The class RingHom is intended to represent homomorphisms between rings.
Currently there is no way to represent more general maps between rings.
It is possible to create a partial homomorphism which can generate
run-time errors when applied to certain values.
The main operations available for RingHoms are application to a
RingElem, and composition: both operations use function application
syntax (see below for details).
Here is a complete list of pseudo-constructors for ring homomorphisms
(some are defined in other files, e.g. QuotientRing.H or FractionField.H).
You should check also the documentation for CanonicalHom which tries to
offer an easy way of building certain simple canonical homomorphisms.
IdentityHom(R) --
where R may be any ring, gives the identity homomorphism on R
ZZEmbeddingHom(R) --
gives the unique homomorphism from RingZZ to the ring R
QQEmbeddingHom(R) -- may be partial
gives the unique homomorphism from RingQQ to the ring R.
Note that the resulting homomorphism may be only partial
(e.g. if Characteristic(R) is not 0, or if the codomain is
not a field).
CoeffEmbeddingHom(P) --
where P is a PolyRing, gives the embedding homomorphism from
the coefficient ring into the polynomial ring.
EmbeddingHom(FrF) --
where FrF is a FractionField, gives the embedding homomorphism
from the base ring into the fracion field (i.e. x |-> x/1)
InducedHom(RmodI, phi) --
where RmodI is a QuotientRing, gives the homomorphism
R/I --> S induced by phi: R --> S
(which must have the base ring of RmodI as its domain,
and whose kernel must contain the defining ideal of RmodI)
InducedHom(FrF, phi) -- may be partial
where FrF is a FractionField, gives the homomorphism induced by
phi (which must have the base ring of FrF as its domain). Note that
the resulting homomorphism may be only partial (e.g. if ker(phi) is
non-trivial, or if the codomain is not a field).
PolyAlgebraHom(Rx, Ry, xImages) -- where
gives the homomorphism from Rx to Ry
mapping coefficients into themselves and mapping the k-th
indeterminate of Rx to the k-th value in xImages (i.e. having
index k-1)
PolyRingHom(Rx, S, CoeffHom, xImages) -- where
Rx is a PolyRing
CoeffHom is a homomorphism whose
domain is CoeffRing(Rx) and codomain is S or CoeffRing(S) (if S is a PolyRing)
xImages is a vector of RingElem or a string
Rx to S
mapping coefficients according to
CoeffHom and mapping the k-th indeterminate of Rx to the k-th
value in xImages (i.e. having index k-1)
A RingHom may applied using natural syntax:
- let phi be an object of type RingHom
- let x be an object of type RingElem
- let n be of type long or int
- let N be an object of type BigInt
- let q be an object of type BigRat
phi(x) applies phi to x; error if owner(x) != domain(phi)
phi(n) applies phi to the image of n in domain(phi)
phi(N) applies phi to the image of N in domain(phi)
phi(q) applies phi to the image of q in domain(phi)
In all cases the result is a RingElem belonging to the codomain of
phi. Currently partial homomorphisms are allowed, so applying a
RingHom could trigger an error (e.g. an induced hom from Q to Z/(3)
applied to 1/3).
If the RingElem supplied belongs to the wrong ring (i.e. not the
domain of the RingHom) then an exception is thrown with code
ERR::BadRingHomArg1. If the argument belongs to the right ring but
lies outside the domain then an exception with code
ERR::BadRingHomArg2 is thrown.
Two RingHoms may be composed using a fairly natural syntax: if we have
two RingHoms phi:R -> S and theta:S -> T then their composition may
be computed using the syntax
theta(phi) the composite homomorphism "apply phi first then theta"
If the domain of the outer homomorphism is different from the codomain of
the inner homomorphism then an exception is thrown with code
ERR::BadCompose.
We may ask for the domain and codomain of a RingHom phi:
domain(phi) gives a const ref to the domain
codomain(phi) gives a const ref to the codomain
Note that the domain and codomain are merely rings, they "forget"
any special ring type (such as SparsePolyRing).
Currently it is not possible to ask for the kernel of a RingHom.
All operations on a RingHom are invisibly converted into member function
calls on a RingHomBase. It is possible to call these member functions
directly: the main difference is that the member functions do not perform
any sanity checking on their arguments (so they should be slightly faster
but if you hand in incompatible arguments, you'll probably get an ugly
crash).
These files contain two "generic" classes (RingHom and RingHomBase), and
a trivial concrete class representing the identity ring homomorphism,
IdentityRingHom. Most of this section is dedicated to the two generic
classes since they represent the primary contribution to the CoCoA
library.
The class RingHom is little more than a "reference counting smart
pointer" class to objects of type RingHomBase; this latter type is
designed to support intrusive reference counting. Beyond its role as a
smart pointer RingHom offers four "function application" syntaxes:
RingElem RingHom::operator()(ConstRefRingElem x) const;
RingElem RingHom::operator()(long n) const;
RingElem RingHom::operator()(const BigInt& N) const;
RingHom RingHom::operator()(const RingHom&) const;
The first three support a natural syntax for applying the homomorphism to a ring element, a small integer, or a large integer. The last offers a fairly natural syntax for creating the composition of two homomorphisms.
