An object of the class PPOrdering represents an "arithmetic" ordering on
the (multiplicative) monoid of power products, i.e. such that the
ordering respects the monoid operation (viz. s < t => r*s < r*t for all
r,s,t in the monoid). In CoCoALib orderings and gradings are
intimately linked (for gradings see also degree).
Currently, the most typical use for a PPOrdering object is as a constructor
argument to a concrete PPMonoid. At the moment there are four
functions which create new PPOrderings:
NewLexOrdering(NumIndets);
NewStdDegLexOrdering(NumIndets);
NewStdDegRevLexOrdering(NumIndets);
NewMatrixOrdering(NumIndets, GradingDim, OrderMatrix);
The first three create respectively lex, DegLex and DevRevLex orderings on the given number of indeterminates. Note the use of "Std" in the names to emphasise that they are only for standard graded polynomial rings.
The last function creates a PPOrdering given a matrix. GradingDim
specifies how many of the rows of OrderMatrix are to be taken as
specifying the grading.
The operations on a PPOrdering object are:
out << PPO; // output the PPO object to channel out
NumIndets(PPO); // number of indeterminates the ordering is intended for
GradingDim(PPO); // the dimension of the grading associated to the ordering
GetMatrix(PPO); // a matrix defining the ordering
IsLex(PPO); // true iff PPO is implemented as lex
IsStdDegLex(PPO); // true iff PPO is implemented as StdDegLex
IsStdDegRevLex(PPO); // true iff PPO is implemented as StdDegRevLex
Thus CoCoALib supports graded polynomial rings with the restriction that the
grading be compatible with the PP ordering: i.e. the grading comprises
simply the first k entries of the "order vector". The GradingDim is merely
the integer k (which may be zero if there is no grading).
A normal CoCoA library user need know no more than this about PPOrderings.
CoCoA Library contributors and the curious should read on.
A recent addition. There is also a member function:
PPO.myMatrixCopy(M) where M is a matrix
It fills M with a matrix which specifies the ordering PPO. Don't use it yet.
The general ideas behind the implementations of PPOrdering and
PPOrderingBase are analogous to those used for ring and RingBase.
PPOrdering is a simple reference counting smart-pointer class, while
PPOrderingBase hosts the intrusive reference count (so that every concrete
derived class will inherit it).
The only remaining observation to make about the simple class PPOrdering is
that I have chosen to disable assignment -- I find it hard to imagine when
it could be useful to be able to assign PPOrderings, and suspect that
allowing assignment is more likely to lead to confusion and poor programming
style.
There are four concrete PPOrderings in the namespace CoCoA::PPO. The
implementations are all simple and straightforward except for the matrix
ordering which is a little longer and messier but still easy enough to
follow.
Must resolve the restrictions on order matrices in NewMatrixOrdering
(matrix must be square).
We need better ways to "compose" PPOrderings, i.e. to build new ones
starting from existing ones. Max knows the sorts of operation needed
here. Something similar to CoCoA4's BlockMatrix command is needed.