GAP (ResClasses) - Chapter 1: Set-Theoretic Unions of Residue Classes

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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

### 1 Set-Theoretic Unions of Residue Classes

#### 1.1 Entering residue classes and set-theoretic unions thereof

##### 1.1-1 ResidueClass

`‣ ResidueClass`( R, m, r ) | ( function )

`‣ ResidueClass`( m, r ) | ( function )

`‣ ResidueClass`( r, m ) | ( function )

Returns: in the three-argument form the residue class r mod m of the ring R, and in the two-argument form the residue class r mod m of the "default ring" (\(\rightarrow\) `DefaultRing` in the GAP Reference Manual) of the arguments.

In the two-argument case, m is taken to be the larger and r is taken to be the smaller of the arguments. For convenience, it is permitted to enclose the argument list in list brackets.

Residue classes have the property `IsResidueClass`. Rings are regarded as residue class 0 (mod 1), and therefore have this property. There are operations `Modulus` and `Residue` to retrieve the modulus m resp. residue r of a residue class.

    gap> ResidueClass(2,3);
    The residue class 2(3) of Z
    gap> ResidueClass(Z_pi([2,5]),2,1);
    The residue class 1(2) of Z_( 2, 5 )
    gap> R := PolynomialRing(GF(2),1);;
    gap> x := Indeterminate(GF(2),1);; SetName(x,"x");
    gap> ResidueClass(R,x+One(R),Zero(R));
    The residue class 0 ( mod x+1 ) of GF(2)[x]

##### 1.1-2 ResidueClassUnion

`‣ ResidueClassUnion`( R, m, r ) | ( function )

`‣ ResidueClassUnion`( R, m, r, included, excluded ) | ( function )

`‣ ResidueClassUnion`( R, cls ) | ( function )

`‣ ResidueClassUnion`( R, cls, included, excluded ) | ( function )

Returns: in the first two cases, the union of the residue classes r[\(i\)] mod m of the ring R, plus / minus finite sets included and excluded of elements of R. In the last two cases, the union of the residue classes cls[\(i\)][1] mod cls[\(i\)][2] of the ring R=ℤ, plus / minus finite sets included and excluded of integers.

For unions of residue classes of the integers, two distinct representations are implemented: in the first representation, a union of residue classes is represented by its modulus m and the list of residues r; this is called the "standard" representation. In the second ("sparse") representation, a union of residue classes \(r_1(m_1) \cup \dots \cup r_k(m_k)\) is represented by the list cls of the pairs `[r_i,m_i]`. One can switch between the two representations by using the operations `StandardRep` and `SparseRep`, respectively. The sparse representation allows more efficient computation in terms of time- and memory requirements when computing with unions of "relatively few" residue classes where the lcm of the moduli is "large"; otherwise the standard representation is advantageous. For rings other than ℤ, presently only the standard representation is available.

    gap> ResidueClassUnion(Integers,5,[1,2],[3,8],[-4,1]);
    (Union of the residue classes 1(5) and 2(5) of Z) U [ 3, 8 ] \ [ -4, 1 ]
    gap> ResidueClassUnion(Integers,[[1,2],[0,40],[2,1200]]);
    Union of the residue classes 1(2), 0(40) and 2(1200) of Z
    gap> ResidueClassUnion(Z_pi([2,3]),8,[3,5]);
    Union of the residue classes 3(8) and 5(8) of Z_( 2, 3 )
    gap> ResidueClassUnion(R,x^2,[One(R),x],[],[One(R)]);
    <union of 2 residue classes (mod x^2) of GF(2)[x]> \ [ 1 ]

When talking about a residue class union in this chapter, we always mean an object as it is returned by this function.

There are operations `Modulus`, `Residues`, `IncludedElements` and `ExcludedElements` to retrieve the components of a residue class union as they have originally been passed as arguments to `ResidueClassUnion`.

The user has the choice between a longer and more descriptive and a shorter and less bulky output format for residue classes and unions thereof:

    gap> ResidueClassUnionViewingFormat("short");
    gap> ResidueClassUnion(Integers,12,[0,1,4,7,8]);
    0(4) U 1(6)
    gap> ResidueClassUnionViewingFormat("long");
    gap> ResidueClassUnion(Integers,12,[0,1,4,7,8]);
    Union of the residue classes 0(4) and 1(6) of Z

##### 1.1-3 AllResidueClassesModulo

`‣ AllResidueClassesModulo`( R, m ) | ( function )

`‣ AllResidueClassesModulo`( m ) | ( function )

Returns: a sorted list of all residue classes (mod m) of the ring R.

