3. Unions of Residue Classes

3.1 Defining unions of residue classes

3.1-1 ResidueClass
> ResidueClass( R, m, r )( function )

The residue class r mod m of the ring R.



gap> A := ResidueClass(Integers,3,2);
The residue class 2(3) of Z
gap> B := ResidueClass(Z_pi([2,5]),2,1);
The residue class 1(2) of Z_( 2, 5 )
gap> B = ResidueClass(Integers,2,1);
false
gap> R := PolynomialRing(GF(7),1);;
gap> x := Indeterminate(GF(7),1);; SetName(x,"x");
gap> C := ResidueClass(R,x+One(R),3*One(R));
The residue class Z(7) ( mod x+Z(7)^0 ) of GF(7)[x]


3.1-2 ResidueClassUnion
> ResidueClassUnion( R, m, r )( function )
> ResidueClassUnion( R, m, r, included, excluded )( function )

The union of the residue classes r[ i ] mod m of the ring R. If the arguments included and excluded are given, they denote lists of elements to be added to resp. to be taken away from the union.



gap> D := ResidueClassUnion(Integers,6,[2,4]);
Union of the residue classes 2(6) and 4(6) of Z
gap> F := 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> G := ResidueClassUnion(R,x,[One(R),5*One(R),6*One(R)],[Zero(R)],[One(R)]);
<union of 3 residue classes (mod x) of GF(7)[x]> U [ 0*Z(7) ] \ [ Z(7)^0 ]
gap> H := ResidueClassUnion(Z_pi([2,3]),8,[3,5]);
<union of 2 residue classes (mod 8) of Z_( 2, 3 )>


The components of a residue class union as given as arguments in ResidueClassUnion can be extracted as follows.

3.1-3 Modulus
> Modulus( U )( method )

Returns: the modulus m of U.

3.1-4 Residues
> Residues( U )( operation )

Returns: the set of residues r of U.

If an element el of the underlying ring of U is congruent to one of the residues of U modulo the modulus of U, it is contained in U, provided that it is not excluded explicitly ( see ExcludedElements (3.1-6) ).

3.1-5 IncludedElements
> IncludedElements( U )( operation )

Returns: the set included of single elements of U.

3.1-6 ExcludedElements
> ExcludedElements( U )( operation )

Returns: the set excluded of single elements explicitly not contained in U.

3.2 Methods for unions of residue classes

3.2-1 String
> String( U )( method )
> String( U, lng )( method )

Returns: a string representation of the residue class union U.

If the argument lng is given its absolute value denotes the length of the returned string; if lng is negative, the result is flushed left, otherwise it is flushed right.



gap> String(C);
"ResidueClassUnion( GF(7)[x], x+Z(7)^0, [ Z(7) ] )"
gap> String(F);
"ResidueClassUnion( Integers, 5, [ 1, 2 ], [ 3, 8 ], [ -4, 1 ] )"
gap> String(H);
"ResidueClassUnion( Z_( 2, 3 ), 8, [ 3, 5 ] )"


3.2-2 Print
> Print( U )( method )

Prints the residue class union U in a way similar to what a user has to write in order to create this object.



gap> Print(C,"\n");
ResidueClassUnion( GF(7)[x], x+Z(7)^0, [ Z(7) ] )
gap> Print(F,"\n");
ResidueClassUnion( Integers, 5, [ 1, 2 ], [ 3, 8 ], [ -4, 1 ] )
gap> Print(H,"\n");
ResidueClassUnion( Z_( 2, 3 ), 8, [ 3, 5 ] )


3.2-3 Display
> Display( U )( method )

Displays the residue class union U in a nice, human-readable form.



gap> Display(F);

The union of the residue classes r ( mod 5 ) of Z for r =

 1 2

and the elements

 3 8

without the elements

 -4  1

gap> Display(G);

The union of the residue classes r ( mod x ) of GF(7)[x] for r =

Z(7)^0  -Z(7)^0 Z(7)^5  

and the element

0*Z(7) 

without the element

Z(7)^0 

gap> Display(H);

The union of the residue classes r ( mod 8 ) of Z_( 2, 3 ) for r =

 3 5



3.2-4 \in
> \in( el, U )( method )

Tests whether el is an element of the residue class union U.



gap> 20 in A;
true
gap> -20 in A;
false
gap> 1/3 in B;
true
gap> 1/5 in B;
false
gap> x in G;
false
gap> Zero(R) in G;
true


3.2-5 IsSubset
> IsSubset( U1, U2 )( method )

Checks whether the residue class union U2 is a subset of the residue class union U1.



gap> IsSubset(A,D);
false
gap> IsSubset(H,ResidueClass(Z_pi([2,3]),16,11));
true


3.2-6 Density
> Density( U )( operation )

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



gap> G;
<union of 3 residue classes (mod x) of GF(7)[x]> U [ 0*Z(7) ] \ [ Z(7)^0 ]
gap> Density(G);
3/7
gap> ResidueClassUnion(Integers,12,[3,5,9]);
Union of the residue classes 3(6) and 5(12) of Z
gap> Density(last);
1/4
gap> 2*last2;
Union of the residue classes 6(12) and 10(24) of Z
gap> Density(last);
1/8


3.2-7 Union
> Union( U1, U2 )( method )
> Union( U, S )( method )

Returns: the union of two residue class unions U1 and U2 resp. of the residue class union U and the finite set S of elements of the ring U is defined over.



