Page 233 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 233
Skew Lattices in Rings
Except for the two cases, ea(esf)bf and fa(esf)be, all of the ua(esf)bv terms reduce to uabv
terms due to the regularity of S. For example,
efa(esf)bfef = efae(esf)bfef = efaebfef = efabfef.
Thus (6.3.1) reduces first to ea(esf)bf + fa(esf)be = eabf + fabe and then to
ea(e f)bf + fa(f e)be = eabf + fabe. (6.3.1ʹ)
Clearly (I) implies (6.3.1ʹ) and when Sʹ is regular, (I) holds so that (6.3.1ʹ) follows. Thus (6.3.1ʹ)
can be replaced by the stronger assertion, (I). Next, expanding (6.3.2ʺ) gives
a(e + f + fe – efe – fef)abc(e + f + fe – efe – fef)c
= a(e + f + fe – efe – fef)b(e + f + fe – efe – fef)c.
Using the identity aeabcec = aebec holding for all regular bands, multiplying out the left side of
(6.3.2ʺ) creates a number of terms that reduce immediately to the corresponding terms on the
right. Of course we have, aeabcec = aeaebecec = aebec and afabcfc = afafbfcfc = afbfec, but
also cases such as
afe(abc)fefc = afef(abc)fefc = afef(b)fefc = afe(b)fefc,
and
aef(abc)fefc = aef(abc)efc = aef(b)efc = aef(b)fefc.
Where this matching fails, (6.3.2ʺ) first reduces to
a(e)abc(f)c + a(e)abc(fe – efe – fef)c + a(fe – efe – fef)abc(f)c
= a(e)b(f)c + a(e)b(fe – efe – fef)c + a(fe – efe – fef)b(f)c.
Using regularity again on products involving the underlined terms, the above reduces to
a(e)abc(f – ef)c – a(ef)abc(f)c = a(e)b(f – ef)c – a(ef)b(f)c
or
a(e)abc(f)c – a(e)abc(ef)c – a(ef)abc(f)c = a(e)b(f)c – a(e)b(ef)c – a(ef)b(f)c.
Regularity gives a(ef)abc(ef)c = a(ef)a(ef)bc(ef)c = a(ef)b(ef)c. (6.3.2) is thus reduced to (II). £
A successful generation of a s-band from a regular band requires that at each stage of
generation the new regular band Sʹ also satisfies (I) and (II) in Theorem 6.3.1 for all
a, b, c, e, f ∈ Sʹ. While (I) and (II) generally need not be passed on to later bands, here is a
strategy for generating s-bands. To this end, a condition C potentially satisfied by regular bands
in rings is s-inductive if
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Except for the two cases, ea(esf)bf and fa(esf)be, all of the ua(esf)bv terms reduce to uabv
terms due to the regularity of S. For example,
efa(esf)bfef = efae(esf)bfef = efaebfef = efabfef.
Thus (6.3.1) reduces first to ea(esf)bf + fa(esf)be = eabf + fabe and then to
ea(e f)bf + fa(f e)be = eabf + fabe. (6.3.1ʹ)
Clearly (I) implies (6.3.1ʹ) and when Sʹ is regular, (I) holds so that (6.3.1ʹ) follows. Thus (6.3.1ʹ)
can be replaced by the stronger assertion, (I). Next, expanding (6.3.2ʺ) gives
a(e + f + fe – efe – fef)abc(e + f + fe – efe – fef)c
= a(e + f + fe – efe – fef)b(e + f + fe – efe – fef)c.
Using the identity aeabcec = aebec holding for all regular bands, multiplying out the left side of
(6.3.2ʺ) creates a number of terms that reduce immediately to the corresponding terms on the
right. Of course we have, aeabcec = aeaebecec = aebec and afabcfc = afafbfcfc = afbfec, but
also cases such as
afe(abc)fefc = afef(abc)fefc = afef(b)fefc = afe(b)fefc,
and
aef(abc)fefc = aef(abc)efc = aef(b)efc = aef(b)fefc.
Where this matching fails, (6.3.2ʺ) first reduces to
a(e)abc(f)c + a(e)abc(fe – efe – fef)c + a(fe – efe – fef)abc(f)c
= a(e)b(f)c + a(e)b(fe – efe – fef)c + a(fe – efe – fef)b(f)c.
Using regularity again on products involving the underlined terms, the above reduces to
a(e)abc(f – ef)c – a(ef)abc(f)c = a(e)b(f – ef)c – a(ef)b(f)c
or
a(e)abc(f)c – a(e)abc(ef)c – a(ef)abc(f)c = a(e)b(f)c – a(e)b(ef)c – a(ef)b(f)c.
Regularity gives a(ef)abc(ef)c = a(ef)a(ef)bc(ef)c = a(ef)b(ef)c. (6.3.2) is thus reduced to (II). £
A successful generation of a s-band from a regular band requires that at each stage of
generation the new regular band Sʹ also satisfies (I) and (II) in Theorem 6.3.1 for all
a, b, c, e, f ∈ Sʹ. While (I) and (II) generally need not be passed on to later bands, here is a
strategy for generating s-bands. To this end, a condition C potentially satisfied by regular bands
in rings is s-inductive if
231