If the argument R is omitted it defaults to the default ring of m -- cf. the documentation of `DefaultRing` in the GAP reference manual. A transversal for the residue classes (mod m) can be obtained by the operation `AllResidues(R,m)`, and their number can be determined by the operation `NumberOfResidues(R,m)`.

    gap> AllResidueClassesModulo(Integers,2);
    [ The residue class 0(2) of Z, The residue class 1(2) of Z ]
    gap> AllResidueClassesModulo(Z_pi(2),2);
    [ The residue class 0(2) of Z_( 2 ), The residue class 1(2) of Z_( 2 ) ]
    gap> AllResidueClassesModulo(R,x);
    [ The residue class 0 ( mod x ) of GF(2)[x],
      The residue class 1 ( mod x ) of GF(2)[x] ]
    gap> AllResidues(R,x^3);
    [ 0, 1, x, x+1, x^2, x^2+1, x^2+x, x^2+x+1 ]
    gap> NumberOfResidues(Z_pi([2,3]),360);
    72

#### 1.2 Methods for residue class unions

There are methods for `Print`, `String` and `Display` which are applicable to residue class unions. There is a method for `in` which tests whether some ring element lies in a given residue class union.

    gap> Print(ResidueClass(1,2),"\n");
    ResidueClassUnion( Integers, 2, [ 1 ] )
    gap> 1 in ResidueClass(1,2);
    true

There are methods for `Union`, `Intersection`, `Difference` and `IsSubset` available for residue class unions. They also accept finite subsets of the base ring as arguments.

    gap> S := Union(ResidueClass(0,2),ResidueClass(0,3));
    Z \ Union of the residue classes 1(6) and 5(6) of Z
    gap> Intersection(S,ResidueClass(0,7));
    Union of the residue classes 0(14) and 21(42) of Z
    gap> Difference(S,ResidueClass(2,4));
    Union of the residue classes 0(4) and 3(6) of Z
    gap> IsSubset(ResidueClass(0,2),ResidueClass(4,8));
    true
    gap> Union(S,[1..10]);
    (Union of the residue classes 0(2) and 3(6) of Z) U [ 1, 5, 7 ]
    gap> Intersection(S,[1..6]);
    [ 2, 3, 4, 6 ]
    gap> Difference(S,[1..6]);
    (Union of the residue classes 0(2) and 3(6) of Z) \ [ 2, 3, 4, 6 ]
    gap> Difference(Integers,[1..10]);
    Z \ <set of cardinality 10>
    gap> IsSubset(S,[1..10]);
    false

If the underlying ring has a residue class ring of a given cardinality \(t\), then a residue class can be written as a disjoint union of \(t\) residue classes with equal moduli:

##### 1.2-1 SplittedClass

`‣ SplittedClass`( cl, t ) | ( operation )

Returns: a partition of the residue class cl into t residue classes with equal moduli, provided that such a partition exists. Otherwise `fail`.

    gap> SplittedClass(ResidueClass(1,2),2);
    [ The residue class 1(4) of Z, The residue class 3(4) of Z ]
    gap> SplittedClass(ResidueClass(Z_pi(3),3,0),2);
    fail

Often one needs a partition of a given residue class union into "few" residue classes. The following operation takes care of this:

##### 1.2-2 AsUnionOfFewClasses

`‣ AsUnionOfFewClasses`( U ) | ( operation )

Returns: a set of disjoint residue classes whose union is equal to U, up to the finite sets `IncludedElements(U)` and `ExcludedElements(U)`.

As the name of the operation suggests, it is taken care that the number of residue classes in the returned list is kept "reasonably small". It is not guaranteed that it is minimal.

    gap> ResidueClassUnionViewingFormat("short");
    gap> AsUnionOfFewClasses(Difference(Integers,ResidueClass(0,30)));
    [ 1(2), 2(6), 4(6), 6(30), 12(30), 18(30), 24(30) ]
    gap> Union(last);
    Z \ 0(30)

One can compute the sets of sums, differences, products and quotients of the elements of a residue class union and an element of the base ring:

    gap> ResidueClass(0,2) + 1;
    1(2)
    gap> ResidueClass(0,2) - 2 = ResidueClass(0,2);
    true
    gap> 3 * ResidueClass(0,2);
    0(6)
    gap> ResidueClass(0,2)/2;
    Integers

##### 1.2-3 PartitionsIntoResidueClasses

`‣ PartitionsIntoResidueClasses`( R, length ) | ( operation )

`‣ PartitionsIntoResidueClasses`( R, length, primes ) | ( operation )

Returns: in the 2-argument version a sorted list of all partitions of the ring R into length residue classes. In the 3-argument version a sorted list of all partitions of the ring R into length residue classes whose moduli have only prime factors in the list primes.

