> | ### 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

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