  
  [1X81 [33X[0;0YAn Example – Residue Class Rings[133X[101X
  
  [33X[0;0YIn  this  chapter,  we  give  an example how [5XGAP[105X can be extended by new data
  structures  and  new  functionality.  In order to focus on the issues of the
  implementation, the mathematics in the example chosen is trivial. Namely, we
  will discuss computations with elements of residue class rings [22Xℤ / nℤ[122X.[133X
  
  [33X[0;0YThe  first  attempt is straightforward (see Section [14X81.1[114X), it deals with the
  implementation  of  the  necessary arithmetic operations. Section [14X81.2[114X deals
  with  the  question  why it might be useful to use an approach that involves
  creating  a  new  data structure and integrating the algorithms dealing with
  these  new  [5XGAP[105X  objects into the system. Section [14X81.3[114X shows how this can be
  done in our example, and Section [14X81.4[114X, the question of further compatibility
  of   the   new  objects  with  known  [5XGAP[105X  objects  is  discussed.  Finally,
  Section [14X81.5[114X  gives  some  hints how to improve the implementation presented
  before.[133X
  
  
  [1X81.1 [33X[0;0YA First Attempt to Implement Elements of Residue Class Rings[133X[101X
  
  [33X[0;0YSuppose  we  want to do computations with elements of a ring [22Xℤ / nℤ[122X, where [22Xn[122X
  is a positive integer.[133X
  
  [33X[0;0YFirst  we  have to decide how to represent the element [22Xk + nℤ[122X in [5XGAP[105X. If the
  modulus [22Xn[122X is fixed then we can use the integer [22Xk[122X. More precisely, we can use
  any  integer [22Xk'[122X such that [22Xk - k'[122X is a multiple of [22Xn[122X. If different moduli are
  likely  to  occur then using a list of the form [22X[ k, n ][122X, or a record of the
  form  [10Xrec(  residue  :=  [3Xk[103X[10X,  modulus  :=  [3Xn[103X[10X  )[110X  is  more appropriate. In the
  following,  let  us  assume  the  list  representation  [22X[  k, n ][122X is chosen.
  Moreover,  we  decide that the residue [22Xk[122X in all such lists satisfies [22X0 ≤ k <
  n[122X,  i.e., the result of adding two residue classes represented by [22X[ k_1, n ][122X
  and  [22X[  k_2, n ][122X (of course with same modulus [22Xn[122X) will be [22X[ k, n ][122X with [22Xk_1 +
  k_2[122X congruent to [22Xk[122X modulo [22Xn[122X and [22X0 ≤ k < n[122X.[133X
  
  [33X[0;0YNow  we  can  implement  the arithmetic operations for residue classes. Note
  that  the result of the [9Xmod[109X operator is normalized as required. The division
  by a noninvertible residue class results in [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xresclass_sum := function( c1, c2 )[127X[104X
    [4X[25X>[125X [27X   if c1[2] <> c2[2] then Error( "different moduli" ); fi;[127X[104X
    [4X[25X>[125X [27X   return [ ( c1[1] + c2[1] ) mod c1[2], c1[2] ];[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27Xresclass_diff := function( c1, c2 )[127X[104X
    [4X[25X>[125X [27X   if c1[2] <> c2[2] then Error( "different moduli" ); fi;[127X[104X
    [4X[25X>[125X [27X   return [ ( c1[1] - c2[1] ) mod c1[2], c1[2] ];[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27Xresclass_prod := function( c1, c2 )[127X[104X
    [4X[25X>[125X [27X   if c1[2] <> c2[2] then Error( "different moduli" ); fi;[127X[104X
    [4X[25X>[125X [27X   return [ ( c1[1] * c2[1] ) mod c1[2], c1[2] ];[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27Xresclass_quo := function( c1, c2 )[127X[104X
    [4X[25X>[125X [27X   local quo;[127X[104X
    [4X[25X>[125X [27X   if c1[2] <> c2[2] then Error( "different moduli" ); fi;[127X[104X
    [4X[25X>[125X [27X   quo:= QuotientMod( c1[1], c2[1], c1[2] );[127X[104X
    [4X[25X>[125X [27X   if quo <> fail then[127X[104X
    [4X[25X>[125X [27X     quo:= [ quo, c1[2] ];[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X   return quo;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWith these functions, we can in principle compute with residue classes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xlist:= List( [ 0 .. 3 ], k -> [ k, 4 ] );[127X[104X
    [4X[28X[ [ 0, 4 ], [ 1, 4 ], [ 2, 4 ], [ 3, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27Xresclass_sum( list[2], list[4] );[127X[104X
    [4X[28X[ 0, 4 ][128X[104X
    [4X[25Xgap>[125X [27Xresclass_diff( list[1], list[2] );[127X[104X
    [4X[28X[ 3, 4 ][128X[104X
    [4X[25Xgap>[125X [27Xresclass_prod( list[2], list[4] );[127X[104X
    [4X[28X[ 3, 4 ][128X[104X
    [4X[25Xgap>[125X [27Xresclass_prod( list[3], list[4] );[127X[104X
    [4X[28X[ 2, 4 ][128X[104X
    [4X[25Xgap>[125X [27XList( list, x -> resclass_quo( list[2], x ) );[127X[104X
    [4X[28X[ fail, [ 1, 4 ], fail, [ 3, 4 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X81.2 [33X[0;0YWhy Proceed in a Different Way?[133X[101X
  
