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@comment -*-texinfo-*@comment $Id: pluconventions.doc,v 1.14 2007/10/31 17:29:40 Singular Exp
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@comment this file contains the type definitions
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@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
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@c ----------------------------@menu
* *-multiplication@value{PSUFFIX}::
* factor@value{PSUFFIX}::
* ideals@value{PSUFFIX}::
* modules@value{PSUFFIX}::
* ordering@value{PSUFFIX}::
* qring@value{PSUFFIX}::
@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{Galgebras} 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{nc_algebra}).
One can perform various computations with polynomials and ideals in
@math{A} and with
vectors and submodules of a free module
@tex
$A^n$.
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@ifinfo
A^n.
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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
@itemize
@value{PSUFFIX}
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
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@item ordering @value{PSUFFIX}
@itemize
@sc{Plural} works with @strong{global} orderings only.
@ifset singularmanual
See @ref{ General definitions for orderings }
@end ifset
@ifclear singularmanual
See @sc{Singular} manual section General definitions for orderings.
@end ifclear
@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