Page 54 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 54
athan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Theorem 2.15. Both decompositions are similar in that they both involve copies of the maximal
lattice, left-handed and right-handed images of S. While external decompositions always occur
and use quotient objects, internal decompositions that use subobjects need not always occur.
One is led to ask the following question. Is symmetry really needed for the all this?
Otherwise put, does a skew lattice S exist for which S/D is countable, or even finite, but S has no
lattice section?
Example 2.2.2 continued. Returning to the non-symmetric Example 2.2.2, a lattice
section is given by the subset {0, a, b, c, d, f, 1}.
Some elementary, albeit useful, cases where symmetry is not needed are as follows.
Theorem 2.2.9. A skew lattice S will have a lattice section T if its maximal lattice image
S/D is a copy of one of the following lattices:
1 1 1
(i) 2 × 2 •• (ii) M3 ••• (iii) N5
• •
0 0 •
0
In these cases, S also has internal copies of S/R and S/L, both occurring as maximal left and
right-handed subalgebras of S.
Proof. First, suppose S/D is a copy of 2 × 2. Let a, b, e, f in S be such that e is in the top D-
class, f lies in the bottom D-class, and a and b are separately in the two incomparable D-classes.
We create a lattice section as follows.
1. Reset a to be e∧a∧e, b to be e∧b∧e and f to be e∧f∧e so that a, b, f < e.
2. Reset a to be f∨a∨f and b to be f∨b∨f so that f < a, b < e is a lattice section.
Next suppose S/D is a copy of M3. Let a, b, c, e, f in S be such that e is in the top D-class, f is
in the bottom D-class, and a, b and c lie separately in the three incomparable D-classes. We
create a lattice section as follows.
1. Reset all x among a, b, c, f to be e∧x∧e so that now a, b, c, f < e.
2. Reset all y among a, b, c to be f∨y∨f so that f < a, b, c < e is a lattice section.
Finally suppose S/D is a copy of N5. Let a, b, c, e, f in S be such that e and f are as before, a, b
and c lie separately in the three middle D-classes, where the D-class of a is incomparable with
the D-classes of b and c, but the D-class of b lies above the D-class of c. We create a lattice
section by first repeating the first two steps so that f < a, b, c < e and then:
3. Reset c to be b∧c∧b so that we also have c < b.
The final assertion follows from Theorem 2.2.8. £
52
Theorem 2.15. Both decompositions are similar in that they both involve copies of the maximal
lattice, left-handed and right-handed images of S. While external decompositions always occur
and use quotient objects, internal decompositions that use subobjects need not always occur.
One is led to ask the following question. Is symmetry really needed for the all this?
Otherwise put, does a skew lattice S exist for which S/D is countable, or even finite, but S has no
lattice section?
Example 2.2.2 continued. Returning to the non-symmetric Example 2.2.2, a lattice
section is given by the subset {0, a, b, c, d, f, 1}.
Some elementary, albeit useful, cases where symmetry is not needed are as follows.
Theorem 2.2.9. A skew lattice S will have a lattice section T if its maximal lattice image
S/D is a copy of one of the following lattices:
1 1 1
(i) 2 × 2 •• (ii) M3 ••• (iii) N5
• •
0 0 •
0
In these cases, S also has internal copies of S/R and S/L, both occurring as maximal left and
right-handed subalgebras of S.
Proof. First, suppose S/D is a copy of 2 × 2. Let a, b, e, f in S be such that e is in the top D-
class, f lies in the bottom D-class, and a and b are separately in the two incomparable D-classes.
We create a lattice section as follows.
1. Reset a to be e∧a∧e, b to be e∧b∧e and f to be e∧f∧e so that a, b, f < e.
2. Reset a to be f∨a∨f and b to be f∨b∨f so that f < a, b < e is a lattice section.
Next suppose S/D is a copy of M3. Let a, b, c, e, f in S be such that e is in the top D-class, f is
in the bottom D-class, and a, b and c lie separately in the three incomparable D-classes. We
create a lattice section as follows.
1. Reset all x among a, b, c, f to be e∧x∧e so that now a, b, c, f < e.
2. Reset all y among a, b, c to be f∨y∨f so that f < a, b, c < e is a lattice section.
Finally suppose S/D is a copy of N5. Let a, b, c, e, f in S be such that e and f are as before, a, b
and c lie separately in the three middle D-classes, where the D-class of a is incomparable with
the D-classes of b and c, but the D-class of b lies above the D-class of c. We create a lattice
section by first repeating the first two steps so that f < a, b, c < e and then:
3. Reset c to be b∧c∧b so that we also have c < b.
The final assertion follows from Theorem 2.2.8. £
52
