a cubic skew lattice in that ring.) When a s-band is generated from a
band S, it is called the s-closure of S. Our next theorem describes what must occur at any stage
in a successful generation of a s-band from a regular band in a ring.
Theorem 6.3.1. Given a regular band S in a ring R and elements e, f ∈ S, S ∪ {esf}
generates a (possibly larger) regular band Sʹ in R if and only if for all a, b, c ∈ S,
I) ea(e f)bf = eabf.
II) a(e – ef)abc(f – ef)c = a(e – ef)b(f – ef)c.
Proof. Observe first that since S is regular, ea(ef + fe – efe – fef)bf = 0 making
ea(–ef)bf = ea(fe – efe – fef)bf, so that ea(esf)bf = ea(ef)bf. Thus if the semigroup Sʹ generated
from S and esf is a regular band, then since ea, bf ≺ esf we have ea(esf)bf = eabf so that (I)
must follow.
In general, elements of the semigroup Sʹ generated from S and esf look like
a0(esf)a1(esf)a2(esf)… an–1(esf)an
with a0, a1, … , an ∈ S1 for n ≥ 1. (The n = 0 case is just a0) For Sʹ to be a regular band, both
and (esf)a(esf)b(e∇f) = (esf)ab(esf) (6.3.1)
or by (6.3.1) [a(esf)b(esf)c]2 = a(esf)b(esf)c (6.3.2)
a(esf)bcab(esf)c = a(esf)b(esf)c (6.3.2ʹ)
must hold for all a, b c ∈ S. Conversely, (6.3.1) guarantees that all elements in Sʹ have the form
a(esf)b(esf)c, (6.3.2) then guarantees that Sʹ is a band, and together with the regularity of S
they guarantee that Sʹ is a regular band. Note in passing that the regularity of S implies that
(6.3.2ʹ) can be replaced by
a(esf)abc(esf)c = a(esf)b(esf)c. (6.3.2ʺ)
Returning to (6.3.1), upon setting Λ = {e, f, fe, –efe, – fef}, we get
(esf)a(esf)b(esf) = ∑u, v ∈ Λ ua(esf)bv.
230
6.3 The question of s-closure
While every s-band in a ring is regular as a band, not every regular band in a ring need
generate a s-band in that ring. Thus far no simple necessary and sufficient criteria for a regular
band in a ring to generate a s-band are known, though cases exist where successful generation is
guaranteed. (By contrast, every regular band S is isomorphic to a regular band Sʹ in some ring
such that Sʹ generates a cubic skew lattice in that ring.) When a s-band is generated from a
band S, it is called the s-closure of S. Our next theorem describes what must occur at any stage
in a successful generation of a s-band from a regular band in a ring.
Theorem 6.3.1. Given a regular band S in a ring R and elements e, f ∈ S, S ∪ {esf}
generates a (possibly larger) regular band Sʹ in R if and only if for all a, b, c ∈ S,
I) ea(e f)bf = eabf.
II) a(e – ef)abc(f – ef)c = a(e – ef)b(f – ef)c.
Proof. Observe first that since S is regular, ea(ef + fe – efe – fef)bf = 0 making
ea(–ef)bf = ea(fe – efe – fef)bf, so that ea(esf)bf = ea(ef)bf. Thus if the semigroup Sʹ generated
from S and esf is a regular band, then since ea, bf ≺ esf we have ea(esf)bf = eabf so that (I)
must follow.
In general, elements of the semigroup Sʹ generated from S and esf look like
a0(esf)a1(esf)a2(esf)… an–1(esf)an
with a0, a1, … , an ∈ S1 for n ≥ 1. (The n = 0 case is just a0) For Sʹ to be a regular band, both
and (esf)a(esf)b(e∇f) = (esf)ab(esf) (6.3.1)
or by (6.3.1) [a(esf)b(esf)c]2 = a(esf)b(esf)c (6.3.2)
a(esf)bcab(esf)c = a(esf)b(esf)c (6.3.2ʹ)
must hold for all a, b c ∈ S. Conversely, (6.3.1) guarantees that all elements in Sʹ have the form
a(esf)b(esf)c, (6.3.2) then guarantees that Sʹ is a band, and together with the regularity of S
they guarantee that Sʹ is a regular band. Note in passing that the regularity of S implies that
(6.3.2ʹ) can be replaced by
a(esf)abc(esf)c = a(esf)b(esf)c. (6.3.2ʺ)
Returning to (6.3.1), upon setting Λ = {e, f, fe, –efe, – fef}, we get
(esf)a(esf)b(esf) = ∑u, v ∈ Λ ua(esf)bv.
230