  [33X[0;0YIt  depends  on  the  computations  we  intended  to do with residue classes
  whether  or  not  the  implementation  described  in the previous section is
  satisfactory for us.[133X
  
  [33X[0;0YProbably  we  are mainly interested in more complex data structures than the
  residue  classes themselves, for example in matrix algebras or matrix groups
  over  a  ring  such as [22Xℤ / 4ℤ[122X. For this, we need functions to add, multiply,
  invert  etc. matrices  of residue classes. Of course this is not a difficult
  task, but it requires to write additional [5XGAP[105X code.[133X
  
  [33X[0;0YAnd  when  we  have  implemented  the  arithmetic operations for matrices of
  residue  classes,  we  might  be  interested  in  domain  operations such as
  computing  the  order of a matrix group over [22Xℤ / 4ℤ[122X, a Sylow [22X2[122X subgroup, and
  so on. The problem is that a residue class represented as a pair [22X[ k, n ][122X is
  not  regarded  as  a  group  element by [5XGAP[105X. We have not yet discussed how a
  matrix of residue classes shall be represented, but if we choose the obvious
  representation  of  a list of lists of our residue classes then also this is
  not  a  valid group element in [5XGAP[105X. Hence we cannot apply the function [2XGroup[102X
  ([14X39.2-1[114X)  to  create  a  group  of residue classes or a group of matrices of
  residue  classes.  This  is  because  [5XGAP[105X assumes that group elements can be
  multiplied  via  the  infix  operator  [10X*[110X (equivalently, via the operation [2X\*[102X
  ([14X31.12-1[114X)).  Note that in fact the multiplication of two lists [22X[ k_1, n ][122X, [22X[
  k_2,  n  ][122X is defined, but we have [22X[ k_1, n ] * [ k_2, n ] = k_1 * k_2 + n *
  n[122X,  the  standard scalar product of two row vectors of same length. That is,
  the  multiplication with [10X*[110X is not compatible with the function [10Xresclass_prod[110X
  introduced  in the previous section. Similarly, ring elements are assumed to
  be  added  via  the infix operator [10X+[110X; the addition of residue classes is not
  compatible with the available addition of row vectors.[133X
  
  [33X[0;0YWhat we have done in the previous section can be described as implementation
  of  a  [21Xstandalone[121X  arithmetic  for  residue  classes.  In  order  to use the
  machinery  of  the  [5XGAP[105X  library  for  creating higher level objects such as
  matrices,  polynomials,  or  domains  over  residue  class rings, we have to
  [21Xintegrate[121X  this implementation into the [5XGAP[105X library. The key step will be to
  create  a  new  kind  of  [5XGAP[105X  objects.  This  will be done in the following
  sections;  there  we assume that residue classes and residue class rings are
  not   yet   available  in  [5XGAP[105X;  in  fact  they  are  available,  and  their
  implementation is very close to what is described here.[133X
  
  
  [1X81.3 [33X[0;0YA Second Attempt to Implement Elements of Residue Class Rings[133X[101X
  
  [33X[0;0YFaced  with  the  problem to implement elements of the rings [22Xℤ / nℤ[122X, we must
  define  the  [13Xtypes[113X  of  these elements as far as is necessary to distinguish
  them from other [5XGAP[105X objects.[133X
  
  [33X[0;0YAs  is  described  in  Chapter [14X13[114X,  the  type of an object comprises several
  aspects of information about this object; the [13Xfamily[113X determines the relation
  of the object to other objects, the [13Xcategories[113X determine what operations the
  object  admits,  the  [13Xrepresentation[113X  determines  how  an object is actually
  represented, and the [13Xattributes[113X describe knowledge about the object.[133X
  
