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 Residue-Class-Wise Affine Mappings, Groups and Monoids over ℤ^2
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<a id="X788EB00B82897762" name="X788EB00B82897762"></a>
<div class="ChapSects"><a href="chap5.html#X788EB00B82897762">5 
 Residue-Class-Wise Affine Mappings, Groups and Monoids over ℤ^2
</a>
<div class="ContSect"> <a href="chap5.html#X781907CA785CC7AC">5.1 
 The definition of residue-class-wise affine mappings of ℤ^d
</a>

</div>
<div class="ContSect"> <a href="chap5.html#X7A39FCF08030AB9B">5.2 
 Entering residue-class-wise affine mappings of ℤ^2
</a>

<div class="ContSSBlock">
 <a href="chap5.html#X790649618012C606">5.2-1 
 RcwaMapping (the general constructor; methods for ℤ^2)
 </a>

 <a href="chap5.html#X7B450EE17B465E02">5.2-2 ClassTransposition (for ℤ^2) </a>

 <a href="chap5.html#X828438127DDAEBB4">5.2-3 ClassRotation (for ℤ^2) </a>

 <a href="chap5.html#X7A14A8F48247E651">5.2-4 ClassShift (for ℤ^2) </a>

</div></div>
<div class="ContSect"> <a href="chap5.html#X8531E39785FFF8A7">5.3 
 Methods for residue-class-wise affine mappings of ℤ^2
</a>

<div class="ContSSBlock">
 <a href="chap5.html#X8408B7837C9EED36">5.3-1 ProjectionsToCoordinates</a>
</div></div>
<div class="ContSect"> <a href="chap5.html#X83A1752F7BE9CE85">5.4 
 Methods for residue-class-wise affine groups and -monoids over ℤ^2
</a>

<div class="ContSSBlock">
 <a href="chap5.html#X79A8F9AD7E839862">5.4-1 
 IsomorphismRcwaGroup (Embeddings of SL(2,ℤ) and GL(2,ℤ))
 </a>

 <a href="chap5.html#X812135EB87527F01">5.4-2 DrawGrid </a>

</div></div>
</div>

<h3>5 
 Residue-Class-Wise Affine Mappings, Groups and Monoids over ℤ^2
</h3>

This chapter describes how to compute with residue-class-wise affine mappings of ℤ^2 and with groups and monoids formed by them.

The rings on which we have defined residue-class-wise affine mappings so far have all been principal ideal domains, and it has been crucial that all nontrivial principal ideals had finite index. However, the rings ℤ^d, d > 1 are not principal ideal domains. Furthermore, their principal ideals have infinite index. Therefore as moduli of residue-class-wise affine mappings we can only use lattices of full rank, for these are precisely the ideals of ℤ^d of finite index. However, on the other hand we can also be more permissive and look at ℤ^d not as a ring, but rather as a free ℤ-module. The consequence of this is that then an affine mapping of ℤ^d is not just given by v ↦ (av+b)/c for some a, b, c ∈ ℤ^d, but rather by v ↦ (vA+b)/c, where A ∈ ℤ^d × d. Also for technical reasons concerning the implementation in GAP, looking at ℤ^d as a free ℤ-module is preferable -- in GAP, <code class="code">Integers^d</code> is not a ring, and multiplying lists of integers means forming their scalar product.

<a id="X781907CA785CC7AC" name="X781907CA785CC7AC"></a>

<h4>5.1 
 The definition of residue-class-wise affine mappings of ℤ^d
</h4>

Let d ∈ ℕ. We call a mapping f: ℤ^d → ℤ^d residue-class-wise affine if there is a lattice L = ℤ^d M where M ∈ ℤ^d × d is a matrix of full rank, such that the restrictions of f to the residue classes r + L ∈ ℤ^d/L are all affine. This means that for any residue class r + L ∈ ℤ^d/L, there is a matrix A_r+L ∈ ℤ^d × d, a vector b_r+L ∈ ℤ^d and a positive integer c_r+L such that the restriction of f to r + L is given by f|_r + L: r + L → ℤ^d, v ↦ (v ⋅ A_r+L + b_r+L)/c_r+L. For reasons of uniqueness, we assume that L is chosen maximal with respect to inclusion, and that no prime factor of c_r+L divides all coefficients of A_r+L and b_r+L.

