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# ResClasses

## Computations with Residue Classes and their Set-Theoretic Unions

( Version 2.1.2 )

January 12, 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

This package for GAP 4 (at least version 4.4) implements set-theoretic unions of residue classes of the ring Z of the integers, of its semilocalisations Z_(pi) at some finite set of primes pi and of the polynomial rings GF(q)[x] for some prime power q as GAP domains.

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

### Copyright

(C) 2003 - 2006 by Stefan Kohl

This package is distributed under the same conditions as the GAP system.

### Contents

1. Preface
2. Semilocalizations of the Integers
2.1 Entering semilocalizations of the integers
2.1-1 Z_pi
2.1-2 IsZ_pi
2.1-3 NoninvertiblePrimes
2.2 Methods for semilocalizations of the integers
3. Unions of Residue Classes
3.1 Entering residue classes and unions thereof
3.1-1 ResidueClass
3.1-2 IsResidueClass
3.1-3 ResidueClassUnion
3.1-4 AllResidueClassesModulo
3.1-5 AllResidues
3.2 Methods for unions of residue classes
3.2-1 Union
3.2-2 Intersection
3.2-3 Difference
3.2-4 SplittedClass
3.2-5 AsUnionOfFewClasses
3.2-6 \+
3.2-7 \-
3.2-8 \*
3.2-9 \/
3.2-10 Density
3.3 The categories and families of unions of residue classes
3.3-1 IsUnionOfResidueClasses
3.3-2 ResidueClassUnionsFamily
4. Unions of Residue Classes with Fixed Representatives
4.1 About unions of residue classes with fixed representatives
4.2 Entering unions of residue classes with fixed representatives
4.2-1 ResidueClassWithFixedRepresentative
4.2-2 ResidueClassUnionWithFixedRepresentatives
4.2-3 AllResidueClassesWithFixedRepresentativesModulo
4.3 Methods for unions of residue classes with fixed representatives
4.3-1 Multiplicity
4.3-2 IsOverlappingFree
4.3-3 AsOrdinaryUnionOfResidueClasses
4.3-4 \in
4.3-5 AsListOfClasses
4.3-6 IsSubset
4.3-7 Density
4.3-8 Union
4.3-9 Intersection
4.3-10 Difference
4.3-11 \+
4.3-12 \-
4.3-13 \*
4.3-14 \/
4.4 The invariant Delta
4.4-1 RepresentativeStabilizingRefinement
4.4-2 Delta
4.4-3 Rho
4.5 The categories of unions of residue classes with fixed rep's
5. Installation and auxiliary functions
5.1 Installation
5.2 Building the manual
5.2-1 ResClassesBuildManual
5.3 The testing routine
5.3-1 ResClassesTest
5.4 Changing the viewing format for residue class unions
5.4-1 ResidueClassUnionViewingFormat

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