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7.1 Getting started with PLURAL

What is and what does PLURAL?

PLURAL is a kernel extension of SINGULAR, providing many algorithms for computations within certain noncommutative algebras (see see G-algebras and Mathematical background (plural) for detailed information on algebras and algorithms).

PLURAL is compatible with 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 SINGULAR to noncommutative rings PLURAL and back.

With PLURAL, one can set up a noncommutative $G$-algebra with a Poincaré-Birkhoff-Witt (PBW) basis, say, $A$ (see G-algebras for step-by-step building instructions and also PLURAL libraries for procedures for setting many important algebras easily).

Functionalities of PLURAL (enlisted in Functions (plural)) are accessible as soon as the basering becomes noncommutative (see ncalgebra).

One can perform various computations with polynomials and ideals in $A$ and with vectors and submodules of a free module $A^n$.

One can work also within factor-algebras of $G$-algebras (see qring (plural) type) by two-sided ideals (see twostd).

What PLURAL does not:
  • PLURAL does not perform computations in free algebra or in its general factor algebras.

    One can only work with $G$-algebras and with their factor-algebras by two-sided ideals.

  • PLURAL requires a monomial ordering but it does not work with local and mixed orderings.

    Right now, one can use only global orderings in PLURAL (see 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).

  • PLURAL does not handle noncommutative parameters.

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

PLURAL conventions

*-multiplication (plural)
    in the noncommutative case, the correct multiplication of y by x must be written as y*x.
    Both expressions yx and xy are equal, since they are interpreted as commutative expressions. See example in poly expressions (plural).
    Note, that PLURAL output consists only of monomials, hence the signs * are omitted.

ideal (plural)
    Under an ideal PLURAL understands a list of generators of a left ideal. For more information see ideal (plural).
    For a two-sided ideal T, use command twostd in order to compute the two-sided Groebner basis of T.

module (plural)
    Under a module PLURAL understands either a fininitely generated left submodule of a free module (of finite rank)
    or a factor module of a free module (of finite rank) by its left submodule (see module (plural) for details).

qring (plural)

    In PLURAL it is only possible to build factor-algebras modulo two-sided ideals (see qring (plural)).


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            User manual for Singular version 3-0-1, October 2005, generated by texi2html.