We call the lattice L the modulus of f, written Mod(f). Further we define the prime set of f as the set of all primes which divide the determinant of at least one of the coefficients A_r+L or which divide the determinant of M, and we call the mapping f class-wise translating if all coefficients A_r+L are identity matrices and all coefficients c_r+L are equal to 1.

For the sake of simplicity, we identify a lattice with the Hermite normal form of the matrix by whose rows it is spanned.

<a id="X7A39FCF08030AB9B" name="X7A39FCF08030AB9B"></a>

<h4>5.2 
 Entering residue-class-wise affine mappings of ℤ^2
</h4>

Residue-class-wise affine mappings of ℤ^2 can be entered using the general constructor <code class="func">RcwaMapping</code> (<a href="chap2.html#X8799551B83644B37">2.2-5</a>) or the more specialized functions <code class="func">ClassTransposition</code> (<a href="chap2.html#X8716A75F7DD1C46B">2.2-3</a>), <code class="func">ClassRotation</code> (<a href="chap2.html#X87EB8C1C87F78A17">2.2-4</a>) and <code class="func">ClassShift</code> (<a href="chap2.html#X86B611BD7EED62A1">2.2-1</a>). The arguments differ only slightly.

<a id="X790649618012C606" name="X790649618012C606"></a>

<h5>5.2-1 
 RcwaMapping (the general constructor; methods for ℤ^2)
 </h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">R</var>, <var class="Arg">L</var>, <var class="Arg">coeffs</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">P1</var>, <var class="Arg">P2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">cycles</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RcwaMapping</code>( <var class="Arg">f</var>, <var class="Arg">g</var> )</td><td class="tdright">( method )</td></tr></table></div>
Returns: an rcwa mapping of ℤ^2.

The above methods return

<dl>
<dt>(a)</dt>
<dd>the rcwa mapping of <code class="code"><var class="Arg">R</var> = Integers^2</code> with modulus <var class="Arg">L</var> and coefficients <var class="Arg">coeffs</var>,

</dd>
<dt>(b)</dt>
<dd>an rcwa permutation which induces a bijection between the partitions <var class="Arg">P1</var> and <var class="Arg">P2</var> of ℤ^2 into residue classes and which is affine on the elements of <var class="Arg">P1</var>,

</dd>
<dt>(c)</dt>
<dd>an rcwa permutation with <q>residue class cycles</q> given by a list <var class="Arg">cycles</var> of lists of pairwise disjoint residue classes of ℤ^2 each of which it permutes cyclically, and

</dd>
<dt>(d)</dt>
<dd>the rcwa mapping of ℤ^2 whose projections to the coordinates are given by <var class="Arg">f</var> and <var class="Arg">g</var>,

</dd>
</dl>
respectively.

The modulus of an rcwa mapping of ℤ^2 is a lattice of full rank. It is represented by a matrix <var class="Arg">L</var> in Hermite normal form, whose rows are the spanning vectors.

A coefficient list for an rcwa mapping of ℤ^2 with modulus <var class="Arg">L</var> consists of |det(<var class="Arg">L</var>)| coefficient triples <code class="code">[</code>A_r+ℤ^2<var class="Arg">L</var>, b_r+ℤ^2<var class="Arg">L</var>, c_r+ℤ^2<var class="Arg">L</var><code class="code">]</code>. The entries A_r+ℤ^2<var class="Arg">L</var> are 2 × 2 integer matrices, the b_r+ℤ^2<var class="Arg">L</var> are elements of ℤ^2, i.e. lists of two integers, and the c_r+ℤ^2<var class="Arg">L</var> are integers. The ordering of the coefficient triples is determined by the ordering of the representatives of the residue classes r+ℤ^2<var class="Arg">L</var> in the sorted list returned by <code class="code">AllResidues(Integers^2,<var class="Arg">L</var>)</code>.

