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# ResClasses

## Set-Theoretic Computations with Residue Classes

Version 2.3.3

August 11, 2006

Stefan Kohl
e-mail: kohl@mathematik.uni-stuttgart.de
WWW: http://www.cip.mathematik.uni-stuttgart.de/~kohlsn
Address:
Institut für Geometrie und Topologie
Universität Stuttgart
70550 Stuttgart
Germany

### Abstract

ResClasses is a package for GAP 4 which implements set-theoretic unions of residue classes of the ring of integers, of its semilocalisations and of univariate polynomial rings over finite fields as GAP domains.

It provides basic functionality for computing with these sets, like forming intersections, unions and differences. It supports these operations also in the case that one of the operands is a finite set of ring elements. The package further implements the mentioned semilocalisations Z_(pi) of Z as GAP domains.

### Copyright

(C) 2003 - 2006 by Stefan Kohl

This package is distributed under the GNU General Public License.

### Contents

1. Preface
2. Unions of Residue Classes
2.1 Entering residue classes and unions thereof
2.1-1 ResidueClass
2.1-2 ResidueClassUnion
2.1-3 AllResidueClassesModulo
2.2 Methods for unions of residue classes
2.2-1 SplittedClass
2.2-2 AsUnionOfFewClasses
2.2-3 Density
2.3 The categories and families of unions of residue classes
2.3-1 IsUnionOfResidueClasses
2.3-2 ResidueClassUnionsFamily
3. Unions of Residue Classes with Fixed Representatives
3.1 Entering unions of residue classes with fixed representatives
3.1-1 ResidueClassWithFixedRep
3.1-2 ResidueClassUnionWithFixedReps
3.1-3 AllResidueClassesWithFixedRepsModulo
3.2 Methods for unions of residue classes with fixed representatives
3.2-1 Multiplicity
3.2-2 Union
3.2-3 Intersection
3.2-4 Difference
3.3 The invariant Delta
3.3-1 Delta
3.3-2 RepresentativeStabilizingRefinement
3.4 The categories of unions of residue classes with fixed rep's
4. Semilocalizations of the Integers
4.1 Entering semilocalizations of the integers
4.1-1 Z_pi
4.2 Methods for semilocalizations of the integers
5. Installation and auxiliary functions
5.1 Requirements
5.2 Installation
5.3 The testing routine
5.3-1 ResClassesTest
5.4 Building the manual
5.4-1 ResClassesBuildManual

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