Singular Manual: PLURAL

| ## 7.1 PLURAL ### A Subsystem for Non-commutative Polynomial G-Algebras What is and what does ? is a kernel extension of , providing many algorithms for computations within non-commutative and algebras (see Mathematical background (plural) for detailed information on algebras and algorithms). It uses the same data structures as , 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 between commutative rings of and non-commutative rings of . With , one can set up a non-commutative -algebra, say , with a Poincaré-Birkhoff-Witt (PBW) basis, (see G-algebras for step-by-step building instructions and also PLURAL libraries for procedures for setting many important algebras easily). Afterwards, one can proceed to the factor-algebra of modulo a two-sided ideal (see twostd (plural)), thus obtaining a -algebra (see qring (plural) type). Functionalities of (enlisted in Functions (plural)) are accessible as soon as the basering becomes non-commutative (see nc_algebra and the library ncalg_lib with many readily predefined algebras). One can perform various computations with polynomials and ideals in and with vectors and submodules of a free module What does not: does not perform computations in the free algebra or in its general factor algebras (instead, these computations can be possibly done by LETTERPLACE). In one can only work with -algebras and with their factor-algebras by two-sided ideals ( -algebras). requires a global monomial ordering (see General definitions for orderings). However, ( Graded commutative algebras (SCA)) provides the possibility of computations in a tensor product of a non-commutative graded commutative algebra (equipped with a global ordering) with a commutative algebra (equipped with any ordering). does not handle non-commutative parameters, i.e. the elements of the coefficient field (or a ring) mutually commute with all variables. Defining parameters, one cannot impose non-commutative relations on them. Moreover, it is impossible to introduce parameters which do not commute with variables. However, olga_lib offers a rich functionality for working within Ore localizations of -algebras and ratgb_lib provides Groebner bases for so-called rational localizations of -algebras. does not yet support rings like as coefficients. conventions *-multiplication (plural) in the non-commutative 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 output consists only of standard monomials, even when the signs `*` are omitted. `ideal` (plural) Unless stated otherwise, an expression of type `ideal` as understood by as a list of generators of a left ideal. For more information see ideal (plural). For a two-sided ideal `T`, use the command twostd (plural) for computing the two-sided Groebner basis of `T`. For a right ideal `I`, use rightstd (letterplace) from `nctools_lib` for computing the right Groebner basis of `I`. `module` (plural) Unless stated otherwise, a `module` as understood by is either a finitely 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). The concrete interpretation left to a function. `qring` (plural) It is only possible to build factor-algebras modulo two-sided ideals (see qring (plural)), which have to be given via their two-sided Groebner basis (see twostd (plural)).

User manual for Singular version 4.3.2, 2023, generated by texi2html.