Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

no text concepts found

Transcript

@comment -*-texinfo-*@comment $Id: pluconventions.doc,v 1.14 2007/10/31 17:29:40 Singular Exp $ @comment this file contains the type definitions @c @c @c @c @c The following directives are necessary for proper compilation with emacs (C-c C-e C-r). Please keep it as it is. Since it is wrapped in `@ignore' and `@end ignore' it does not harm `tex' or `makeinfo' but is a great help in editing this file (emacs 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 @c @c @c @c @c @c @c [email protected] * *[email protected]{PSUFFIX}:: * [email protected]{PSUFFIX}:: * [email protected]{PSUFFIX}:: * [email protected]{PSUFFIX}:: * [email protected]{PSUFFIX}:: * [email protected]{PSUFFIX}:: @end menu @c [email protected] @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 [email protected]'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 $A^n$. @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 @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 @c @c @c @c @c @c @c @c @c @c @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