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# ResClasses

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

( Version 1.1.1 )

February 3, 2004

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 and provides basic functionality for computing with these sets (intersection, union, difference etc., any of these also when one of the operands is a finite set of elements). It also implements the above-mentioned rings Z_(pi) as GAP domains.

### Copyright

(C) 2003 by Stefan Kohl

We adopt the copyright regulations of GAP as detailed in the copyright notice in the GAP manual.

### Contents

1. Preface
2. Semilocalizations of the Integers
2.1 Defining 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
2.2-1 \in
2.2-2 Intersection
2.2-3 IsSubset
2.2-4 StandardAssociate
2.2-5 GcdOp
2.2-6 LcmOp
2.2-7 Factors
2.2-8 IsUnit
3. Unions of Residue Classes
3.1 Defining unions of residue classes
3.1-1 ResidueClass
3.1-2 ResidueClassUnion
3.1-3 Modulus
3.1-4 Residues
3.1-5 IncludedElements
3.1-6 ExcludedElements
3.2 Methods for unions of residue classes
3.2-1 String
3.2-2 Print
3.2-3 Display
3.2-4 \in
3.2-5 IsSubset
3.2-6 Density
3.2-7 Union
3.2-8 Intersection
3.2-9 Difference
3.2-10 Iterator
3.2-11 NextIterator
3.2-12 \+
3.2-13 \-
3.2-14 \*
3.2-15 \/
3.3 The categories and families of unions of residue classes
3.3-1 IsUnionOfResidueClasses
3.3-2 IsUnionOfResidueClassesOfZ
3.3-3 IsUnionOfResidueClassesOfZ_pi
3.3-4 IsUnionOfResidueClassesOfZorZ_pi
3.3-5 IsUnionOfResidueClassesOfGFqx
3.3-6 ResidueClassUnionsFamily
4. Installation and auxiliary functions
4.1 Installation
4.2 Building the manual
4.2-1 ResClassesBuildManual
4.3 The testing routine
4.3-1 ResClassesTest
4.4 Changing the viewing format for residue class unions
4.4-1 ResidueClassUnionViewingFormat

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