    gap> PartitionsIntoResidueClasses(Integers,4);
    [ [ 0(2), 1(4), 3(8), 7(8) ], [ 0(2), 3(4), 1(8), 5(8) ],
      [ 0(2), 1(6), 3(6), 5(6) ], [ 1(2), 0(4), 2(8), 6(8) ],
      [ 1(2), 2(4), 0(8), 4(8) ], [ 1(2), 0(6), 2(6), 4(6) ],
      [ 0(3), 1(3), 2(6), 5(6) ], [ 0(3), 2(3), 1(6), 4(6) ],
      [ 1(3), 2(3), 0(6), 3(6) ], [ 0(4), 1(4), 2(4), 3(4) ] ]

##### 1.2-4 RandomPartitionIntoResidueClasses

`‣ RandomPartitionIntoResidueClasses`( R, length, primes ) | ( operation )

Returns: a "random" partition of the ring R into length residue classes whose moduli have only prime factors in primes, respectively `fail` if no such partition exists.

    gap> RandomPartitionIntoResidueClasses(Integers,30,[2,3,5,7]);
    [ 0(7), 2(7), 5(7), 3(14), 10(14), 1(21), 8(21), 15(21), 18(21), 20(21),
      6(63), 13(63), 25(63), 27(63), 32(63), 34(63), 46(63), 48(63), 53(63),
      55(63), 4(126), 67(126), 137(189), 74(567), 200(567), 263(567),
      389(567), 452(567), 11(1134), 578(1134) ]
    gap> Union(last);
    Integers
    gap> Sum(List(last2,Density));
    1

##### 1.2-5 CoverByResidueClasses

`‣ CoverByResidueClasses`( Integers, moduli ) | ( method )

`‣ CoversByResidueClasses`( Integers, moduli ) | ( method )

Returns: in the first form a cover of the integers by residue classes with moduli moduli if such cover exists, and `fail` otherwise; in the second form a list of all covers of the integers by residue classes with moduli moduli.

Since there are often very many such covers, computing all of them can take a lot of time and memory.

    gap> CoverByResidueClasses(Integers,[2,3,4,6,8,12]);
    [ 0(2), 0(3), 1(4), 1(6), 3(8), 11(12) ]
    gap> Union(last);
    Integers
    gap> CoversByResidueClasses(Integers,[2,3,3,6]);
    [ [ 0(2), 0(3), 1(3), 5(6) ], [ 0(2), 0(3), 2(3), 1(6) ],
      [ 0(2), 1(3), 2(3), 3(6) ], [ 1(2), 0(3), 1(3), 2(6) ],
      [ 1(2), 0(3), 2(3), 4(6) ], [ 1(2), 1(3), 2(3), 0(6) ] ]
    gap> List(last,Union);
    [ Integers, Integers, Integers, Integers, Integers, Integers ]

##### 1.2-6 Density

`‣ Density`( U ) | ( operation )

Returns: the natural density of U as a subset of the underlying ring.

The natural density of a residue class \(r(m)\) of a ring \(R\) is defined by \(1/|R/mR|\), and the natural density of a union \(U\) of finitely many residue classes is defined by the sum of the densities of the elements of a partition of \(U\) into finitely many residue classes.

    gap> Density(ResidueClass(0,2));
    1/2
    gap> Density(Difference(Integers,ResidueClass(0,5)));
    4/5

For looping over residue class unions of the integers, there are methods for the operations `Iterator` and `NextIterator`.

#### 1.3 On residue class unions of \(ℤ^2\)

Residue class unions of \(ℤ^2\) are treated similar as those of any other ring. Also there is roughly the same functionality available for them. However there are some differences and a few additional features, which are described in this section.

The elements of \(ℤ^2\) are represented as lists of length 2 with integer entries. The modulus of a residue class union of \(ℤ^2\) is a lattice. This lattice is stored as a \(2 \times 2\) integer matrix of full rank in Hermite normal form, whose rows are the spanning vectors. Residue classes of \(ℤ^2\) modulo principal ideals are presently not implemented. Residue class unions of \(ℤ^2\) can be multiplied by matrices of full rank from the right. A snippet of a residue class union of \(ℤ^2\) is shown in "ASCII art" when one `Display`'s it with option `AsGrid`. We give some illustrative examples:

