GNU Free Documentation License, Version 1.2

- Examples
- User documentation
- Maintainer documentation for PPOrdering
- Bugs, shortcomings and other ideas

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`

. If you want to use an ordering to compare power
products then see `PPMonoid`

.

Currently, the most typical use for a `PPOrdering`

object is as an
argument to a constructor of a concrete `PPMonoid`

or `PolyRing`

,
so see below **Convenience constructors**.

These are the functions which create new `PPOrdering`

s:

`lex(NumIndets)`

-- GradingDim = 0`xel(NumIndets)`

-- GradingDim = 0`StdDegLex(NumIndets)`

-- GradingDim = 1`StdDegRevLex(NumIndets)`

-- GradingDim = 1`NewMatrixOrdering(OrderMatrix, GradingDim)`

The first three create respectively `lex`

, `StdDegLex`

and
`StdDegRevLex`

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 (*i.e.* each indet has degree 1).

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. Then entries of the given matrix must be integers
(and the ring must have characteristic zero).

For convenience there is also the class `PPOrderingCtor`

which provides
a handy interface for creating `PPMonoid`

and `SparsePolyRing`

, so that
`lex`

, `xel`

, `StdDegLex`

, `StdDegRevLex`

may be used as shortcuts instead
of the proper constructors, *e.g.*

NewPolyRing(RingQQ(), symbols("a","b","c","d"), lex);

is the same as

NewPolyRing(RingQQ(), symbols("a","b","c","d"), lex(4));

`IsStdGraded(PPO)`

-- true iff`PPO`

is standard graded (or "degree compatible")`IsLex(PPO)`

-- true iff`PPO`

is implemented as lex`IsXel(PPO)`

-- true iff`PPO`

is implemented as xel`IsStdDegLex(PPO)`

-- true iff`PPO`

is implemented as StdDegLex`IsStdDegRevLex(PPO)`

-- true iff`PPO`

is implemented as StdDegRevLex`IsMatrixOrdering(PPO)`

-- true iff`PPO`

is implemented as MatrixOrdering`IsTermOrdering(PPO)`

-- true iff`PPO`

is a term ordering

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`OrdMat(PPO)`

-- a (constant) matrix defining the ordering`GradingDim(PPO)`

-- the dimension of the grading associated to the ordering`GradingMat(PPO)`

-- the matrix defining the grading associated to the ordering

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 `PPOrdering`

s.
CoCoA Library contributors and the curious should read on.

A `PPOrdering`

is just a smart pointer to an instance of a class
derived from `PPOrderingBase`

; so `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).

There are four concrete `PPOrdering`

s 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.

The class `PPOrderingCtor`

is just a simple "trick" to allow for
a convenient user interface. The mem fn `operator()`

, with arg the
actual number of indets, is used to generate an actual ordering.

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.

2021-02-21: added `xel`