#15976: IntegerLattice class
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Reporter: malb | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.2
Component: linear algebra | Resolution:
Keywords: | Merged in:
Authors: Martin Albrecht | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/malb/15976_integer_lattice | a13b4aad3e9578866922d01932e6abac02687718
Dependencies: | Stopgaps:
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Comment (by SimonKing):
Replying to [comment:7 malb]:
> However, the next complaint is that elements created by `some_element()`
do not live in the lattice but in '''Z'''^n^. I am not sure I want my
lattice vectors to live in the lattice, but if I wanted to do that, I'd
have to define my lattice as a proper submodule of '''Z'''^n^?
There is something called "facade". I didn't try to fully understand it,
but I think the idea is: If you have the set of prime numbers, then its
elements are just natural numbers. That's to say, the parent of the
elements of `Primes()` is `ZZ`, not `Primes()`:
{{{
sage: P = Primes()
sage: p = P.an_element()
sage: p
43
sage: p.parent()
Integer Ring
}}}
This is done by telling that `Primes()` is a "facade set":
{{{
sage: P.category()
Category of facade infinite enumerated sets
}}}
Your situation could be similar: You have a lattice `L` in 'RR^n`, and the
parent of the elements of `L` wouldn't be `L` but `RR^n` (but please not
`ZZ^n`, since you need the embedding!).
So, could be that using
{{{
sage: Category.join([FacadeSets(),CommutativeAdditiveGroups()])
Category of facade commutative additive groups
}}}
to initialise your lattice's category would suffice.
> I'd view the lattice somewhat analogously to an ideal in a multivariate
ring, would we want our ideal elements to live anywhere but the ring?
No, see above. But for historical reasons, we have
{{{
sage: P.<x,y> = QQ[]
sage: I = P*[x^2+y,y^2+x]
sage: x^2 in I
False
sage: x^2+y in I
True
sage: I.category()
Category of ring ideals in Multivariate Polynomial Ring in x, y over
Rational Field
}}}
without facade. I think today it would be implemented using facades.
> I don't want to cache my representations or make lattices unique. It is
useful to have two objects representing the same lattice: one with a good
basis and one with a bad basis (that's the secret and public key
respectively in many crypto systems).
That's why I suggested `CachedRepresentation` rather than
`UniqueRepresentation`. If you use `CachedRepresentation` than starting
twice with the same basis yields identical lattices, but starting with
different bases will yield non-identical lattices that may still be equal.
Only `UniqueRepresentation` would imply a "comparison by identity".
--
Ticket URL: <http://trac.sagemath.org/ticket/15976#comment:8>
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