Page 227 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 227
VI: Skew Lattices in Rings
images S/L and S/R of S. Each e ∈S factors as eLeR where eL = ete ∈ TL and eR = tee ∈ TR
where te is the unique element in T∩De. Put otherwise, eL and eR are the unique elements in De
related to e and te in the following D-class picture.
eR eL = ete
L
tee = eR R L
te
Due to S being regular
ef = (eLeR)(fLfR) = (eLfL)(eRfR)
holds for all e, f ∈ S. Indeed,
ef = efefef = (eLeR)(fLfR)ef(eLeR)(fLfR) = eLfLefeRfR = (eLfL)(eRfR)
with both latter reductions due to regularly. Multiplication on S thus decomposes internally into
separate products on TL and TR so that S is isomorphic to a sub-band of TL × TR. We call the
factorization e = eLeR the internal Kimura factorization of e relative to T.
Even more is true if S is a s-band in a ring. In this case T is a lattice section in that for
all e, f in T, e s f = e ○ f is in T also making (T; ○, •) a lattice that meets each D-class exactly
once, so that T ≅ S/D. (If u ∈ T is in the join D-class of e and f, then u ≥ both e, f and so u ≥ e ○ f
follows. But Since u and e ○ f lie in a common D-class, u = e ○ f.) Given a s-band S with a
lattice section T, a modified join operation ∨T can be defined such that (S, ∨T, •) is a distributive,
symmetric skew lattice. To begin, TL and TR are in fact skew lattices. Indeed, suppose that
a, b ∈ TL. Certainly asb ∈ S, but since a and b lie in the left regular band TL, we also have
asb = a○b. Let ta, tb ∈ T be such that a L ta and b L tb. By left regularity, we have
abta = ab, atbta = atb, batb = ba and btatb = bta. Hence
(a○b)(ta○tb) = (a + b – ab)(ta + tb – tatb) = a + b – ab = a○b
Similarly, (ta○tb)(a○b) = ta○tb. Thus a○b ∈ TL, and TL is a skew lattice in S as claimed.
Likewise, TR is a skew lattice in S. The internal Kimura decomposition of the band S with
respect to T enables us to define an operation ∨T on S by setting
e ∨T f = (eL ○ fL)(eR ○ fR).
Clearly (e ∨T f)L = eL ○ fL and (e ∨T f)R = eR ○ fR. It follows that (S, ∨T, •) is an “internal” fibered
product of the skew lattices (TL, ○, •) and (TR, ○, •) over their common sublattice T and thus is a
skew lattice. We will call ∨T the associative join on S induced from T. Cleary ∨T is dependent
225
images S/L and S/R of S. Each e ∈S factors as eLeR where eL = ete ∈ TL and eR = tee ∈ TR
where te is the unique element in T∩De. Put otherwise, eL and eR are the unique elements in De
related to e and te in the following D-class picture.
eR eL = ete
L
tee = eR R L
te
Due to S being regular
ef = (eLeR)(fLfR) = (eLfL)(eRfR)
holds for all e, f ∈ S. Indeed,
ef = efefef = (eLeR)(fLfR)ef(eLeR)(fLfR) = eLfLefeRfR = (eLfL)(eRfR)
with both latter reductions due to regularly. Multiplication on S thus decomposes internally into
separate products on TL and TR so that S is isomorphic to a sub-band of TL × TR. We call the
factorization e = eLeR the internal Kimura factorization of e relative to T.
Even more is true if S is a s-band in a ring. In this case T is a lattice section in that for
all e, f in T, e s f = e ○ f is in T also making (T; ○, •) a lattice that meets each D-class exactly
once, so that T ≅ S/D. (If u ∈ T is in the join D-class of e and f, then u ≥ both e, f and so u ≥ e ○ f
follows. But Since u and e ○ f lie in a common D-class, u = e ○ f.) Given a s-band S with a
lattice section T, a modified join operation ∨T can be defined such that (S, ∨T, •) is a distributive,
symmetric skew lattice. To begin, TL and TR are in fact skew lattices. Indeed, suppose that
a, b ∈ TL. Certainly asb ∈ S, but since a and b lie in the left regular band TL, we also have
asb = a○b. Let ta, tb ∈ T be such that a L ta and b L tb. By left regularity, we have
abta = ab, atbta = atb, batb = ba and btatb = bta. Hence
(a○b)(ta○tb) = (a + b – ab)(ta + tb – tatb) = a + b – ab = a○b
Similarly, (ta○tb)(a○b) = ta○tb. Thus a○b ∈ TL, and TL is a skew lattice in S as claimed.
Likewise, TR is a skew lattice in S. The internal Kimura decomposition of the band S with
respect to T enables us to define an operation ∨T on S by setting
e ∨T f = (eL ○ fL)(eR ○ fR).
Clearly (e ∨T f)L = eL ○ fL and (e ∨T f)R = eR ○ fR. It follows that (S, ∨T, •) is an “internal” fibered
product of the skew lattices (TL, ○, •) and (TR, ○, •) over their common sublattice T and thus is a
skew lattice. We will call ∨T the associative join on S induced from T. Cleary ∨T is dependent
225
