Page 225 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 225
VI: Skew Lattices in Rings
Proof. This identity implies that of Lemma 6.2.7, making s associative. On the other hand,
replacing a by aba and b by bab in the latter gives
[babc – cbab]2aba – aba[bab – cbab]2aba = c[ab – ba]2 – c[ab – ba]2c.
Regularity first gives aba[babc – cbab]2aba = aba(c – c)aba =0
and then [babc – cbab]2aba = (babc – cbab)(c – c)aba = 0.
Thus c(ab – ba)2c = c(ab – ba)2 and by (6.2.7), a(bc – cb)2a = (bc – cb)2a. Permuting variables
in c(ab – ba)2c = c(ab – ba)2 gives a(bc – cb)2a = a(bc – cb)2 from which a[b, c]2 = [b, c]2a
follows. £
The associativity of s thus reduces to cases of possible commutation: does [x, y]2 always
produce elements lying in the center of the subring S± generated from S?
The associativity of s: the Green’s relations L and R
We next consider connections between the associativity of s and the equivalences L and
R that refine D. Recall that L is a right congruence on S in that aLb implies acLbc for all c ∈S
while R is a left congruence on S. If S is regular, then both L and R are full congruences. In
particular, L and R are multiplicative congruences on all s-bands. We turn to the status of L
and R as s-congruences on a s-band. But first let aRsb denote the conjunction, asb = b and
bsa = a, and similarly let aLsb denote asb = a and bsa = b.
Lemma 6.2.9. In a s-band, aLb if and only if aRsb and similarly aRb if and only if
aLsb. In general, aLb implies (csa)(csb) = csa for all c and aRb implies
(asc)(bsc) = bsc for all c.
Proof. Expanding, asb = b and bsa = a reduce to a = aba + bab – ba and
b = aba + bab – ab. Multiplying on the left by a and b respectively, yields a = ab and b = ba,
that is aLb. Conversely, if aLb under the ring multiplication, then asb reduces to b and bsa
reduces to a. In general, for all c ∈S, (csa)(csb) = (c + a + ac – aca – cac)(c + b + bc – bcb –
cbc) which with the assistance of L as a congruence on the multiplicative band expands as
c + (ac + a – acb) + (ac) – (ac + aca – acb) – (cac) = c + a + ac – aca – cac = csa
Similarly, (csb)(csa) = (csb) so that csa L csb. The case for R is similar. £
223
Proof. This identity implies that of Lemma 6.2.7, making s associative. On the other hand,
replacing a by aba and b by bab in the latter gives
[babc – cbab]2aba – aba[bab – cbab]2aba = c[ab – ba]2 – c[ab – ba]2c.
Regularity first gives aba[babc – cbab]2aba = aba(c – c)aba =0
and then [babc – cbab]2aba = (babc – cbab)(c – c)aba = 0.
Thus c(ab – ba)2c = c(ab – ba)2 and by (6.2.7), a(bc – cb)2a = (bc – cb)2a. Permuting variables
in c(ab – ba)2c = c(ab – ba)2 gives a(bc – cb)2a = a(bc – cb)2 from which a[b, c]2 = [b, c]2a
follows. £
The associativity of s thus reduces to cases of possible commutation: does [x, y]2 always
produce elements lying in the center of the subring S± generated from S?
The associativity of s: the Green’s relations L and R
We next consider connections between the associativity of s and the equivalences L and
R that refine D. Recall that L is a right congruence on S in that aLb implies acLbc for all c ∈S
while R is a left congruence on S. If S is regular, then both L and R are full congruences. In
particular, L and R are multiplicative congruences on all s-bands. We turn to the status of L
and R as s-congruences on a s-band. But first let aRsb denote the conjunction, asb = b and
bsa = a, and similarly let aLsb denote asb = a and bsa = b.
Lemma 6.2.9. In a s-band, aLb if and only if aRsb and similarly aRb if and only if
aLsb. In general, aLb implies (csa)(csb) = csa for all c and aRb implies
(asc)(bsc) = bsc for all c.
Proof. Expanding, asb = b and bsa = a reduce to a = aba + bab – ba and
b = aba + bab – ab. Multiplying on the left by a and b respectively, yields a = ab and b = ba,
that is aLb. Conversely, if aLb under the ring multiplication, then asb reduces to b and bsa
reduces to a. In general, for all c ∈S, (csa)(csb) = (c + a + ac – aca – cac)(c + b + bc – bcb –
cbc) which with the assistance of L as a congruence on the multiplicative band expands as
c + (ac + a – acb) + (ac) – (ac + aca – acb) – (cac) = c + a + ac – aca – cac = csa
Similarly, (csb)(csa) = (csb) so that csa L csb. The case for R is similar. £
223
