Page 184 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 184
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
a∨(aʹ∧b) = aʹ since both are images of aʹ∧b in A, which is the unique B-coset within A.
Likewise, (b∨cʹ)∧c = cʹ. No new elements are created thus far from {a, aʹ, b, c, cʹ} giving us:
A: a − a′
B: b − a′ ∧ b − b ∨ c′
C: c − c′
At this stage it follows that the subalgebra formed from {a, aʹ, b, c, cʹ} only has {a, aʹ} in the top
L-class and {c, cʹ} in the bottom L-class. Continuing, step-by-step, in both directions we get
− a′ − a − a′ − a − a′ − a − a′ −
– [a ∧ (b ∧ c′)] ∨ c′ – a ∧ (b ∧ c′) – b ∨ c′ – b – a′ ∧ b – (a′ ∧ b) ∨ c – a ∧ [(a′ ∧ b) ∨ c] –
A C A CA C A C
. − c′ − c − c′ − c − c′ − c − c′ −
Expansion thus continues in B. If repetition never occurs, we obtain a copy of Xω. Otherwise, a
cyclic closure arises and we have obtained a copy of some Xn. The left-handed case follows from
this. The right-handed case is similar.
Clearly, a categorical skew lattice contains none of these algebras. Conversely, if a skew
lattice S contains copies of none of them, then neither does S/R or S/L since every skew chain
with finitely many D-classes in either S/R or S/L can be lifted up into S. Thus both S/L and S/R
are categorical, and hence so it S which is embedded in their product. £
Corollary 5.4.6. Distributive skew lattices contain no copies of Xn or Yn for n ≥ 2. £
Comment: If these skew chains seem familiar it is because they are precisely the
maximal skew chains within the skew diamonds arising in the classification of all symmetric
skew lattices. Xn and Yn for all n ≥ 2 in particular arise in the various non-categorical cases.
Strictly categorical skew lattices
We turn to categorical skew lattices that are strictly categorical in that for each skew chain
of D-classes A > B > C, each A-coset in B has nonempty intersection with each C-coset in B,
making both B an entire AC-component and empty coset bijections unnecessary. Examples are:
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a∨(aʹ∧b) = aʹ since both are images of aʹ∧b in A, which is the unique B-coset within A.
Likewise, (b∨cʹ)∧c = cʹ. No new elements are created thus far from {a, aʹ, b, c, cʹ} giving us:
A: a − a′
B: b − a′ ∧ b − b ∨ c′
C: c − c′
At this stage it follows that the subalgebra formed from {a, aʹ, b, c, cʹ} only has {a, aʹ} in the top
L-class and {c, cʹ} in the bottom L-class. Continuing, step-by-step, in both directions we get
− a′ − a − a′ − a − a′ − a − a′ −
– [a ∧ (b ∧ c′)] ∨ c′ – a ∧ (b ∧ c′) – b ∨ c′ – b – a′ ∧ b – (a′ ∧ b) ∨ c – a ∧ [(a′ ∧ b) ∨ c] –
A C A CA C A C
. − c′ − c − c′ − c − c′ − c − c′ −
Expansion thus continues in B. If repetition never occurs, we obtain a copy of Xω. Otherwise, a
cyclic closure arises and we have obtained a copy of some Xn. The left-handed case follows from
this. The right-handed case is similar.
Clearly, a categorical skew lattice contains none of these algebras. Conversely, if a skew
lattice S contains copies of none of them, then neither does S/R or S/L since every skew chain
with finitely many D-classes in either S/R or S/L can be lifted up into S. Thus both S/L and S/R
are categorical, and hence so it S which is embedded in their product. £
Corollary 5.4.6. Distributive skew lattices contain no copies of Xn or Yn for n ≥ 2. £
Comment: If these skew chains seem familiar it is because they are precisely the
maximal skew chains within the skew diamonds arising in the classification of all symmetric
skew lattices. Xn and Yn for all n ≥ 2 in particular arise in the various non-categorical cases.
Strictly categorical skew lattices
We turn to categorical skew lattices that are strictly categorical in that for each skew chain
of D-classes A > B > C, each A-coset in B has nonempty intersection with each C-coset in B,
making both B an entire AC-component and empty coset bijections unnecessary. Examples are:
182
