Page 78 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 78
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Given a lattice T and a separable, rectangular functor K from (T, ≥) to the category RP, a
normal ∧-band may be constructed on S = ∪s∈TK(s) by setting
a∧b = K(s, s∧t)[a] ∧ K(t, s∧t)[b], for a ∈ K(s) and b ∈ K(t).
This is, of course, the Yamada-Kimura construction. One would like to be able to use injections
to define a join operation ∨ on S and thus turn this normal band into a normal skew lattice. To do
so, requires a key concept from Section 2.4.
Given s, t ∈ T with join n ∈ T we say that projections K(n, s) and K(n, t) are orthogonal
in K(n) if (i) for each a ∈ K(s), its pre-image K(n, s)–1[a] in K(n) lies in the image of a unique
injection k(n, t, j) from K(t) to K(n) and similarly (ii) for each b ∈ K(t), its pre-image
K(n, t )–1[b] in K(n) lies in the image of a unique injection k(n, s, j) from K(s) to K(n).
Our observations above combined with earlier results in this section yield the following
extension of the description of normal bands by Yamada and Kimura [1958]:
Theorem 2.5.7. Let K be a separable, rectangular functor defined on a lattice T such
that for every join situation n = s∨t in T the projections K(n, s) and K(n, t) are orthogonal in
K(n). Then S = ∪s∈TK(s) becomes a normal skew lattice with K providing the system of natural
projections and coprojections, if given a in K(s) and b in K(t), their meet and join are defined by
a∧b = K(s, m)[a] ∧ K(t, m)[b] and a∨b = k(n, s, i)[a] ∨ k(n, t, j)[b]
where m = s∧t, n = s∨t, the image of k(n, s, i) contains K(n, t)–1[b] and the image of k(n, t, j)
contains K(n, s)–1[a]. Conversely, every normal skew lattice arises in this fashion. £
2.6 Decompositions of normal, symmetric skew lattices
Given rectangular skew lattices I and B, let A be their direct product I × B. A normal
primitive skew lattice PI,B with D-class structure A > B is given by letting B be a full A-coset in
itself and for each i ∈I, letting {i} × B be a full B-coset in A, and using the coset bijections:
ϕi: {i} × B → B and ϕi−1: B →{i} × B given by ϕi (i, b) = b and ϕi−1(b) = (i, b) . Clearly
PI,B ≅ I0 × B where I0 is just I with a zero element 0 adjoined, so that I > {0}. Our first
decomposition result states that for normal, primitive skew lattices this is essentially all there is.
Lemma 2.6.1. Let P be a normal, primitive skew lattice with D-class structure A > B,
and let I be a set of indices for the B-cosets Ai in A. Then a rectangular skew lattice structure
exists on I such that I0 × B ≅ P. Given b ∈B, I can be given as the image set b∨A∨b of b in A, in
which case I0 ≅ Ib = I∪{b} = b∨S∨b. An isomorphism θ: Ib × B ≅ S is given by θ(x, y) = y∨x∨y,
the unique image of y in B∨x∨B in S for all (x, y) ∈ Ib × B.
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Given a lattice T and a separable, rectangular functor K from (T, ≥) to the category RP, a
normal ∧-band may be constructed on S = ∪s∈TK(s) by setting
a∧b = K(s, s∧t)[a] ∧ K(t, s∧t)[b], for a ∈ K(s) and b ∈ K(t).
This is, of course, the Yamada-Kimura construction. One would like to be able to use injections
to define a join operation ∨ on S and thus turn this normal band into a normal skew lattice. To do
so, requires a key concept from Section 2.4.
Given s, t ∈ T with join n ∈ T we say that projections K(n, s) and K(n, t) are orthogonal
in K(n) if (i) for each a ∈ K(s), its pre-image K(n, s)–1[a] in K(n) lies in the image of a unique
injection k(n, t, j) from K(t) to K(n) and similarly (ii) for each b ∈ K(t), its pre-image
K(n, t )–1[b] in K(n) lies in the image of a unique injection k(n, s, j) from K(s) to K(n).
Our observations above combined with earlier results in this section yield the following
extension of the description of normal bands by Yamada and Kimura [1958]:
Theorem 2.5.7. Let K be a separable, rectangular functor defined on a lattice T such
that for every join situation n = s∨t in T the projections K(n, s) and K(n, t) are orthogonal in
K(n). Then S = ∪s∈TK(s) becomes a normal skew lattice with K providing the system of natural
projections and coprojections, if given a in K(s) and b in K(t), their meet and join are defined by
a∧b = K(s, m)[a] ∧ K(t, m)[b] and a∨b = k(n, s, i)[a] ∨ k(n, t, j)[b]
where m = s∧t, n = s∨t, the image of k(n, s, i) contains K(n, t)–1[b] and the image of k(n, t, j)
contains K(n, s)–1[a]. Conversely, every normal skew lattice arises in this fashion. £
2.6 Decompositions of normal, symmetric skew lattices
Given rectangular skew lattices I and B, let A be their direct product I × B. A normal
primitive skew lattice PI,B with D-class structure A > B is given by letting B be a full A-coset in
itself and for each i ∈I, letting {i} × B be a full B-coset in A, and using the coset bijections:
ϕi: {i} × B → B and ϕi−1: B →{i} × B given by ϕi (i, b) = b and ϕi−1(b) = (i, b) . Clearly
PI,B ≅ I0 × B where I0 is just I with a zero element 0 adjoined, so that I > {0}. Our first
decomposition result states that for normal, primitive skew lattices this is essentially all there is.
Lemma 2.6.1. Let P be a normal, primitive skew lattice with D-class structure A > B,
and let I be a set of indices for the B-cosets Ai in A. Then a rectangular skew lattice structure
exists on I such that I0 × B ≅ P. Given b ∈B, I can be given as the image set b∨A∨b of b in A, in
which case I0 ≅ Ib = I∪{b} = b∨S∨b. An isomorphism θ: Ib × B ≅ S is given by θ(x, y) = y∨x∨y,
the unique image of y in B∨x∨B in S for all (x, y) ∈ Ib × B.
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