    gap> R := Integers^2;
    ( Integers^2 )
    gap> 5*R+[2,3];
    (2,3)+(5,0)Z+(0,5)Z
    gap> Difference(R,last);
    Z^2 \ (2,3)+(5,0)Z+(0,5)Z
    gap> Density(last);
    24/25
    gap> L1 := [[2,1],[-1,2]];;
    gap> L2 := [[6,2],[0,6]];;
    gap> AllResidueClassesModulo(R,L1); # The modulus is transformed to HNF.
    [ (0,0)+(1,3)Z+(0,5)Z, (0,1)+(1,3)Z+(0,5)Z, (0,2)+(1,3)Z+(0,5)Z,
      (0,3)+(1,3)Z+(0,5)Z, (0,4)+(1,3)Z+(0,5)Z ]
    gap> cl1 := ResidueClass(R,L1,[0,0]);
    (0,0)+(1,3)Z+(0,5)Z
    gap> cl2 := ResidueClass(R,L2,[0,0]);
    (0,0)+(6,2)Z+(0,6)Z
    gap> cl3 := Intersection(cl1,cl2);
    (0,0)+(6,8)Z+(0,30)Z
    gap> S1 := Difference(cl1,cl2);
    <union of 35 residue classes (mod (6,8)Z+(0,30)Z)>
    gap> S2 := Difference(cl2,cl1);
    <union of 4 residue classes (mod (6,8)Z+(0,30)Z)>
    gap> Display(S1); # The set is written as union of "few" residue classes:
    (0,5)+(1,3)Z+(0,10)Z U (1,3)+(2,6)Z+(0,10)Z U (2,6)+(6,8)Z+(0,10)Z U
    (4,2)+(6,8)Z+(0,10)Z U (0,10)+(6,8)Z+(0,30)Z U (0,20)+(6,8)Z+(0,30)Z
    gap> Display(S2);
    (0,6)+(6,8)Z+(0,30)Z U (0,12)+(6,8)Z+(0,30)Z U (0,18)+(6,8)Z+(0,30)Z
     U (0,24)+(6,8)Z+(0,30)Z
    gap> cls := AsUnionOfFewClasses(S1);
    [ (0,5)+(1,3)Z+(0,10)Z, (1,3)+(2,6)Z+(0,10)Z, (2,6)+(6,8)Z+(0,10)Z,
      (4,2)+(6,8)Z+(0,10)Z, (0,10)+(6,8)Z+(0,30)Z, (0,20)+(6,8)Z+(0,30)Z ]
    gap> Union(cls) = S1;
    true
    gap> S3 := S1*[[3,5],[2,4]];
    <union of 35 residue classes (mod (2,46)Z+(0,180)Z)>
    gap> Display(S1:AsGrid);
        *    *    *    *    *    *    *    *    *    *    *    *    *    *
     *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
       *    *    *    *    *    *    *    *    *    *    *    *    *    *
    *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
      *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
        *    *    *    *         *    *    *    *    *         *    *    *
     *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
       *    *    *    *    *    *    *    *    *    *    *    *    *    *
    *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
      *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
        *    *    *    *    *    *    *    *    *    *    *    *    *    *
     *    *    *         *    *    *    *    *         *    *    *    *    *
       *    *    *    *    *    *    *    *    *    *    *    *    *    *
    *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
      *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
        *    *    *    *    *    *    *    *    *    *    *    *    *    *
     *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
       *         *    *    *    *    *         *    *    *    *    *
    *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
      *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
        *    *    *    *    *    *    *    *    *    *    *    *    *    *
     *    *    *    *    *    *    *    *    *    *    *    *    *    *    *
       *    *    *    *    *    *    *    *    *    *    *    *    *    *
         *    *    *    *    *         *    *    *    *    *         *    *

Note that in GAP multiplying lists of integers means computing their scalar product as vectors. The consequence is that technically the free module \(ℤ^2\) is not a ring in GAP.

#### 1.4 The categories and families of residue class unions

##### 1.4-1 IsResidueClassUnion

`‣ IsResidueClassUnion`( U ) | ( filter )

`‣ IsResidueClassUnionOfZ`( U ) | ( filter )

`‣ IsResidueClassUnionOfZxZ`( U ) | ( filter )

`‣ IsResidueClassUnionOfZ_pi`( U ) | ( filter )

`‣ IsResidueClassUnionOfGFqx`( U ) | ( filter )

Returns: `true` if U is a residue class union, a residue class union of ℤ, a residue class union of \(ℤ^2\), a residue class union of a semilocalization of ℤ or a residue class union of a polynomial ring in one variable over a finite field, respectively, and `false` otherwise.

Often the same methods can be used for residue class unions of the ring of integers and of its semilocalizations. For this reason, there is a category `IsResidueClassUnionOfZorZ_pi` which is the union of `IsResidueClassUnionOfZ` and `IsResidueClassUnionOfZ_pi`. The internal representation of residue class unions is called `IsResidueClassUnionResidueListRep`. There are methods available for `ExtRepOfObj` and `ObjByExtRep`.

##### 1.4-2 ResidueClassUnionsFamily

`‣ ResidueClassUnionsFamily`( R ) | ( function )

`‣ ResidueClassUnionsFamily`( R, fixedreps ) | ( function )

Returns: the family of residue class unions or the family of unions of residue classes with fixed representatives of the ring R, depending on whether fixedreps is present and `true` or not.

The ring R can be retrieved as `UnderlyingRing(ResidueClassUnionsFamily(R))`. There is no coercion between residue class unions or unions of residue classes with fixed representatives which belong to different families. Unions of residue classes with fixed representatives are described in the next chapter.

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