  [33X[0;0YFirst  of  all,  we  must  decide  about the [13Xfamily[113X of each residue class. A
  natural  way  to  do  this is to put the elements of each ring [22Xℤ / nℤ[122X into a
  family  of their own. This means that for example elements of [22Xℤ / 3ℤ[122X and [22Xℤ /
  9ℤ[122X  lie  in different families. So the only interesting relation between the
  families  of  two  residue classes is equality; binary arithmetic operations
  with  two  residue  classes  will  be  admissible only if their families are
  equal.  Note that in the naive approach in Section [14X81.1[114X, we had to take care
  of  different moduli by a check in each function; these checks may disappear
  in the new approach because of our choice of families.[133X
  
  [33X[0;0YNote  that  we  do  not  need  to tell [5XGAP[105X anything about the above decision
  concerning  the families of the objects that we are going to implement, that
  is,  the  [13Xdeclaration  part[113X  (see [14X79.19[114X)  of  the  little [5XGAP[105X package we are
  writing  contains  nothing  about  the  distribution of the new objects into
  families.  (The  actual  construction  of  a  family happens in the function
  [10XMyZmodnZ[110X shown below.)[133X
  
  [33X[0;0YSecond,  we  want to describe methods to add or multiply two elements in [22Xℤ /
  nℤ[122X,  and  these  methods  shall  be not applicable to other [5XGAP[105X objects. The
  natural  way to do this is to create a new [13Xcategory[113X in which all elements of
  all rings [22Xℤ / nℤ[122X lie. This is done as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDeclareCategory( "IsMyZmodnZObj", IsScalar );[127X[104X
    [4X[25Xgap>[125X [27Xcat:= CategoryCollections( IsMyZmodnZObj );;[127X[104X
    [4X[25Xgap>[125X [27Xcat:= CategoryCollections( cat );;[127X[104X
    [4X[25Xgap>[125X [27Xcat:= CategoryCollections( cat );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  all elements in the rings [22Xℤ / nℤ[122X will lie in the category [10XIsMyZmodnZObj[110X,
  which is a subcategory of [2XIsScalar[102X ([14X31.14-20[114X). The latter means that one can
  add,  subtract,  multiply  and divide two such elements that lie in the same
  family,  with  the obvious restriction that the second operand of a division
  must  be  invertible.  (The name [10XIsMyZmodnZObj[110X is chosen because [2XIsZmodnZObj[102X
  ([14X14.5-4[114X) is already defined in [5XGAP[105X, for an implementation of residue classes
  that is very similar to the one developed in this manual chapter. Using this
  different name, one can simply enter the [5XGAP[105X code of this chapter into a [5XGAP[105X
  session,  either  interactively  or  by  reading  a file with this code, and
  experiment after each step whether the expected behaviour has been achieved,
  and what is still missing.)[133X
  
  [33X[0;0YThe  next lines of [5XGAP[105X code above create the categories [10XCategoryCollections(
  IsMyZmodnZObj  )[110X  and  two  higher levels of collections categories of this,
  which  will  be  needed  later;  it  is important to create these categories
  before collections of the objects in [10XIsMyZmodnZObj[110X actually arise.[133X
  
  [33X[0;0YNote   that  the  only  difference  between  [2XDeclareCategory[102X  ([14X79.18-1[114X)  and
  [2XNewCategory[102X  ([14X79.1-1[114X)  is  that  in  a  call to [2XDeclareCategory[102X ([14X79.18-1[114X), a
  variable corresponding to the first argument is set to the new category, and
  this    variable    is   read-only   (see [14X79.18[114X).   The   same   holds   for
  [2XDeclareRepresentation[102X ([14X79.18-2[114X) and [2XNewRepresentation[102X ([14X79.2-1[114X) etc.[133X
  
  [33X[0;0YThere  is  no  analogue of categories in the implementation in Section [14X81.1[114X,
  since  there  it was not necessary to distinguish residue classes from other
  [5XGAP[105X objects. Note that the functions there assumed that their arguments were
  residue  classes,  and  the user was responsible not to call them with other
  arguments.  Thus  an  important  aspect of types is to describe arguments of
  functions explicitly.[133X
  