gap> I := ResidueClassUnion(Integers,6,[1,5]);
Union of the residue classes 1(6) and 5(6) of Z
gap> J := ResidueClassUnion(Integers,5,[1,2,3,4]);
Union of the residue classes 1(5), 2(5), 3(5) and 4(5) of Z
gap> K := Union(I,J);
<union of 26 residue classes (mod 30) of Z>
gap> Residues(K);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 
  25, 26, 27, 28, 29 ]
gap> Union(K,[0]);
<union of 26 residue classes (mod 30) of Z> U [ 0 ]
gap> Union(D,I);
Union of the residue classes 1(3) and 2(3) of Z


3.2-8 Intersection
> Intersection( U1, U2 )( method )
> Intersection( U, S )( method )

Returns: the intersection of two residue class unions U1 and U2 resp. of the residue class union U and the finite set S of elements of the ring U is defined over.



gap> L := Intersection(I,J);
<union of 8 residue classes (mod 30) of Z>
gap> Display(L);

The union of the residue classes r ( mod 30 ) of Z for r =

  1  7 11 13 17 19 23 29

gap> cl := List([1..25],i->ResidueClass(Integers,Primes[i],i));;
gap> cl_int := Intersection(cl);
The residue class 941584379775558526136539054851975983(
2305567963945518424753102147331756070) of Z
gap> List(Primes{[1..25]},p->Representative(cl_int) mod p);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 
  22, 23, 24, 25 ]


3.2-9 Difference
> Difference( U1, U2 )( method )
> Difference( U, S )( method )

Returns: the difference of two residue class unions U1 and U2 resp. of the residue class union U and the finite set S of elements of the ring U is defined over.



gap> M := Difference(I,J);
Union of the residue classes 5(30) and 25(30) of Z
gap> N := Difference(J,I);
<union of 16 residue classes (mod 30) of Z>
gap> Display(N);

The union of the residue classes r ( mod 30 ) of Z for r =

  2  3  4  6  8  9 12 14 16 18 21 22 24 26 27 28

gap> Difference(Integers,[1,2,3]);
Z \ [ 1, 2, 3 ]
gap> Difference(Z_pi([2,3,7]),[1/5,1/55]);
Z_( 2, 3, 7 ) \ [ 1/55, 1/5 ]


3.2-10 Iterator
> Iterator( U )( method )

Returns: an iterator of the residue class union U.

Currently, iterators are only implemented for residue class unions of the integers.



gap> it := Iterator(F);
<iterator of a residue class union of Z>


3.2-11 NextIterator
> NextIterator( iter )( method )

Returns: the next element delivered by the iterator iter of some residue class union.

Currently, iterators are only implemented for residue class unions of the integers.



gap> l := List([1..16],i->NextIterator(it));
[ 3, 8, 2, -3, 6, -9, 7, -8, 11, -14, 12, -13, 16, -19, 17, -18 ]
gap> Difference(l,Intersection(F,[-20..20]));
[  ]


3.2-12 \+
> \+( U, x )( method )
> \+( x, U )( method )

Returns: the set of sums u + x, u in U.



gap> Display(L+1);

The union of the residue classes r ( mod 30 ) of Z for r =

  0  2  8 12 14 18 20 24

gap> L+30 = L;
true


3.2-13 \-
> \-( U, x )( method )
> \-( x, U )( method )
> \-( U )( method )

Returns: the set of differences u - x, u in U resp. the set of differences x - u, u in U resp. the set of the additive inverses of the elements of U.



gap> F-7;
(Union of the residue classes 0(5) and 4(5) of Z) U [ -4, 1 ] \ [ -11, -6 ]
gap> -L = L;
true
gap> -C;
The residue class Z(7)^4 ( mod x+Z(7)^0 ) of GF(7)[x]


3.2-14 \*
> \*( U, x )( method )
> \*( x, U )( method )

Returns: the set of products x * u, u in U.



gap> 2*A;
The residue class 4(6) of Z
gap> D*17;
Union of the residue classes 34(102) and 68(102) of Z
gap> x^3*C;
The residue class Z(7)*x^3 ( mod x^4+x^3 ) of GF(7)[x]
gap> 2*Difference(Integers,[1,2,3]);
(The residue class 0(2) of Z) \ [ 2, 4, 6 ]


3.2-15 \/
> \/( U, x )( method )

Returns: the set of quotients u/x, u in U.

If the result is not a subset of the underlying ring, the method gives up.



gap> D/2;
Union of the residue classes 1(3) and 2(3) of Z
gap> M/5;
Union of the residue classes 1(6) and 5(6) of Z


3.3 The categories and families of unions of residue classes

3.3-1 IsUnionOfResidueClasses
> IsUnionOfResidueClasses( U )( filter )

The category of all unions of residue classes.

3.3-2 IsUnionOfResidueClassesOfZ
> IsUnionOfResidueClassesOfZ( U )( filter )

The category of all unions of residue classes of Z.

3.3-3 IsUnionOfResidueClassesOfZ_pi
> IsUnionOfResidueClassesOfZ_pi( U )( filter )

The category of all unions of residue classes of some semilocalization Z_(pi) of the integers.

3.3-4 IsUnionOfResidueClassesOfZorZ_pi
> IsUnionOfResidueClassesOfZorZ_pi( U )( filter )

The category of all unions of residue classes of Z or Z_(pi).

3.3-5 IsUnionOfResidueClassesOfGFqx
> IsUnionOfResidueClassesOfGFqx( U )( filter )

The category of all unions of residue classes of some polynomial ring GF( q )[ x ].

3.3-6 ResidueClassUnionsFamily
> ResidueClassUnionsFamily( R )( function )

The family of unions of residue classes of the ring R.

The ring R can be accessed as UnderlyingRing( ResidueClassUnionsFamily( R ) ).




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