The methods for the operation <code class="code">RcwaMapping</code> perform a number of argument checks, which can be skipped by using <code class="code">RcwaMappingNC</code> instead.

Last but not least, regarding Method (d) it should be mentioned that only very special rcwa mappings of ℤ^2 have projections to coordinates.

<div class="example"><pre>

gap> R := Integers^2;;
gap> twice := RcwaMapping(R,[[1,0],[0,1]],
> [[[[2,0],[0,2]],[0,0],1]]); # method (a)
Rcwa mapping of Z^2: (m,n) -> (2m,2n)
gap> [4,5]^twice;
[ 8, 10 ]
gap> twice1 := RcwaMapping(R,[[1,0],[0,1]],
> [[[[2,0],[0,1]],[0,0],1]]); # method (a)
Rcwa mapping of Z^2: (m,n) -> (2m,n)
gap> [4,5]^twice1;
[ 8, 5 ]
gap> Image(twice1);
(0,0)+(2,0)Z+(0,1)Z
gap> hyperbolic := RcwaMapping(R,[[1,0],[0,2]],
> [[[[4,0],[0,1]],[0, 0],2],
> [[[4,0],[0,1]],[2,-1],2]]); # method (a)
<rcwa mapping of Z^2 with modulus (1,0)Z+(0,2)Z>
gap> IsBijective(hyperbolic);
true
gap> Display(hyperbolic);

Rcwa permutation of Z^2 with modulus (1,0)Z+(0,2)Z

 /
 | (2m,n/2) if (m,n) in (0,0)+(1,0)Z+(0,2)Z
(m,n) |-> < (2m+1,(n-1)/2) if (m,n) in (0,1)+(1,0)Z+(0,2)Z
 |
 \

gap> Trajectory(hyperbolic,[0,10000],20);
[ [ 0, 10000 ], [ 0, 5000 ], [ 0, 2500 ], [ 0, 1250 ], [ 0, 625 ],
 [ 1, 312 ], [ 2, 156 ], [ 4, 78 ], [ 8, 39 ], [ 17, 19 ], [ 35, 9 ],
 [ 71, 4 ], [ 142, 2 ], [ 284, 1 ], [ 569, 0 ], [ 1138, 0 ],
 [ 2276, 0 ], [ 4552, 0 ], [ 9104, 0 ], [ 18208, 0 ] ]
gap> P1 := AllResidueClassesModulo(R,[[2,1],[0,2]]);
[ (0,0)+(2,1)Z+(0,2)Z, (0,1)+(2,1)Z+(0,2)Z, (1,0)+(2,1)Z+(0,2)Z,
 (1,1)+(2,1)Z+(0,2)Z ]
gap> P2 := AllResidueClassesModulo(R,[[1,0],[0,4]]);
[ (0,0)+(1,0)Z+(0,4)Z, (0,1)+(1,0)Z+(0,4)Z, (0,2)+(1,0)Z+(0,4)Z,
 (0,3)+(1,0)Z+(0,4)Z ]
gap> g := RcwaMapping(P1,P2); # method (b)
<rcwa permutation of Z^2 with modulus (2,1)Z+(0,2)Z>
gap> P1^g = P2;
true
gap> Display(g:AsTable);

