Page 28 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
a(a1ba2)aʹ = (aa1a)b(aʹa2aʹ) = abaʹ
and likewise a(a3ca4)aʹ = acaʹ, so that abaʹ = acaʹ. From this (i) - (iii) follow. To see (iv),
clearly a ≥ aba in the cell containing AbA. Assuming b ∈ Bi, then aca = aba for all c ∈ Bi, so
that aba is this unique bi ≤ a in Bi. That bab = b follows from a ≻ b. £
Given A > B and cell Bi in B, Bi is called an A-cell in B and αi: A → Bi is defined by
αi(a) = aba for all a ∈A and any b ∈Bi is the cell-map of A into Bi. Its image in B is the coset
AbA, since by Corollary 1.2.8 again, abaʹ = aaʹbaaʹ for all a, aʹ ∈ A and b ∈ B.
Application. All left regular bands with just two D-classes A > B are constructed as
follows.
(1) Given sets A and B, partition B into disjoint nonempty subsets {Bi ⊆ B ⎜i ∈ I}.
(2) For each cell Bi of the partition, choose a function αi: A → Bi.
(3) Define multiplication on S = A ∪ B by first imposing left zero multiplications on A
and B separately, setting ba = b for all a ∈ A and b ∈ B, and finally by setting ab = αi(a)
if b ∈ Bi.
X In the resulting band S = A ∪ B, the cell decomposition of B is the given partition B =∪Bi and
the cell-maps from A to B are precisely the αi: A → Bi ⊆ B for each i ∈ I.
Cell decompositions and cell-maps also determine the multiplication of elements in
incomparable D-classes in a left regular band. But first a definition: given D-classes A > B in a
band with a ∈A, the image of a in B is the set {b ∈ B ⎜a ≥ b}. Put otherwise, this image is
aBa = {aba ⎜b ∈ B}, or when S is left regular, aB = {ab ⎜b ∈ B}. In the next situation, both D-
classes A and B can be incomparable
Proposition 1.2.15. Given a left regular band S and D-classes A and B, with meet D-
class M, and with a ∈A and b ∈B, then:
(1) The image bMb of b in M lies in a unique cell Mi of the A-decomposition of M.
(2) If αi: A → Mi ⊆ M is the corresponding cell-map, then ab = αi(a).
(3) Dual remarks hold for the image class of a in M and the product ba.
Conversely, given two left regular bands consisting of D-classes A > M and B > M respectively,
such that A∩B = ∅, with M the same for both bands with A > M and B > M satisfying (1) – (3),
then the multiplications on each band extend uniquely to a left regular multiplication on their
union A∪B∪M such that M is the meet class of A and B. (Dual constructions and observations
exist for the right regular case. The general case follows using fibered products.)
Proof. Given b > bʹ ∈M, abʹ = a(bbʹ) = (ab)bʹ = ab. Thus all such bʹ < b lie in the same cell of
the A-decomposition of M and (1) – (3) are now clear. For the second part, given (1) – (3), one
needs to show that all possible multiplications of abm (and bam), amb (and bma), mab (and mba),
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a(a1ba2)aʹ = (aa1a)b(aʹa2aʹ) = abaʹ
and likewise a(a3ca4)aʹ = acaʹ, so that abaʹ = acaʹ. From this (i) - (iii) follow. To see (iv),
clearly a ≥ aba in the cell containing AbA. Assuming b ∈ Bi, then aca = aba for all c ∈ Bi, so
that aba is this unique bi ≤ a in Bi. That bab = b follows from a ≻ b. £
Given A > B and cell Bi in B, Bi is called an A-cell in B and αi: A → Bi is defined by
αi(a) = aba for all a ∈A and any b ∈Bi is the cell-map of A into Bi. Its image in B is the coset
AbA, since by Corollary 1.2.8 again, abaʹ = aaʹbaaʹ for all a, aʹ ∈ A and b ∈ B.
Application. All left regular bands with just two D-classes A > B are constructed as
follows.
(1) Given sets A and B, partition B into disjoint nonempty subsets {Bi ⊆ B ⎜i ∈ I}.
(2) For each cell Bi of the partition, choose a function αi: A → Bi.
(3) Define multiplication on S = A ∪ B by first imposing left zero multiplications on A
and B separately, setting ba = b for all a ∈ A and b ∈ B, and finally by setting ab = αi(a)
if b ∈ Bi.
X In the resulting band S = A ∪ B, the cell decomposition of B is the given partition B =∪Bi and
the cell-maps from A to B are precisely the αi: A → Bi ⊆ B for each i ∈ I.
Cell decompositions and cell-maps also determine the multiplication of elements in
incomparable D-classes in a left regular band. But first a definition: given D-classes A > B in a
band with a ∈A, the image of a in B is the set {b ∈ B ⎜a ≥ b}. Put otherwise, this image is
aBa = {aba ⎜b ∈ B}, or when S is left regular, aB = {ab ⎜b ∈ B}. In the next situation, both D-
classes A and B can be incomparable
Proposition 1.2.15. Given a left regular band S and D-classes A and B, with meet D-
class M, and with a ∈A and b ∈B, then:
(1) The image bMb of b in M lies in a unique cell Mi of the A-decomposition of M.
(2) If αi: A → Mi ⊆ M is the corresponding cell-map, then ab = αi(a).
(3) Dual remarks hold for the image class of a in M and the product ba.
Conversely, given two left regular bands consisting of D-classes A > M and B > M respectively,
such that A∩B = ∅, with M the same for both bands with A > M and B > M satisfying (1) – (3),
then the multiplications on each band extend uniquely to a left regular multiplication on their
union A∪B∪M such that M is the meet class of A and B. (Dual constructions and observations
exist for the right regular case. The general case follows using fibered products.)
Proof. Given b > bʹ ∈M, abʹ = a(bbʹ) = (ab)bʹ = ab. Thus all such bʹ < b lie in the same cell of
the A-decomposition of M and (1) – (3) are now clear. For the second part, given (1) – (3), one
needs to show that all possible multiplications of abm (and bam), amb (and bma), mab (and mba),
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