### 7.5.21 olga_lib

Library:
  olga.lib
Purpose:
  Ore-localization in G-Algebras
Author:
  Johannes Hoffmann, email: johannes.hoffmann at math.rwth-aachen.de

Overview:
  Let A be a G-algebra.

  Current localization types:
  Type 0: monoidal
  - represented by a list of polys g_1,...,g_k that have to be contained in a commutative polynomial subring of A generated by a subset of the variables of A
  Type 1: geometric
  - only for algebras with an even number of variables where the first half induces a commutative polynomial subring B of A
  - represented by an ideal p, which has to be a prime ideal in B Type 2: rational
  - represented by an intvec v = [i_1,...,i_k] in the range 1..nvars(basering)

  Localization data is an int specifying the type and a def with the corresponding information.

  A fraction is represented as a vector with four entries: [s,r,p,t] Here, s^{-1}r is the left fraction representation, pt^{-1} is the right one. If s or t is zero, it means that the corresponding representation is not set. If both are zero, the fraction is not valid.

  A detailed description along with further examples can be found in our paper: Johannes Hoffmann, Viktor Levandovskyy:
  Constructive Arithmetics in Ore Localizations of Domains https://arxiv.org/abs/1712.01773

Procedures:

> 7.5.21.0. locStatus |  | report on the status/validity of the given localization data
> 7.5.21.0. testLocData |  | check if the given data specifies a denominator set
> 7.5.21.0. isInS |  | determine if a polynomial is in a denominator set
> 7.5.21.0. fracStatus |  | report on the status/validity of the given fraction wrt. to the given localization data
> 7.5.21.0. testFraction |  | check if the given vector is a representation of a fraction in the specified localization
> 7.5.21.0. leftOre |  | compute left Ore data
> 7.5.21.0. rightOre |  | compute right Ore data
> 7.5.21.0. convertRightToLeftFraction |  | determine a left fraction representation of a given fraction
> 7.5.21.0. convertLeftToRightFraction |  | determine a right fraction representation of a given fraction
> 7.5.21.0. addLeftFractions |  | add two left fractions in the specified localization
> 7.5.21.0. multiplyLeftFractions |  | multiply two left fractions in the specified localization
> 7.5.21.0. areEqualLeftFractions |  | check if two given fractions are equal
> 7.5.21.0. isInvertibleLeftFraction |  | check if a fraction is invertible in the specified localization (NOTE: check description for specific behaviour)
> 7.5.21.0. invertLeftFraction |  | invert a fraction in the specified localization (NOTE: check description for specific behaviour)
> 7.5.21.0. isZeroFraction |  | determine if the given fraction is equal to zero
> 7.5.21.0. isOneFraction |  | determine if the given fraction is equal to one
> 7.5.21.0. normalizeMonoidal |  | determine a normal form for monoidal localization data
> 7.5.21.0. normalizeRational |  | determine a normal form for rational localization data
> 7.5.21.0. testOlga |  | execute a series of internal testing procedures
> 7.5.21.0. testOlgaExamples |  | execute the examples of all procedures in this library

User manual for Singular version 4.4.1, 2025, generated by texi2html.