Rcwa permutation of Z^2 with modulus (2,1)Z+(0,2)Z

 [m,n] mod (2,1)Z+(0,2)Z | Image of [m,n]
-----------------------------+-------------------------------------------
[0,0] | [m/2,-m+2n]
[0,1] | [m/2,-m+2n-1]
[1,0] | [(m-1)/2,-m+2n+3]
[1,1] | [(m-1)/2,-m+2n+2]

gap> classes := List([[[0,0],[[2,1],[0,2]]],[[1,0],[[2,1],[0,4]]],
> [[1,1],[[4,2],[0,4]]]],ResidueClass);
[ (0,0)+(2,1)Z+(0,2)Z, (1,0)+(2,1)Z+(0,4)Z, (1,1)+(4,2)Z+(0,4)Z ]
gap> g := RcwaMapping([classes]); # method (c)
<rcwa permutation of Z^2 with modulus (4,2)Z+(0,4)Z, of order 3>
gap> Permutation(g,classes);
(1,2,3)
gap> Support(g);
(0,0)+(2,1)Z+(0,2)Z U (1,0)+(2,1)Z+(0,4)Z U (1,1)+(4,2)Z+(0,4)Z
gap> Display(g);

Rcwa permutation of Z^2 with modulus (4,2)Z+(0,4)Z, of order 3

 /
 | (m+1,(-m+4n)/2) if (m,n) in (0,0)+(2,1)Z+(0,2)Z
 | (2m-1,(m+2n+1)/2) if (m,n) in (1,0)+(2,1)Z+(0,4)Z
(m,n) |-> < ((m-1)/2,(n-1)/2) if (m,n) in (1,1)+(4,2)Z+(0,4)Z
 | (m,n) otherwise
 |
 \

gap> g := RcwaMapping(ClassTransposition(0,2,1,2),
> ClassReflection(0,2)); # method (d)
<rcwa mapping of Z^2 with modulus (2,0)Z+(0,2)Z>
gap> Display(g);

Rcwa mapping of Z^2 with modulus (2,0)Z+(0,2)Z

 /
 | (m+1,-n) if (m,n) in (0,0)+(2,0)Z+(0,2)Z
 | (m+1,n) if (m,n) in (0,1)+(2,0)Z+(0,2)Z
(m,n) |-> < (m-1,-n) if (m,n) in (1,0)+(2,0)Z+(0,2)Z
 | (m-1,n) if (m,n) in (1,1)+(2,0)Z+(0,2)Z
 |
 \

gap> g^2;
IdentityMapping( ( Integers^2 ) )
gap> List(ProjectionsToCoordinates(g),Factorization);
[ [ ( 0(2), 1(2) ) ], [ ClassReflection( 0(2) ) ] ]

</pre></div>

<a id="X7B450EE17B465E02" name="X7B450EE17B465E02"></a>

<h5>5.2-2 ClassTransposition (for ℤ^2) </h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassTransposition</code>( <var class="Arg">r1</var>, <var class="Arg">L1</var>, <var class="Arg">r2</var>, <var class="Arg">L2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassTransposition</code>( <var class="Arg">cl1</var>, <var class="Arg">cl2</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the class transposition τ_r_1+ℤ^2L_1,r_2+ℤ^2L_2.

Let d ∈ ℕ, and let L_1, L_2 ∈ ℤ^d × d be matrices of full rank which are in Hermite normal form. Further let r_1 + ℤ^d L_1 and r_2 + ℤ^d L_2 be disjoint residue classes, and assume that the representatives r_1 and r_2 are reduced modulo ℤ^d L_1 and ℤ^d L_2, respectively. Then we define the class transposition τ_r_1+ℤ^d L_1, r_2+ℤ^d L_2 ∈ Sym(ℤ^d) as the involution which interchanges r_1 + k L_1 and r_2 + k L_2 for all k ∈ ℤ^d.

The class transposition τ_r_1+ℤ^d L_1, r_2+ℤ^d L_2 interchanges the residue classes r_1+ℤ^d L_1 and r_2+ℤ^d L_2, and fixes the complement of their union pointwise. The set of all class transpositions of ℤ^d generates the simple group CT(ℤ^d) (cf. <a href="chapBib.html#biBKohl13">[Koh13]</a>).

