          ### 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 4.3.2, 2023, generated by texi2html.