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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