2. Lattices
2.1
Points in a lattice in dimension one generated by
have the form of
. Rewrite
as:
![{\displaystyle \gcd(a,b)(ma'+nb'),a'={\frac {a}{\gcd(a,b)}},b'={\frac {b}{\gcd(a,b)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee76cffea782eac092b12bb6c9de721c3d9e004)
By definition a' is coprime with b'. By Bézout's identity we know that there exist integers m and n such that
. It follows that for all
, we have
.
Since
must be an integer, we conclude that all
must have form
. However, this means that a dimension-1 lattice generated by
can be generated equivalently by
.
Therefore, we see that all dimension-1 lattices generated by a set of vectors can be equivalently generated by a set of one vector. One can achieve this by repeatedly replacing one pair of vectors within the set with an "equivalent" vector, reducing the total number of vectors in the set by one for each step, until only one vector is remaining. Thus, all the lattices in dimension one can be generated by
.
2.2
Assume the clause following the "iff" is true, and that all vectors in
is expressible in the form
.
Define
, and also set
. Now define
. We see that
is true, therefore
by definition. This means that for all
,
, since it is true that
and is therefore expressible as
.
Similarly, assume that the premise is true and that if lattice
generated by
is full, then by definition for all
there exists some
such that
. From the definition of the lattice we know that a point in
is expressible as
.
From the fact that
, we have
. Define
, and we see that
must be in
since
. Therefore, if
is full, then all
can be expressed as
.
2.3
The lattice generated by
is not full; adding vectors
gives
, which is a multiple of the third vector,
. We note that a vector of the form
such that
is not in the lattice since it is not possible to combine vectors such that the first two coordinates remain equal while the third varies. Because every integer multiple of such a vector will remain in the same form, the lattice is therefore not full.
2.4
Arithmetic with vectors first give
, and then
. We then do
, and finally
.
This means that a subset of the lattice (We denote this
can be written as a combination of vectors
. Since
, we can see that given any vector
, we have
, which means
, meaning the lattice given is full.
3. Determinant and Divisor
3.1
Given that
, then for any
, we also have
.
From this, we see that a lattice point
in
corresponds to point
in colattice
. We also see that lattice point
in
corresponds to point
in colattice
. Therefore, we see that every point in
corresponds to a point in
if
, and thus they are equal.
Similarly, if
, then the points in
and
corresponding to their original point in
before the translation must differ by some sum of vectors
, because if that was not the case, the two colattices would not be equal to L.
From this, we have
. Subtract
from both sides to get the desired result.
3.2
Given that
, assume
. There is at least one
in the intersection that belongs to both
and
. Without loss of generality, write
. Subtracting
and
, respectively, yield vectors
and
, which should both be in
by definition. However, this implies that
, producing a contradiction, therefore
must be false, and thus
implies
.
Equivalently, this implies that
.
3.3
This lattice
, when generated by
, contains all
such that
since both vectors used to generate the lattice obey this. Collatices
,
,
obey
respectively due to the additional y-value contributed by the shifting vector, and thus it is impossible for points in each lattice to overlap and they are therefore distinct. Because a lattice must be within the
space, a point in a lattice must have integer coordinates. However,
cannot possibly be anything other than
, thus we see that the three colattices cover the entire
space. Therefore, we see that there are no more colattices of
.
3.4
As in 4.2, a lattice in
is full iff a subset
of it can be generated by
, where
, such that
is in the
position and
. (If it cannot be generated by one such set, then at least one vector in
does not satisfy the prerequisites of a "full" lattice. Similarly, if it can be generated by one such set, then for
,
as shown earlier in 2.
We find that
due to the fact that such a lattice represents a regular grid in
, which must be finite. If it is infinite,
cannot possibly be a full lattice as the product used to compute
is not an infinite product. Therefore, a lattice is full iff its determinant is finite.
3.5
Given lattice
generated by
and
, the lattice
generated by
is a homothety of
about the origin by a factor of
. The determinant of this lattice will clearly be an integer. Now apply a dilation about the origin by a factor of
. Every point in the
space of
now corresponds to
points in the
space of
. Therefore, every colattice of
corresponds to
colattices of
. Therefore,
, and we see that it must, therefore, be divisible by
since
.
4. Finite Generation
4.1
Given that
,
will contain more points than
for some arbitrary region of
. In such a region, the number of colattices that
can take on is equal to the number that
can take on, minus the number of points in
but not in
. Take a smallest region containing all vectors
such that
describe all distinct colattices of L_2. Here, every point not in
represents a distinct colattice, but for
, its number of distinct colattices is less than that of
as
contains more points (and therefore less empty spaces). Therefore, the determinant of
must be lower than that of
.
Oh the other hand, if the determinant of
is infinite, then the determinant of
is either also infinite, or takes on a finite value. The lattice containing all lattice points in
space would be a superset of all
, and yet have a determinant of 1.
4.2
A lattice in
is full iff a subset
of it can be generated by
, where
, such that
is in the
position and
. (If it cannot be generated by one such set, then at least one vector in
does not satisfy the prerequisites of a "full" lattice. Similarly, if it can be generated by one such set, then for
,
.
4.3
Given a finitely generated subset of the full lattice, the remaining points may be accounted for by the addition of vectors to the finite set. As full lattices are ordered, only a finite number of vectors is needed or allowed to account for all points in the lattice, thus only a finite number of vectors is needed to completely describe the lattice.
5. Isomorphism Types of Lattices
5.1
5.2
5.3
Given 5.6, we conclude that the two lattices are isomorphic as their determinants are both 15 and divisors both 1.
5.4
Assume
is generated by
and
is generated by
. Since
, by 5.1 we see that the lattices are not isomorphic, since the divisors would be equal if they were.
5.5
5.6
5.7
6. Canonical Form
6.1
As the isomorphism effectively relies on a projection of
, the signature is (1,1,0) as the GCD of 2 and 3 is 1.
6.2
The signature can be found by breaking down the lattice to the combination of
, giving the signature