@comment -*-texinfo-*-
@comment $Id$
@comment this file describes polynomial graded commutative algebras and their implementation in Singular

@node Graded commutative algebras (SCA),LETTERPLACE,Mathematical background (plural),Non-commutative subsystem
@section Graded commutative algebras (SCA)
@cindex Graded commutative algebras
@cindex super-commutative algebras
@cindex SCA

This section describes basic mathematical notions, definition, and a little bit the implementation of
the experimental non-commutative kernel extension @sc{Sca} of @sc{Singular}
which improves performance of many algorithms in graded commutative algebras.

In order to improve performance of @sc{Singular} in specific non-commutative algebras
one can extend the internal implementation for them in a virtual-method-overloading-like manner.
At the moment graded commutative algebras (SCA) and in particular exterior algebras are implemented this way.

Note that grade commutative algebras require no special user actions apart from 
defining an appropriate non-commutative GR-algebra in @sc{Singular}. 
Upon doing that, the supper-commutative structure will be automatically detected and 
special multiplication will be used. 
Moreover, in most SCA-aware (e.g. @code{std}) algorithms special internal improvements will be used
(otherwise standard generic non-commutative implementations will be used).

All considered algebras are assumed to be associative @math{K}-algebras for some ground field @math{K}.

@itemize
@item Definition

Polynomial graded commutative algebras are factors of tensor products of commutative algebras 
with an exterior algebra over a ground field @math{K}.

These algebras can be naturally endowed with a @math{Z_2}-grading, where anti-commutative algebra generators 
have degree @math{1} and commutative algebra generators (and naturally scalars) have degree @math{0}. 
In this particular case they may be considered as super-commutative algebras.

@item GR-algebra representation

A graded commutative algebra with @math{n} commutative and @math{m} anti-commutative algebra generators can be represented 
as factors of the following GR-algebra by some two-sided ideal: 
@tex
$$K\left\langle x_1, \ldots, x_n; y_1, \ldots, y_m \mid y_j * y_i = - y_i y_j, j > i \right\rangle / \left\langle y_1^2, \ldots, y_m^2  \right\rangle.$$
@end tex 

@item Distinctive features

Graded commutative algebras have zero divisors iff @math{m > 0}: @math{y_i * y_i = 0}.

Unlike other non-commutative algebras one may use any monomial ordering
where only the non-commutative variables are required to be global.
In particular, commutative variables are allowed to be local. 
This means that one can work in tensor products of any commutative ring with an exterior algebra.

@item Example of defining graded commutative algebras in @sc{Singular}

Given a commutative polynomial ring @math{r}, super-commutative structure on it can be introduced as follows:

@smallexample
@c example
LIB "nctools.lib";
ring r = 0,(a, b, x,y,z, Q, W),(lp(2), dp(3), Dp(2));
// Let us make variables x = var(3), ..., z = var(5) to be anti-commutative
// and add additionally a quotient ideal:
def S = superCommutative(3, 5, ideal(a*W + b*Q*x + z) ); setring S; S;
ideal I = a*x*y + z*Q + b, y*Q + a; I;
std(I); // Groebner basis is used here since > is global
kill r;
// Let's do the same but this time with some local commutative variables:
ring r = 0,(a, b, x,y,z, Q, W),(dp(1), ds(1), lp(3), ds(2));
def S = superCommutative(3, 5, ideal(a*W + b*Q*x + z) ); setring S; S;
ideal I = a*x*y + z*Q + b, y*Q + a; I;
std(I);
@c example
@end smallexample
@end itemize

See example of @code{superCommutative} from the library @code{nctools.lib}.

Reference: Ph.D thesis by Oleksandr Motsak (2010).