  [33X[0;0YThird,  we  must  decide  about  the  [13Xrepresentation[113X of our objects. This is
  something we know already from Section [14X81.1[114X, where we chose a list of length
  two.  Here  we  may choose between two essentially different representations
  for  the  new  [5XGAP[105X  objects,  namely  as  [21Xcomponent  object[121X (record-like) or
  [21Xpositional  object[121X  (list-like).  We  decide  to  store  the modulus of each
  residue  class in its family, and to encode the element [22Xk + nℤ[122X by the unique
  residue  in  the range [22X[ 0 .. n-1 ][122X that is congruent to [22Xk[122X modulo [22Xn[122X, and the
  object  itself  is chosen to be a positional object with this residue at the
  first and only position (see [14X79.11[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDeclareRepresentation("IsMyModulusRep", IsPositionalObjectRep, [1]);[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  fourth  ingredients  of  a  type,  [13Xattributes[113X,  are  usually  of  minor
  importance  for  element objects. In particular, we do not need to introduce
  special attributes for residue classes.[133X
  
  [33X[0;0YHaving defined what the new objects shall look like, we now declare a global
  function  (see [14X79.19[114X),  to  create  an  element  when family and residue are
  given.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDeclareGlobalFunction( "MyZmodnZObj" );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we have declared what we need, and we can start to implement the missing
  methods   resp.   functions;   so  the  following  command  belongs  to  the
  [13Ximplementation part[113X of our package (see [14X79.19[114X).[133X
  
  [33X[0;0YThe  probably  most  interesting  function is the one to construct a residue
  class.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallGlobalFunction( MyZmodnZObj, function( Fam, residue )[127X[104X
    [4X[25X>[125X [27X   return Objectify( NewType( Fam, IsMyZmodnZObj and IsMyModulusRep ),[127X[104X
    [4X[25X>[125X [27X                     [ residue mod Fam!.modulus ] );[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  we  normalize  [10Xresidue[110X explicitly using [9Xmod[109X; we assumed that the
  modulus  is  stored  in [10XFam[110X, so we must take care of this below. If [10XFam[110X is a
  family  of  residue  classes, and [10Xresidue[110X is an integer, [10XMyZmodnZObj[110X returns
  the  corresponding  object  in  the  family  [10XFam[110X, which lies in the category
  [10XIsMyZmodnZObj[110X and in the representation [10XIsMyModulusRep[110X.[133X
  
  [33X[0;0Y[10XMyZmodnZObj[110X needs an appropriate family as first argument, so let us see how
  to  get  our  hands  on  this.  Of course we could write a handy function to
  create  such  a family for given modulus, but we choose another way. In fact
  we  do not really want to call [10XMyZmodnZObj[110X explicitly when we want to create
  residue  classes. For example, if we want to enter a matrix of residues then
  usually  we  start  with  a matrix of corresponding integers, and it is more
  elegant to do the conversion via multiplying the matrix with the identity of
  the  required  ring [22Xℤ / nℤ[122X; this is also done for the conversion of integral
  matrices  to  finite  field  matrices.  (Note that we will have to install a
  method  for  this.)  So  it is often sufficient to access this identity, for
  example  via  [10XOne(  MyZmodnZ(  [3Xn[103X[10X  )  )[110X,  where  [10XMyZmodnZ[110X  returns  a  domain
  representing the ring [22Xℤ / nℤ[122X when called with the argument [22Xn[122X. We decide that
  constructing  this  ring is a natural place where the creation of the family
  can  be  hidden,  and  implement  the  function.  (Note that the declaration
  belongs  to  the  declaration  part,  and  the  installation  belongs to the
  implementation part, see [14X79.19[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDeclareGlobalFunction( "MyZmodnZ" );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallGlobalFunction( MyZmodnZ, function( n )[127X[104X
    [4X[25X>[125X [27X   local F, R;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   if not IsPosInt( n ) then[127X[104X
    [4X[25X>[125X [27X     Error( "<n> must be a positive integer" );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   # Construct the family of element objects of our ring.[127X[104X
    [4X[25X>[125X [27X   F:= NewFamily( Concatenation( "MyZmod", String( n ), "Z" ),[127X[104X
    [4X[25X>[125X [27X                  IsMyZmodnZObj );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   # Install the data.[127X[104X
    [4X[25X>[125X [27X   F!.modulus:= n;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   # Make the domain.[127X[104X
    [4X[25X>[125X [27X   R:= RingWithOneByGenerators( [ MyZmodnZObj( F, 1 ) ] );[127X[104X
    [4X[25X>[125X [27X   SetIsWholeFamily( R, true );[127X[104X
    [4X[25X>[125X [27X   SetName( R, Concatenation( "(Integers mod ", String(n), ")" ) );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   # Return the ring.[127X[104X
    [4X[25X>[125X [27X   return R;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNote that the modulus [10Xn[110X is stored in the component [10Xmodulus[110X of the family, as
  is  assumed  by  [10XMyZmodnZ[110X.  Thus it is not necessary to store the modulus in
  each  element. When storing [10Xn[110X with the [10X!.[110X operator as value of the component
  [10Xmodulus[110X,  we  used  that  all  families are in fact represented as component
  objects (see [14X79.10[114X).[133X
  