In the four-argument form, the arguments <var class="Arg">r1</var>, <var class="Arg">L1</var>, <var class="Arg">r2</var> and <var class="Arg">L2</var> stand for r_1, L_1, r_2 and L_2, respectively. In the two-argument form, the arguments <var class="Arg">cl1</var> and <var class="Arg">cl2</var> stand for the residue classes r_1+ℤ^2 L_1 and r_2+ℤ^2 L_2, respectively. Enclosing the argument list in list brackets is permitted. The residue classes r_1+ℤ^2 L_1 and r_2+ℤ^2 L_2 are stored as an attribute <code class="code">TransposedClasses</code>.

There is also a method for <code class="code">SplittedClassTransposition</code> available for class transpositions of ℤ^2. This method takes as first argument the class transposition, and as second argument a list of two integers. These integers are the numbers of parts into which the class transposition is to be sliced in each dimension. Note that the product of the returned class transpositions is not always equal to the class transposition passed as first argument. However this equality holds if the first entry of the second argument is 1.

<div class="example"><pre>

gap> ct := ClassTransposition([0,0],[[2,1],[0,2]],[1,0],[[2,1],[0,4]]);
( (0,0)+(2,1)Z+(0,2)Z, (1,0)+(2,1)Z+(0,4)Z )
gap> Display(ct);

Rcwa permutation of Z^2 with modulus (2,1)Z+(0,4)Z, of order 2

 /
 | (m+1,(-m+4n)/2) if (m,n) in (0,0)+(2,1)Z+(0,2)Z
(m,n) |-> < (m-1,(m+2n-1)/4) if (m,n) in (1,0)+(2,1)Z+(0,4)Z
 | (m,n) otherwise
 \

gap> TransposedClasses(ct);
[ (0,0)+(2,1)Z+(0,2)Z, (1,0)+(2,1)Z+(0,4)Z ]
gap> ct = ClassTransposition(last);
true
gap> SplittedClassTransposition(ct,[1,2]);
[ ( (0,0)+(2,1)Z+(0,4)Z, (1,0)+(2,1)Z+(0,8)Z ),
 ( (0,2)+(2,1)Z+(0,4)Z, (1,4)+(2,1)Z+(0,8)Z ) ]
gap> Product(last) = ct;
true
gap> SplittedClassTransposition(ct,[2,1]);
[ ( (0,0)+(4,0)Z+(0,2)Z, (1,0)+(4,2)Z+(0,4)Z ),
 ( (2,1)+(4,0)Z+(0,2)Z, (3,1)+(4,2)Z+(0,4)Z ) ]
gap> Product(last) = ct;
false

</pre></div>

<a id="X828438127DDAEBB4" name="X828438127DDAEBB4"></a>

<h5>5.2-3 ClassRotation (for ℤ^2) </h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassRotation</code>( <var class="Arg">r</var>, <var class="Arg">L</var>, <var class="Arg">u</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassRotation</code>( <var class="Arg">cl</var>, <var class="Arg">u</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the class rotation ρ_r(m),u.

Let d ∈ ℕ. Given a residue class r+ℤ^dL and a matrix u ∈ GL(d,ℤ), the class rotation ρ_r+ℤ^dL,u is the rcwa mapping which maps v ∈ r+ℤ^dL to vu + r(1-u) and which fixes ℤ^d ∖ r+ℤ^dL pointwise. In the two-argument form, the argument <var class="Arg">cl</var> stands for the residue class r+ℤ^dL. Enclosing the argument list in list brackets is permitted. The argument <var class="Arg">u</var> is stored as an attribute <code class="code">RotationFactor</code>.

<div class="example"><pre>

gap> interchange := ClassRotation([0,0],[[1,0],[0,1]],[[0,1],[1,0]]);
ClassRotation( Z^2, [ [ 0, 1 ], [ 1, 0 ] ] )
gap> Display(interchange);
Rcwa permutation of Z^2: (m,n) -> (n,m)
gap> classes := AllResidueClassesModulo(Integers^2,[[2,1],[0,3]]);
[ (0,0)+(2,1)Z+(0,3)Z, (0,1)+(2,1)Z+(0,3)Z, (0,2)+(2,1)Z+(0,3)Z,
 (1,0)+(2,1)Z+(0,3)Z, (1,1)+(2,1)Z+(0,3)Z, (1,2)+(2,1)Z+(0,3)Z ]
gap> transvection := ClassRotation(classes[5],[[1,1],[0,1]]);
ClassRotation((1,1)+(2,1)Z+(0,3)Z,[[1,1],[0,1]])
gap> Display(transvection);

