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# ResClasses

## Set-Theoretic Computations with Residue Classes

4.7.4

24 September 2025

Stefan Kohl
Email: sk239@st-andrews.ac.uk
Homepage: https://stefan-kohl.github.io/

### Abstract

ResClasses is a package for GAP 4 which provides a fully-featured and easy-to-use implementation of set-theoretic unions of residue classes of the integers and of a few other rings.

The class of sets which ResClasses can deal with includes the open and the closed sets in the topology on the respective ring which is induced by taking the set of all residue classes as a basis, as far as the usual restrictions imposed by the finiteness of computing resources permit this.

The package further provides slightly more specialized functionality for unions of residue classes with distinguished representatives and signed moduli.

The ResClasses package is used in a group theoretical context by the RCWA package [Koh16].

### Copyright

© 2003 - 2017 by Stefan Kohl.

ResClasses is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the License, or (at your option) any later version.

ResClasses is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

For a copy of the GNU General Public License, see the file `GPL` in the `etc` directory of the GAP distribution or see https://www.gnu.org/licenses/gpl.html.

### Contents

1 Set-Theoretic Unions of Residue Classes
1.1 Entering residue classes and set-theoretic unions thereof
1.1-1 ResidueClass
1.1-2 ResidueClassUnion
1.1-3 AllResidueClassesModulo
1.2 Methods for residue class unions
1.2-1 SplittedClass
1.2-2 AsUnionOfFewClasses
1.2-3 PartitionsIntoResidueClasses
1.2-4 RandomPartitionIntoResidueClasses
1.2-5 CoverByResidueClasses
1.2-6 Density
1.3 On residue class unions of ℤ^2
1.4 The categories and families of residue class unions
1.4-1 IsResidueClassUnion
1.4-2 ResidueClassUnionsFamily
2 Unions of Residue Classes with Fixed Representatives
2.1 Entering unions of residue classes with fixed representatives
2.1-1 ResidueClassWithFixedRepresentative
2.1-2 UnionOfResidueClassesWithFixedReps
2.1-3 AllResidueClassesWithFixedRepsModulo
2.2 Methods for unions of residue classes with fixed representatives
2.2-1 Multiplicity
2.2-2 Union
2.2-3 Intersection
2.2-4 Difference
2.3 The invariant Delta
2.3-1 Delta
2.3-2 RepresentativeStabilizingRefinement
2.4 The categories of unions of residue classes with fixed rep's
3 Semilocalizations of the Integers
3.1 Entering semilocalizations of the integers
3.1-1 Z_pi
3.2 Methods for semilocalizations of the integers
4 Installation and Auxiliary Functions
4.1 Requirements
4.2 Installation
4.3 The testing routines
4.3-1 ResClassesTest
4.3-2 ResClassesTestExamples
4.4 Creating timestamped logfiles
4.4-1 LogToDatedFile
4.5 DownloadFile, SendEmail and EmailLogFile
4.5-1 DownloadFile
4.5-2 SendEmail
4.5-3 EmailLogFile
4.6 Creating bitmap pictures
4.6-1 SaveAsBitmapPicture
4.6-2 DrawLineNC
4.7 Some general utility functions
References
Index
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