Page 185 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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V: Further Topics in Skew Lattices
a) Normal skew lattices characterized by the identity x∧y∧z∧w = x∧z∧y∧w;
equivalently, each subset [e]↓ = {x ∈S⎪e ≥ x} = {e∧x∧e⎪x ∈S} is a sublattice.
b) Conormal skew lattices that satisfy x∨y∨z∨w = x∨z∨y∨w; equivalently, every
subset [a]↑ = {x ∈S⎪x ≥ e} = {e∨x∨e⎪x ∈S} is a sublattice.
c) Primitive skew lattices consisting of two D-classes: A > B and rectangular skew
lattices. Any algebra in the variety that primitive skew lattices generate.
Their significance is due in part to skew Boolean algebras being normal as skew lattices.
Theorem 5.4.7. Let A > B > C be a strictly categorical skew chain. Then:
i) For any a ∈A, all images of a in B lie in a unique C-coset in B.
ii) For any c ∈C, all images of c in B lie in a unique A-coset in B.
iii) Given a > c with a ∈A and c ∈C, a unique b ∈B exists such that a > b > c. This
b lies jointly in the C-coset in B containing all images of a in B and in the A-
coset in B containing all images of c in B. (It is the midpoint of a and c in B.)
Proof. (i) Without loss of generality we assume that C is a full B-coset within itself. If
a∧C∧a = {c ∈C⎪a > c} is the image set of a in C parameterizing the A-cosets in C and if a > b in
B, then the set {c∨b∨c⎪c ∈ a∧C∧a} of all images of a in the C-coset C∨b∨C in B, parameterizes
the A-C cosets in B lying in C∨b∨C (that is, the coset intersections A∧b∧A ∩ C∨b∨C in C∨b∨C)
that are inverse images of the A-cosets in C relative to the coset bijection of C∨b∨C onto C. (See
Theorem 2.4.14 and its preceding discussion.) By assumption, all A-cosets X in B are in bijective
correspondence with these A-C cosets under the map X → X ∩ C∨b∨C. Thus each x in
{c∨b∨c⎪c ∈ a∧C∧a} is the (necessarily) unique image of a in the A-coset in B to which x
belongs and as we traverse through these x’s, every such A-coset occurs as A∧x∧A. Thus all
images of a in B lie in the C-coset C∨b∨C in B.
In like fashion one verifies (ii).
Finally, given a > c with a ∈A and c ∈C, a unique AC-coset U exists that is the
intersection of the A-coset containing all images of c in B and the C-coset containing all images
of a in B. In particular U contains any b in B such that a > b > c. Such a b exists in B, e.g.,
b = a∧(c∨u∨c)∧a for any u in B. But being in a single AC-coset in B, at most one such b is in U.
£
In the terminology of Section 2.4, the A-cosets in B are orthogonal to the C-cosets in B.
All this leads to the following multiple characterization of strictly categorical skew lattices:
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a) Normal skew lattices characterized by the identity x∧y∧z∧w = x∧z∧y∧w;
equivalently, each subset [e]↓ = {x ∈S⎪e ≥ x} = {e∧x∧e⎪x ∈S} is a sublattice.
b) Conormal skew lattices that satisfy x∨y∨z∨w = x∨z∨y∨w; equivalently, every
subset [a]↑ = {x ∈S⎪x ≥ e} = {e∨x∨e⎪x ∈S} is a sublattice.
c) Primitive skew lattices consisting of two D-classes: A > B and rectangular skew
lattices. Any algebra in the variety that primitive skew lattices generate.
Their significance is due in part to skew Boolean algebras being normal as skew lattices.
Theorem 5.4.7. Let A > B > C be a strictly categorical skew chain. Then:
i) For any a ∈A, all images of a in B lie in a unique C-coset in B.
ii) For any c ∈C, all images of c in B lie in a unique A-coset in B.
iii) Given a > c with a ∈A and c ∈C, a unique b ∈B exists such that a > b > c. This
b lies jointly in the C-coset in B containing all images of a in B and in the A-
coset in B containing all images of c in B. (It is the midpoint of a and c in B.)
Proof. (i) Without loss of generality we assume that C is a full B-coset within itself. If
a∧C∧a = {c ∈C⎪a > c} is the image set of a in C parameterizing the A-cosets in C and if a > b in
B, then the set {c∨b∨c⎪c ∈ a∧C∧a} of all images of a in the C-coset C∨b∨C in B, parameterizes
the A-C cosets in B lying in C∨b∨C (that is, the coset intersections A∧b∧A ∩ C∨b∨C in C∨b∨C)
that are inverse images of the A-cosets in C relative to the coset bijection of C∨b∨C onto C. (See
Theorem 2.4.14 and its preceding discussion.) By assumption, all A-cosets X in B are in bijective
correspondence with these A-C cosets under the map X → X ∩ C∨b∨C. Thus each x in
{c∨b∨c⎪c ∈ a∧C∧a} is the (necessarily) unique image of a in the A-coset in B to which x
belongs and as we traverse through these x’s, every such A-coset occurs as A∧x∧A. Thus all
images of a in B lie in the C-coset C∨b∨C in B.
In like fashion one verifies (ii).
Finally, given a > c with a ∈A and c ∈C, a unique AC-coset U exists that is the
intersection of the A-coset containing all images of c in B and the C-coset containing all images
of a in B. In particular U contains any b in B such that a > b > c. Such a b exists in B, e.g.,
b = a∧(c∨u∨c)∧a for any u in B. But being in a single AC-coset in B, at most one such b is in U.
£
In the terminology of Section 2.4, the A-cosets in B are orthogonal to the C-cosets in B.
All this leads to the following multiple characterization of strictly categorical skew lattices:
183