Tame rcwa permutation of Z^2 with modulus (2,1)Z+(0,3)Z, of order infinity

 /
 | (m,(3m+2n-3)/2) if (m,n) in (1,1)+(2,1)Z+(0,3)Z
(m,n) |-> < (m,n) otherwise
 |
 \

</pre></div>

<a id="X7A14A8F48247E651" name="X7A14A8F48247E651"></a>

<h5>5.2-4 ClassShift (for ℤ^2) </h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassShift</code>( <var class="Arg">r</var>, <var class="Arg">L</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClassShift</code>( <var class="Arg">cl</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: the class shift ν_r+ℤ^dL,k.

Let d ∈ ℕ. Given a residue class r+ℤ^dL and an integer k ∈ {1, dots, d}, the class shift ν_r+ℤ^dL,k is the rcwa mapping which maps v ∈ r+ℤ^dL to v + L_k and which fixes ℤ^d ∖ r+ℤ^dL pointwise. Here L_k denotes the kth row of L.

In the two-argument form, the argument <var class="Arg">cl</var> stands for the residue class r+ℤ^dL. Enclosing the argument list in list brackets is permitted.

<div class="example"><pre>

gap> shift1 := ClassShift([0,0],[[1,0],[0,1]],1);
ClassShift( Z^2, 1 )
gap> Display(shift1);
Tame rcwa permutation of Z^2: (m,n) -> (m+1,n)
gap> s := ClassShift(ResidueClass([1,1],[[2,1],[0,2]]),2);
ClassShift((1,1)+(2,1)Z+(0,2)Z,2)
gap> Display(s);

Tame rcwa permutation of Z^2 with modulus (2,1)Z+(0,2)Z, of order infinity

 /
 | (m,n+2) if (m,n) in (1,1)+(2,1)Z+(0,2)Z
(m,n) |-> < (m,n) if (m,n) in (0,0)+(2,0)Z+(0,1)Z U
 | (1,0)+(2,1)Z+(0,2)Z
 \

</pre></div>

As for other rings, class transpositions, class rotations and class shifts of ℤ^2 have the distinguishing properties <code class="code">IsClassTransposition</code>, <code class="code">IsClassRotation</code> and <code class="code">IsClassShift</code>.

<a id="X8531E39785FFF8A7" name="X8531E39785FFF8A7"></a>

<h4>5.3 
 Methods for residue-class-wise affine mappings of ℤ^2
</h4>

There are methods available for rcwa mappings of ℤ^2 for the following general operations:

<dl>
<dt> Output </dt>
<dd><code class="code">View</code>, <code class="code">Display</code>, <code class="code">Print</code>, <code class="code">String</code>, <code class="code">LaTeXStringRcwaMapping</code>, <code class="code">LaTeXAndXDVI</code>.

</dd>
<dt> Access to components </dt>
<dd><code class="code">Modulus</code>, <code class="code">Coefficients</code>.

</dd>
<dt> Attributes </dt>
<dd><code class="code">Support</code> / <code class="code">MovedPoints</code>, <code class="code">Order</code>, <code class="code">Multiplier</code>, <code class="code">Divisor</code>, <code class="code">PrimeSet</code>, <code class="code">One</code>, <code class="code">Zero</code>.

</dd>
<dt> Properties </dt>
<dd><code class="code">IsInjective</code>, <code class="code">IsSurjective</code>, <code class="code">IsBijective</code>, <code class="code">IsTame</code>, <code class="code">IsIntegral</code>, <code class="code">IsBalanced</code>, <code class="code">IsClassWiseOrderPreserving</code>, <code class="code">IsOne</code>, <code class="code">IsZero</code>.

