README

This is a C++ library that helps to write code that reasonably
expresses various concepts from abstract algebra. Using this you can
write algorithms that act on generic group elements, and then plug in
specifics, and get optimized code for your specific usage.

Overall structure:

All of the mapping definitions are contained inside of Ops
structures. These ops define all the operations necessary for the
semigroup/monoid/group/ring elements. To review:

Semigroup: composition (*)
Monoid: composition (*), and identity
Group: composition (*), inverse (/), and identity (id)
Ring: multiply (*), add (+), negate(-), multiplicative identity (id),
      additive identity (zero)
Field: multiply (*), inverse (/), add (+), negate (-), multiplicative
       identity (id), additive identity (zero)

The operation structure will also have a ::element typedef which
defines the object type that will be contained, but that will most
often just be passed in as a template parameter. A good, simple
example to look at is BasicOps in basic.h. For something a bit more
complicated, see modn.h which implements integers mod N in both a
known-at-compilation and dynamic methods.

Most operation structures do not have a variable way of working, and
will have a Ops::instance object. However, some will require a
parameter to construct, like the dynamic integer mod N ops structure.

The operation structure will also generally be used with
groups/rings/etc, and will have typedefs in place for the
GroupElt/RingElt/etc that make use of the operation, via ::group,
::ring, etc.

A GroupElt contains both a representation of the group element it is
storing, as well as a reference to the operation structure so that it
knows how to compose that group element with other elements. (Same
goes for the other *Elt classes.) Operators are overridden to make
usage seamless. Also due to implicit casting, you can do something
like:

  typedef IntegerModNOps<5>::ring Mod5Ring;
  auto v = Mod5Ring(4) + 7;

and have v become the element 1.

Implemented types:

Rational -- stores an int numerator/denominator, supports + and *
Word<T> -- stores an ordered list of elements of type T
Trace<T> -- stores a Word<T> and specifies cyclic equality
Monomial<T> -- stores a map of T -> exponent
Polynomial<R, S> -- stores a map of S monoids -> R rings. Addition and
                    multiplication work as expected for polynomials
                    with coefficients in R and monomials in S. (R and
                    S are operations structures.)
DenseMatrix<Ops> -- stores a full matrix of elements from Ops::ring.

Implemented operation structures:

BasicOps<T> -- just uses the regular +, * semantics on the given type,
               and uses the literal 1 for identity and 0 for zero. A
               type has to be constructable from those literals to be
               used.
IntegerModNOps<N, T> -- integer operations mod N, which is given at
                        compile time.
IntegerModOps<T> -- integer operations mod N, which is given in the
                    constructor. Note that this ops structure needs to
                    be carefully passed to group elements, since there
                    is no default ::instance
MonomialOps<T> -- Monomial<T> as the element. semigroup and monoid typedefs
PolynomialOps<R, S> -- Polynomial<T> as the element. ring typedef.
DenseMatrixOps<Ops> -- DenseMatrix<Ops> as the element
DenseMatrixNSpace<N, Ops> -- defines a GL(N) space of matrices that
                             contain DenseMatrix elements in Ops::ring

See test.cc for demonstrations of a bunch of these, along with the
intended usage of all of these types.