@comment -*-texinfo-*-
@comment $Id: pluconventions.doc,v 1.13 2006-07-17 14:32:22 Singular Exp $
@comment this file contains the type definitions

@c The following directives are necessary for proper compilation
@c with emacs (C-c C-e C-r).  Please keep it as it is.  Since it
@c is wrapped in `@ignore' and `@end ignore' it does not harm `tex' or
@c `makeinfo' but is a great help in editing this file (emacs
@c ignores the conditionals).

@ignore
%**start
\input texinfo.tex
@setfilename plureference.info

@c @settitel PLURAL
@c @node Top, Getting started with PLURAL, (dir), (dir)
@menu
*  Getting started with PLURAL ::
@end menu
@c *  PLURAL conventions ::
@node Getting started with PLURAL,  , Top, Top
@chapter Getting started with PLURAL
%**end
@end ignore


@c @c -----------------------------
@c @menu
@c * *-multiplication@value{PSUFFIX}::
@c * factor@value{PSUFFIX}::
@c * ideals@value{PSUFFIX}::
@c * modules@value{PSUFFIX}::
@c * ordering@value{PSUFFIX}::
@c * qring@value{PSUFFIX}::
@c @end menu

@c -----------------------------------------------

@table @strong
@item @strong{What is and what does @sc{Plural}?}

@sc{Plural} is a kernel extension of @sc{Singular},
providing many algorithms for computations within certain noncommutative algebras
(see see @ref{G-algebras} and @ref{Mathematical background @value{PSUFFIX}} for detailed information on algebras and algorithms).

@sc{Plural} is compatible with @sc{Singular}, since it uses the
same data structures, sometimes interpreting them in a different way
and/or modifying them for its own purposes.
In spite of such a difference, one can always transfer
objects from commutative rings of @sc{Singular}
to noncommutative rings @sc{Plural} and back.


With @sc{Plural}, one can set up a noncommutative @math{G}-algebra with
a Poincar@'e-Birkhoff-Witt (PBW) basis, say, @math{A} (see @ref{G-algebras} for step-by-step building instructions and also @ref{PLURAL libraries}
for procedures for setting many important algebras easily).

Functionalities of @sc{Plural} (enlisted in @ref{Functions  @value{PSUFFIX}}) are accessible as soon as the basering becomes noncommutative (see @ref{ncalgebra}).

One can perform various computations with polynomials and ideals in @math{A} and with
vectors and submodules of a free module
@tex
$A^n$.
@end tex
@ifinfo
A^n.
@end ifinfo

One can work also within factor-algebras of @math{G}-algebras (see @ref{qring @value{PSUFFIX}} type)
by two-sided ideals (see @ref{twostd}).
@end table

@table @strong
@item @strong{What @sc{Plural} does not:}
@itemize

@item
@sc{Plural} does not perform computations in free algebra or in its general factor algebras.

One can only work with @math{G}-algebras and with their factor-algebras
by two-sided ideals.

@item @sc{Plural} requires a monomial ordering but it does not work with local and mixed orderings.

Right now, one can use only global orderings in @sc{Plural}
(see @ref{General definitions for orderings}).

This will be enhaced in the future by providing the possibility of computations in a tensor product of a noncommutative algebra (with a global ordering)
@* with a commutative algebra (with any ordering).

@item @sc{Plural} does not handle noncommutative parameters.

Defining parameters, one @strong{cannot} impose noncommutative relations
on them. Moreover, it is impossible to introduce
@* parameters which do not commute with variables.

@end itemize
@end table


@table @strong
@item @sc{Plural} conventions

@item *-multiplication  @value{PSUFFIX}
@itemize

in the noncommutative case, the correct multiplication of @code{y} by
 @code{x} must be written as @code{y*x}.
@* Both expressions @code{yx} and @code{xy} are equal, since they are
interpreted as commutative expressions. See example in @ref{poly expressions @value{PSUFFIX}}.
@* Note, that @sc{Plural} output consists only of monomials, hence the signs @code{*} are omitted.

@end itemize

@item   @code{ideal} @value{PSUFFIX}
@itemize

Under an @code{ideal} @sc{Plural} understands a list of generators
of a @strong{left} ideal. For more information see @ref{ideal @value{PSUFFIX}}.
 @* For a @strong{two-sided ideal} @code{T}, use command @ref{twostd} in order to compute
the two-sided Groebner basis of @code{T}.
@c ( at the same time it is a left Groebner basis).
@end itemize

@item @code{module} @value{PSUFFIX}
@itemize
Under a @code{module} @sc{Plural} understands
@strong{either} a fininitely generated @strong{left} submodule of a free module (of finite rank)
@* @strong{or} a factor module of a free module (of finite rank) by its left submodule (see @ref{module @value{PSUFFIX}} for details).
@end itemize

@c @item ordering @value{PSUFFIX}
@c @itemize
@c @sc{Plural} works with @strong{global} orderings  only.
@c @ifset singularmanual
@c See @ref{ General definitions for orderings }
@c @end ifset
@c @ifclear singularmanual
@c See @sc{Singular} manual section General definitions for orderings.
@c @end ifclear
@c @end itemize

@item @code{qring} @value{PSUFFIX}

@itemize
In @sc{Plural} it is only possible to build factor-algebras modulo @strong{two-sided} ideals (see @ref{qring @value{PSUFFIX}}).
@end itemize
@end table