</dd>
<dt> Action on ℤ^d </dt>
<dd><code class="code">^</code> (for points / finite sets / residue class unions), <code class="code">Trajectory</code>, <code class="code">ShortCycles</code>, <code class="code">Multpk</code>, <code class="code">ClassWiseOrderPreservingOn</code>, <code class="code">ClassWiseOrderReversingOn</code>, <code class="code">ClassWiseConstantOn</code>.

</dd>
<dt> Arithmetical operations </dt>
<dd><code class="code">=</code>, <code class="code">*</code> (multiplication / composition and multiplication by a 2 × 2 matrix or an integer), <code class="code">^</code> (exponentiation and conjugation), <code class="code">Inverse</code>, <code class="code">+</code> (addition of a constant).

</dd>
</dl>
The above operations are documented either in the GAP Reference Manual or earlier in this manual. The operations which are special for rcwa mappings of ℤ^2 are described in the sequel.

<a id="X8408B7837C9EED36" name="X8408B7837C9EED36"></a>

<h5>5.3-1 ProjectionsToCoordinates</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectionsToCoordinates</code>( <var class="Arg">f</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: the projections of the rcwa mapping <var class="Arg">f</var> of ℤ^2 to the coordinates if such projections exist, and <code class="code">fail</code> otherwise.

An rcwa mapping can be projected to the first / second coordinate if and only if the first / second coordinate of the image of a point depends only on the first / second coordinate of the preimage. Note that this is a very strong and restrictive condition.

<div class="example"><pre>

gap> f := RcwaMapping(ClassTransposition(0,2,1,2),ClassReflection(0,2));;
gap> Display(f);

Rcwa mapping of Z^2 with modulus (2,0)Z+(0,2)Z

 /
 | (m+1,-n) if (m,n) in (0,0)+(2,0)Z+(0,2)Z
 | (m+1,n) if (m,n) in (0,1)+(2,0)Z+(0,2)Z
(m,n) |-> < (m-1,-n) if (m,n) in (1,0)+(2,0)Z+(0,2)Z
 | (m-1,n) if (m,n) in (1,1)+(2,0)Z+(0,2)Z
 |
 \

gap> List(ProjectionsToCoordinates(f),Factorization);
[ [ ( 0(2), 1(2) ) ], [ ClassReflection( 0(2) ) ] ]

</pre></div>

<a id="X83A1752F7BE9CE85" name="X83A1752F7BE9CE85"></a>

<h4>5.4 
 Methods for residue-class-wise affine groups and -monoids over ℤ^2
</h4>

Residue-class-wise affine groups over ℤ^2 can be entered by <code class="code">Group</code>, <code class="code">GroupByGenerators</code> and <code class="code">GroupWithGenerators</code>, like any groups in GAP. Likewise, residue-class-wise affine monoids over ℤ^2 can be entered by <code class="code">Monoid</code> and <code class="code">MonoidByGenerators</code>. The groups RCWA(ℤ^2) and CT(ℤ^2) are entered as <code class="code">RCWA(Integers^2)</code> and <code class="code">CT(Integers^2)</code>, respectively. The monoid Rcwa(ℤ^2) is entered as <code class="code">Rcwa(Integers^2)</code>.

There are methods provided for the operations <code class="code">Size</code>, <code class="code">IsIntegral</code>, <code class="code">IsClassWiseTranslating</code>, <code class="code">IsTame</code>, <code class="code">Modulus</code>, <code class="code">Multiplier</code> and <code class="code">Divisor</code>.

