Page 224 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 224
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Proof. Multiplying out (a + b + ba – aba – bab)c(a + b + ba – aba – bab) and cancelling yields
(aca + acb – acab) + (bca + bcb – bcab) – (abca + abcb – abcab) = aca + bca + bcb – bcab – abca
where the underlined terms vanish collectively by identity (6.2.2). £
Lemma 6.2.7. Given a, b, c in a s-band S, as(bsc) = (asb) sc if and only if
[b, c]2a – a[b, c]2a = c[a, b]2 – c[a, b]2c. (6.2.7)
Proof. as (bsc) = a + (b + c + cb – bcb – cbc) + (ba + ca + cba – bcba – cbca)
– (aba + aca + acba – abcba – acbca) – (bsc)a(bsc)
while (asb)sc = (a + b + ba – aba – bab) + c + (ca + cb + cba – caba – cbab)
– (cac + cbc + cbac – cabac – cbabc) – (asb)c(asb).
Equating as(bsc) with (asb)sc and then canceling common terms yields
– bcb – bcba – cbca – aca – acba + abcba + acbca – (bsc)a(bsc)
= – bab – caba – cbab – cac – cbac + cabac + cbabc – (asb)c(asb).
Applying the previous lemma gives
– bcb – bcba – cbca – aca – acba + abcba + acbca + (bcab – bab – cab – cac + cabc)
= – bab – caba – cbab – cac – cbc – cbac + cabac + cbabc + (abca – aca – bca – bcb + bcab)
which reduces to
– bcba – cbca – acba + abcba + acbca – cab + cabc
= – caba – cbab – cbac + cabac + cbabc + abca – bca.
Adding cab + bca to both sides, grouping the aXa terms on the left and the cYc terms on the right
and then factoring gives,
bca – bcba – cbca – a(bc – cb)2a = cab – caba – cbab – c(ab – ba)2c,
Adding cba to both sides, then grouping and factoring once again gives
(bc – cb)2a – a(bc – cb)2a = c(ab – ba)2 – c(ab – ba)2c
which is the statement of the lemma. £
Theorem 6.2.8. A s-band S is associative if and only if for all a, b, c ∈ S,
a[b, c]2 = [b, c]2a.
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Proof. Multiplying out (a + b + ba – aba – bab)c(a + b + ba – aba – bab) and cancelling yields
(aca + acb – acab) + (bca + bcb – bcab) – (abca + abcb – abcab) = aca + bca + bcb – bcab – abca
where the underlined terms vanish collectively by identity (6.2.2). £
Lemma 6.2.7. Given a, b, c in a s-band S, as(bsc) = (asb) sc if and only if
[b, c]2a – a[b, c]2a = c[a, b]2 – c[a, b]2c. (6.2.7)
Proof. as (bsc) = a + (b + c + cb – bcb – cbc) + (ba + ca + cba – bcba – cbca)
– (aba + aca + acba – abcba – acbca) – (bsc)a(bsc)
while (asb)sc = (a + b + ba – aba – bab) + c + (ca + cb + cba – caba – cbab)
– (cac + cbc + cbac – cabac – cbabc) – (asb)c(asb).
Equating as(bsc) with (asb)sc and then canceling common terms yields
– bcb – bcba – cbca – aca – acba + abcba + acbca – (bsc)a(bsc)
= – bab – caba – cbab – cac – cbac + cabac + cbabc – (asb)c(asb).
Applying the previous lemma gives
– bcb – bcba – cbca – aca – acba + abcba + acbca + (bcab – bab – cab – cac + cabc)
= – bab – caba – cbab – cac – cbc – cbac + cabac + cbabc + (abca – aca – bca – bcb + bcab)
which reduces to
– bcba – cbca – acba + abcba + acbca – cab + cabc
= – caba – cbab – cbac + cabac + cbabc + abca – bca.
Adding cab + bca to both sides, grouping the aXa terms on the left and the cYc terms on the right
and then factoring gives,
bca – bcba – cbca – a(bc – cb)2a = cab – caba – cbab – c(ab – ba)2c,
Adding cba to both sides, then grouping and factoring once again gives
(bc – cb)2a – a(bc – cb)2a = c(ab – ba)2 – c(ab – ba)2c
which is the statement of the lemma. £
Theorem 6.2.8. A s-band S is associative if and only if for all a, b, c ∈ S,
a[b, c]2 = [b, c]2a.
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