Page 64 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
(i) I, J and C are nonempty sets.
(ii) G is a group and µ: G × C → C is a group action of group G on C.
(iii) θ: J × I → G is a map.
From this data construct a P-graph by first setting A = I × C and B = J × C. The B-cosets in A are
the Ai = {i} × C and the A-cosets in B are the Bj = {j} × C. Coset bijections are given by
φji(i, c) = (j, θ(j, i)c). From this data, a right or left primitive skew lattice is constructed to be
denoted by P[I, J, C, G, µ, θ] or Pʹ[ I, J, C, G, µ, θ].
Every right [left] primitive skew lattice has a coordinatization. Given a P-graph
representation, P ≅ P[Ai, φji, Bj], let C = A0 for some common index 0 in I ∩ J.
A0 Ai A0
↓ ϕ00 ϕ −1 ↓ ϕ ji ϕ −1
0i 0j
B0 B j
Next, for each pair (j, i), let θ(j, i) be the permutation ϕj0–1ϕjiϕi0–1ϕ00 of A0 and then let G denote
the permutation group on A0 generated collectively by the various θ(j, i). Here the
coordinatization is normalized in that θ({0} × I ∪ J × {0}) = 1 in G.
In the case where G = C and u is group multiplication, P has a coordinatization with
group translations. In this case the data reduces to the indexing sets I, J, the group G and map
θ: I ×J → G, and the skew lattice is denoted by P[I, J, G, θ]. By the above discussion we may
assume that 0 ∈ I ∩ J, and θ({0} × I ∪ J × {0}) = 1 in G (or = 0 in the case of additive notation).
An instance of this is given by any maximal right [left] primitive skew lattice in a ring.
Example 2.4.1. Given a ring R and an idempotent e ∈ E(R), the R-set of e in R is the set
Re = e + eR(1 – e) = {x ∈ R⎪ex = x and xe = e}
It is the maximal right-0 band (xy = y) in R containing e. Re also forms a left-0 band under ○,
and thus is a rectangular skew lattice in R. Let f be a second idempotent in R such that e > f.
Then Rf forms a second rectangular skew lattice and the union Pe>f = Re ∪ Rf is a right primitive
skew lattice in R with upper D-class Re and lower D-class Rf. Its coordinatization is given as
follows. To begin, notice that the group A = eR(1 – e) acts simply transitively on Re and that the
group B = fR(1 – f) acts simply transitively on Rf, both under the operation of addition. A and B
share the common subgroup
G = A ∩ B = eR(1 – e) ∩ fR(1 – f) = fR(1 – e).
A splits as fR(1 – e) ⊕ (e – f)R(1 – e) and B splits as fR(1 – e) ⊕ fR(e – f). The various
summands are naturally arranged in the following array format:
62
(i) I, J and C are nonempty sets.
(ii) G is a group and µ: G × C → C is a group action of group G on C.
(iii) θ: J × I → G is a map.
From this data construct a P-graph by first setting A = I × C and B = J × C. The B-cosets in A are
the Ai = {i} × C and the A-cosets in B are the Bj = {j} × C. Coset bijections are given by
φji(i, c) = (j, θ(j, i)c). From this data, a right or left primitive skew lattice is constructed to be
denoted by P[I, J, C, G, µ, θ] or Pʹ[ I, J, C, G, µ, θ].
Every right [left] primitive skew lattice has a coordinatization. Given a P-graph
representation, P ≅ P[Ai, φji, Bj], let C = A0 for some common index 0 in I ∩ J.
A0 Ai A0
↓ ϕ00 ϕ −1 ↓ ϕ ji ϕ −1
0i 0j
B0 B j
Next, for each pair (j, i), let θ(j, i) be the permutation ϕj0–1ϕjiϕi0–1ϕ00 of A0 and then let G denote
the permutation group on A0 generated collectively by the various θ(j, i). Here the
coordinatization is normalized in that θ({0} × I ∪ J × {0}) = 1 in G.
In the case where G = C and u is group multiplication, P has a coordinatization with
group translations. In this case the data reduces to the indexing sets I, J, the group G and map
θ: I ×J → G, and the skew lattice is denoted by P[I, J, G, θ]. By the above discussion we may
assume that 0 ∈ I ∩ J, and θ({0} × I ∪ J × {0}) = 1 in G (or = 0 in the case of additive notation).
An instance of this is given by any maximal right [left] primitive skew lattice in a ring.
Example 2.4.1. Given a ring R and an idempotent e ∈ E(R), the R-set of e in R is the set
Re = e + eR(1 – e) = {x ∈ R⎪ex = x and xe = e}
It is the maximal right-0 band (xy = y) in R containing e. Re also forms a left-0 band under ○,
and thus is a rectangular skew lattice in R. Let f be a second idempotent in R such that e > f.
Then Rf forms a second rectangular skew lattice and the union Pe>f = Re ∪ Rf is a right primitive
skew lattice in R with upper D-class Re and lower D-class Rf. Its coordinatization is given as
follows. To begin, notice that the group A = eR(1 – e) acts simply transitively on Re and that the
group B = fR(1 – f) acts simply transitively on Rf, both under the operation of addition. A and B
share the common subgroup
G = A ∩ B = eR(1 – e) ∩ fR(1 – f) = fR(1 – e).
A splits as fR(1 – e) ⊕ (e – f)R(1 – e) and B splits as fR(1 – e) ⊕ fR(e – f). The various
summands are naturally arranged in the following array format:
62