There are methods for <code class="func">IsomorphismRcwaGroup</code> (<a href="chap3.html#X7EB8A301790290C7">3.1-1</a>) which embed the groups SL(2,ℤ) and GL(2,ℤ) into RCWA(ℤ^2) in such a way that the support of the image is a specified residue class:

<a id="X79A8F9AD7E839862" name="X79A8F9AD7E839862"></a>

<h5>5.4-1 
 IsomorphismRcwaGroup (Embeddings of SL(2,ℤ) and GL(2,ℤ))
 </h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismRcwaGroup</code>( <var class="Arg">sl2z</var>, <var class="Arg">cl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismRcwaGroup</code>( <var class="Arg">gl2z</var>, <var class="Arg">cl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a monomorphism from <var class="Arg">sl2z</var> respectively <var class="Arg">gl2z</var> to RCWA(ℤ^2), such that the support of the image is the residue class <var class="Arg">cl</var> and the generators are affine on <var class="Arg">cl</var>.

<div class="example"><pre>

gap> sl := SL(2,Integers);
SL(2,Integers)
gap> phi := IsomorphismRcwaGroup(sl,ResidueClass([1,0],[[2,2],[0,3]]));
[ [ [ 0, 1 ], [ -1, 0 ] ], [ [ 1, 1 ], [ 0, 1 ] ] ] ->
[ ClassRotation((1,0)+(2,2)Z+(0,3)Z,[[0,1],[-1,0]]),
 ClassRotation((1,0)+(2,2)Z+(0,3)Z,[[1,1],[0,1]]) ]
gap> Support(Image(phi));
(1,0)+(2,2)Z+(0,3)Z
gap> gl := GL(2,Integers);
GL(2,Integers)
gap> phi := IsomorphismRcwaGroup(gl,ResidueClass([1,0],[[2,2],[0,3]]));
[ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, 0 ], [ 0, 1 ] ],
 [ [ 1, 1 ], [ 0, 1 ] ] ] ->
[ ClassRotation((1,0)+(2,2)Z+(0,3)Z,[[0,1],[1,0]]),
 ClassRotation((1,0)+(2,2)Z+(0,3)Z,[[-1,0],[0,1]]),
 ClassRotation((1,0)+(2,2)Z+(0,3)Z,[[1,1],[0,1]]) ]
gap> [[-47,-37],[61,48]]^phi;
ClassRotation((1,0)+(2,2)Z+(0,3)Z,[[-47,-37],[61,48]])
gap> Display(last:AsTable);

Rcwa permutation of Z^2 with modulus (2,2)Z+(0,3)Z, of order 6

 [m,n] mod (2,2)Z+(0,3)Z | Image of [m,n]
-----------------------------+-------------------------------------------
[0,0] [0,1] [0,2] [1,1] |
[1,2] | [m,n]
[1,0] | [(-263m+122n+266)/3,(-1147m+532n+1147)/6]

</pre></div>

The function <code class="func">DrawOrbitPicture</code> (<a href="chap3.html#X7D9DFAC97F9F0891">3.3-3</a>) can also be used to depict orbits under the action of rcwa groups over ℤ^2. Further there is a function which depicts residue class unions of ℤ^2 and partitions of ℤ^2 into such:

<a id="X812135EB87527F01" name="X812135EB87527F01"></a>

<h5>5.4-2 DrawGrid </h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DrawGrid</code>( <var class="Arg">U</var>, <var class="Arg">yrange</var>, <var class="Arg">xrange</var>, <var class="Arg">filename</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DrawGrid</code>( <var class="Arg">P</var>, <var class="Arg">yrange</var>, <var class="Arg">xrange</var>, <var class="Arg">filename</var> )</td><td class="tdright">( function )</td></tr></table></div>
Returns: nothing.

This function depicts the residue class union <var class="Arg">U</var> of ℤ^2 or the partition <var class="Arg">P</var> of ℤ^2 into residue class unions, respectively. The arguments <var class="Arg">yrange</var> and <var class="Arg">xrange</var> are the coordinate ranges of the rectangular snippet to be drawn, and the argument <var class="Arg">filename</var> is the name, i.e. the full path name, of the output file. If the first argument is a residue class union, the output picture is black-and-white, where black pixels represent members of <var class="Arg">U</var> and white pixels represent non-members. If the first argument is a partition of ℤ^2 into residue class unions, the produced picture is colored, and different colors are used to denote membership in different parts.

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