Page 183 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 183
V: Further Topics in Skew Lattices
{ } { }b1, b2 b3, b4 ... b2n–1, b2n and b2n, b1 b2, b3 ... b2n−2, b2n−1
and
{ } { }b2k+1, b2k+2 and b2k , b2k+1 when n = ω.
Clearly Bn is a single component. We denote the left-handed skew chain thus determined by Xn
and its right-handed dual by Yn for n ≤ ω. Their shared Hasse diagrams are as follows.
a1 − a2 a1−a2
2: b1 − b2 − b3 − b4 − b1
1: b1 − b2 ACA C
c1−c2
c1 − c2
a1 − a2 a1 − a2
n: b1 − b2 − .. − b2n –1 − b2n − (b1 ) ω: ... − b−2 − b−1 − b0 − b1 −A b2 − ...
A C C A C A C A C C
c1 − c2 c1 − c2
In line with our remarks above, instances of left-handed operations on X2 are given by
a1 ∨ c2 = a1 ∨ a2 = a2, a1 ∧ b4 = b3 ∧ b4 = b3 and b1 ∨ c2 = b1 ∨ b4 = b4.
Except for X1 and Y1, none of these skew lattices is categorical. Indeed,
a1 > b1 > c1, a2∧c1∧a2 = c2, c2∨a1∨c2 = a2,
but a2∧b1∧a2 = b2, while c2∨b1∨c2 is either b2n or b0. Thus, except for X1 and Y1, none of these
is distributive. Note also that each Xn is generated from a1, c2 and any bi. Thus no Xn contains a
copy of a lower Xm as a subalgebra. Similar remarks hold for the Yn.
Theorem 5.4.5. A left-handed skew lattice is categorical if and only if it contains no
copy of Xn for 2 ≤ n ≤ ω. Dually, a right-handed skew lattice is categorical if and only if it
contains no copy of Yn for 2 ≤ n ≤ ω. In general, a skew lattice is categorical if and only if it
contains no copy of any of these algebras.
Proof: We begin with a left-handed noncategorical skew chain S. Without loss of generality we
may assume that A is a full B-coset in itself and that C is a full B-coset in itself. Let a > b > c in
S with aʹ L a and cʹ L c being such that aʹ ∧ c = cʹ in C, a ∨ cʹ = aʹ in A so that a, aʹ correspond
to c, cʹ under a coset bijection between A and C, but aʹ ∧ b ≠ b ∨ cʹ in B where A, B, C
are respective L-classes. The first new elements formed are aʹ∧b and b∨cʹ in B. Note that
181
{ } { }b1, b2 b3, b4 ... b2n–1, b2n and b2n, b1 b2, b3 ... b2n−2, b2n−1
and
{ } { }b2k+1, b2k+2 and b2k , b2k+1 when n = ω.
Clearly Bn is a single component. We denote the left-handed skew chain thus determined by Xn
and its right-handed dual by Yn for n ≤ ω. Their shared Hasse diagrams are as follows.
a1 − a2 a1−a2
2: b1 − b2 − b3 − b4 − b1
1: b1 − b2 ACA C
c1−c2
c1 − c2
a1 − a2 a1 − a2
n: b1 − b2 − .. − b2n –1 − b2n − (b1 ) ω: ... − b−2 − b−1 − b0 − b1 −A b2 − ...
A C C A C A C A C C
c1 − c2 c1 − c2
In line with our remarks above, instances of left-handed operations on X2 are given by
a1 ∨ c2 = a1 ∨ a2 = a2, a1 ∧ b4 = b3 ∧ b4 = b3 and b1 ∨ c2 = b1 ∨ b4 = b4.
Except for X1 and Y1, none of these skew lattices is categorical. Indeed,
a1 > b1 > c1, a2∧c1∧a2 = c2, c2∨a1∨c2 = a2,
but a2∧b1∧a2 = b2, while c2∨b1∨c2 is either b2n or b0. Thus, except for X1 and Y1, none of these
is distributive. Note also that each Xn is generated from a1, c2 and any bi. Thus no Xn contains a
copy of a lower Xm as a subalgebra. Similar remarks hold for the Yn.
Theorem 5.4.5. A left-handed skew lattice is categorical if and only if it contains no
copy of Xn for 2 ≤ n ≤ ω. Dually, a right-handed skew lattice is categorical if and only if it
contains no copy of Yn for 2 ≤ n ≤ ω. In general, a skew lattice is categorical if and only if it
contains no copy of any of these algebras.
Proof: We begin with a left-handed noncategorical skew chain S. Without loss of generality we
may assume that A is a full B-coset in itself and that C is a full B-coset in itself. Let a > b > c in
S with aʹ L a and cʹ L c being such that aʹ ∧ c = cʹ in C, a ∨ cʹ = aʹ in A so that a, aʹ correspond
to c, cʹ under a coset bijection between A and C, but aʹ ∧ b ≠ b ∨ cʹ in B where A, B, C
are respective L-classes. The first new elements formed are aʹ∧b and b∨cʹ in B. Note that
181
