Page 52 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 52
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Proof. If S is a right-handed symmetric skew lattice, then it satisfies the identities of the above
theorem plus the identities x∨y∨x = x∨y and x∧y∧x = y∧x from which the displayed identities
follow. Conversely, if the displayed identities hold on a skew lattice S, then one has
x∧y∧x = x∧(y∧x) = x∧[x∧y∧(x∨y)] = x∧y∧(x∨y) = y∧x
and similarly x∨y∨x = x∨y so that S is right-handed. But using x∨y∨x = x∨y and x∧y∧x = y∧x,
the displayed identities can be transformed back into the identities of the previous theorem. £
Our interest in symmetry is due further to the following three theorems.
Theorem 2.2.6. All skew lattices in rings (using multiplication and the circle operation)
are symmetric.
Proof. Obviously a + b – ab = b + a – ba if and only if ab = ba. £
A lattice section in a skew lattice S is any sublattice T of S having nonempty intersection
with each D-class of S, in which case, T ≅ S/D.
Theorem 2.2.7. If S is a symmetric skew lattice for which S/D is countable, then S has a
lattice section.
Proof. Let D1, D2, … be a listing of all D-classes. Pick x1 ∈ D1. By Theorem 2.2.1 we know
that in every D-class of S elements exist that commute with x. By symmetry we know that such
commuting is under both operations and, moreover, that the set of all such elements is closed
under both operations. Thus S1 = {y ∈ S⎮y commutes with x} is a sub-skew lattice of S that has
nonempty intersection with each D-class of S and for which Dx1 = {x1} as a D-class in S1. If S1
is not a lattice (and hence not a lattice section of S) we find an element x2 in D2∩S1 and thus not
in Dx1 in S1. The argument repeats. We thus obtain a descending chain of sub-skew lattices
S1 ⊇ S2 ⊇ S3 ⊇ …
each having nonempty intersection with each D-class of S, and such that Dj ∩ Si = {xj} for all
i ≥ j where each xj commutes with all xi that precede it. Clearly, the full intersection n≥1Sn is a
lattice section of S. £
A left-handed section of S is any left-handed sub-skew lattice SL of S such that its
intersection with each D-class in S is an L-class. A right-handed section SR of S is similarly
defined. An internal (subobject) variant of the Kimura decomposition is the following result:
50
Proof. If S is a right-handed symmetric skew lattice, then it satisfies the identities of the above
theorem plus the identities x∨y∨x = x∨y and x∧y∧x = y∧x from which the displayed identities
follow. Conversely, if the displayed identities hold on a skew lattice S, then one has
x∧y∧x = x∧(y∧x) = x∧[x∧y∧(x∨y)] = x∧y∧(x∨y) = y∧x
and similarly x∨y∨x = x∨y so that S is right-handed. But using x∨y∨x = x∨y and x∧y∧x = y∧x,
the displayed identities can be transformed back into the identities of the previous theorem. £
Our interest in symmetry is due further to the following three theorems.
Theorem 2.2.6. All skew lattices in rings (using multiplication and the circle operation)
are symmetric.
Proof. Obviously a + b – ab = b + a – ba if and only if ab = ba. £
A lattice section in a skew lattice S is any sublattice T of S having nonempty intersection
with each D-class of S, in which case, T ≅ S/D.
Theorem 2.2.7. If S is a symmetric skew lattice for which S/D is countable, then S has a
lattice section.
Proof. Let D1, D2, … be a listing of all D-classes. Pick x1 ∈ D1. By Theorem 2.2.1 we know
that in every D-class of S elements exist that commute with x. By symmetry we know that such
commuting is under both operations and, moreover, that the set of all such elements is closed
under both operations. Thus S1 = {y ∈ S⎮y commutes with x} is a sub-skew lattice of S that has
nonempty intersection with each D-class of S and for which Dx1 = {x1} as a D-class in S1. If S1
is not a lattice (and hence not a lattice section of S) we find an element x2 in D2∩S1 and thus not
in Dx1 in S1. The argument repeats. We thus obtain a descending chain of sub-skew lattices
S1 ⊇ S2 ⊇ S3 ⊇ …
each having nonempty intersection with each D-class of S, and such that Dj ∩ Si = {xj} for all
i ≥ j where each xj commutes with all xi that precede it. Clearly, the full intersection n≥1Sn is a
lattice section of S. £
A left-handed section of S is any left-handed sub-skew lattice SL of S such that its
intersection with each D-class in S is an L-class. A right-handed section SR of S is similarly
defined. An internal (subobject) variant of the Kimura decomposition is the following result:
50