The class RingHomBase is a purely abstract class which is used to specify
the interface which any concrete ring homomorphism class must offer. In
particular this base class already includes an intrusive reference counter,
as required by RingHom. It also includes two private data members
myDomainValue and myCodomainValue which store the domain and codomain
rings. Note that these data fields are plain rings and so "forget" any
special ring type which the domain or codomain may have had. Originally I
had hoped to preserve any special ring type information, but this seemed to
lead to a confusing and complex implementation (which probably would never
have worked as I hoped). The two ring fields may be read using the accessor
functions:
const ring& myDomain() const;
const ring& myCodomain() const;
A concrete class implementing a ring homomorphism must supply definition for the following (pure virtual) functions:
virtual void myApply(RingBase::RawValue& image, RingBase::ConstRawValue arg) const;
virtual void myOutputSelf(std::ostream& out) const;
DO NOTE THAT the two arguments to myApply normally belong to
DIFFERENT rings. arg belongs to myDomain() whereas image
belongs to myCodomain(). The function myOutputSelf should
print out a useful description of the homomorphism.
Cannot compute a kernel of a RingHom.
Arranging for domain(phi) and codomain(phi) to preserve C++ type
information about the respective rings (e.g. PolyRing or FractionField
rather than simply ring), appears to be difficult to achieve in any
reasonable manner. I've decided that it is much simpler just to discard
all special type information, and return simply rings. If the user knows
something more, he can use a "cast" function like AsFractionField. Even
if it were feasible to maintain such C++ type info, there would have to
n-squared cases to cover all possible combinations of domain and codomain.
We should implement more special cases: e.g. same vars different coeff ring, PP --> PP, other... Also need some way of handling canonical homomorphisms.
Some special cases of homomorphic embeddings R --> S: (may belong with the special types of ring to which they are associated)
This all needs to be sorted out!
How to decide whether a value can be mapped into the current_ring?
If the rings are marked as being equivalent isomorphically then we can just use the obvious isomorphism. A more interesting case is when a value resides in a ring which is a natural subring of the current_ring e.g. Z inside Q(sqrt(2))[x,y,z].
One could argue that to create Q(sqrt(2))[x,y,z] we had to follow this path
Z --> fraction field Q
Q --> polynomial ring (1 indet) or DUP extension Q[gensym]
Q[gensym] --> quotient by gensym^2-2 to get Q(sqrt(2))
Q(sqrt(2)) --> polynomial ring (3 indets) Q(sqrt(2))[x,y,z]
From this it ought to be easy to identify natural embeddings of Z,
Q, and (possibly) Q(sqrt(2)) in Q(sqrt(2))[x,y,z]. We do
not get an embedding for Q[gensym] since we had to generate the symbol
gensym and no one else can create the same gensym. Because of this
it is not altogether clear that an independently created copy of
Q(sqrt(2)) can be embedded automatically, since that copy would
have a different symbol/gensym. Now if the algebraic extension were
achieved directly...
Would we want Q[x]/(x^2-2) to be regarded as isomorphically equivalent
to Q[y]/(y^2-2)? In fact there are two possible isoms: x <---> y
and x <---> -y. I think that these should not be viewed as isom
automatically, especially as there is more than one possible choice.
In contrast, if R = Q[x]/(x^2-2), and S = Q[x]/(36-18x^2), and
T = Q[x]/(x^2-2). It is clear that Q[x] can be mapped into
each of R, S and T in a natural way. Of course, in each
case x stands for sqrt(2), and it wouldn't be too hard to spot
that R and T are identical; it is not quite as simple to see
that R and S are isom. Presumably with a little more effort
one could create examples where it could be jolly hard to spot that
two such rings are just the same ring. For this reason, I think no
attempt should be made to spot such natural isoms between
independent rings. Had T been created from R (e.g. by
making copy via assignment) then they would no longer be independent,
and a natural isom could be deduced automatically. Now I think about
it, a facility to make a copy of a ring WITHOUT the natural isom
should be made available.
There is also a need for a way to specify that one ring embeds
naturally into another (and via which homomorphism), or indeed that
they are isomorphic. Isomorphism could be expressed by giving two
inverse homs -- the system could then check that the homs are inverse
on the generators, how it would check that the maps are homs is not so
clear (perhaps the only maps which can be created are homs). Oooops,
this would allow one to declare that Z and Q (or Z[x] and
Q[x]) are isom..... need to think more about this!
A similar mechanism will be needed for modules (and vector spaces). A module should naturally embed into a vector space over the fraction field of the base ring....
Conceivably someone might want to change the natural embedding between two rings. So a means of finding out what the natural embedding is will be necessary, and also a way replacing it.
There is also a general question of retracting values into subrings.
Suppose I have computed 2 in Q(x), can I get the integer 2 from
this? In this case I think the user must indicate explicitly that a
retraction is to occur. Obviously retraction can only be into rings
on the way to where the value currently resides.
Other points to note:
Q(x) = Z(x) = FrF(Z[x]) == FrF(FrF(Z)[x])
Q(alpha) = FrF(Z[alpha]) though denoms in Q(alpha) can be taken in Z
Q[alpha]/I_alpha = FrF(Z[alpha]/I_alpha) BUT the ideal on LHS is an ideal inside Q[alpha] whereas that on RHS is in Z[alpha]. Furthermore Z[alpha]/I_alpha is hairy if the min poly of alpha is not monic!