  [33X[0;0YWe  see that we can use [2XRingWithOneByGenerators[102X ([14X56.3-3[114X) to construct a ring
  with  one  if  we  have  the  appropriate  generators.  The construction via
  [2XRingWithOneByGenerators[102X ([14X56.3-3[114X) makes sure that [2XIsRingWithOne[102X ([14X56.3-1[114X) (and
  [2XIsRing[102X ([14X56.1-1[114X)) is [9Xtrue[109X for each output of [10XMyZmodnZ[110X. So the main problem is
  to  create  the  identity element of the ring, which in our case suffices to
  generate  the ring. In order to create this element via [10XMyZmodnZObj[110X, we have
  to construct its family first, at each call of [10XMyZmodnZ[110X.[133X
  
  [33X[0;0YAlso  note that we may enter known information about the ring. Here we store
  that  it  contains  the whole family of elements; this is useful for example
  when we want to check the membership of an element in the ring, which can be
  decided from the type of the element if the ring contains its whole elements
  family.  Giving  a  name  to  the  ring  causes  that it will be printed via
  printing the name. (By the way: This name [10X(Integers mod [3Xn[103X[10X)[110X looks like a call
  to  [2X\mod[102X ([14X31.12-1[114X) with the arguments [2XIntegers[102X ([14X14[114X) and [3Xn[103X; a construction of
  the  ring  via  this call seems to be more natural than by calling [10XMyZmodnZ[110X;
  later  we  shall  install  a  [2X\mod[102X  ([14X31.12-1[114X)  method in order to admit this
  construction.)[133X
  
  [33X[0;0YNow we can read the above code into [5XGAP[105X, and the following works already.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:= MyZmodnZ( 4 );[127X[104X
    [4X[28X(Integers mod 4)[128X[104X
    [4X[25Xgap>[125X [27XIsRing( R );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgens:= GeneratorsOfRingWithOne( R );[127X[104X
    [4X[28X[ <object> ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YBut  of course this means just to ask for the information we have explicitly
  stored in the ring. Already the questions whether the ring is finite and how
  many  elements  it  has,  cannot  be  answered  by  [5XGAP[105X. Clearly we know the
  answers,  and  we  could store them in the ring, by setting the value of the
  property  [2XIsFinite[102X  ([14X30.4-2[114X)  to  [9Xtrue[109X  and  the value of the attribute [2XSize[102X
  ([14X30.4-6[114X)  to  [3Xn[103X (the argument of the call to [10XMyZmodnZ[110X). If we do not want to
  do so then [5XGAP[105X could only try to find out the number of elements of the ring
  via forming the closure of the generators under addition and multiplication,
  but  up to now, [5XGAP[105X does not know how to add or multiply two elements of our
  ring.[133X
  
  [33X[0;0YSo  we  must install some methods for arithmetic and other operations if the
  elements are to behave as we want.[133X
  
  [33X[0;0YWe  start with a method for showing elements nicely on the screen. There are
  different  operations  for  this  purpose.  One of them is [2XPrintObj[102X ([14X6.3-5[114X),
  which  is  called  for  each  argument in an explicit call to [2XPrint[102X ([14X6.3-4[114X).
  Another  one is [2XViewObj[102X ([14X6.3-5[114X), which is called in the read-eval-print loop
  for  each  object.  [2XViewObj[102X  ([14X6.3-5[114X)  shall produce short and human readable
  information  about  the  object  in question, whereas [2XPrintObj[102X ([14X6.3-5[114X) shall
  produce  information  that  may be longer and is (if reasonable) readable by
  [5XGAP[105X.  We cannot satisfy the latter requirement for a [2XPrintObj[102X ([14X6.3-5[114X) method
  because  there  is  no  way  to  make a family [5XGAP[105X readable. So we decide to
  display  the  expression  [10X(  k  mod  n  )[110X for an object that is given by the
  residue  [10Xk[110X  and  the  modulus  [10Xn[110X,  which  would be fine as a [2XViewObj[102X ([14X6.3-5[114X)
  method.  Since  the default for [2XViewObj[102X ([14X6.3-5[114X) is to call [2XPrintObj[102X ([14X6.3-5[114X),
  and  since no other [2XViewObj[102X ([14X6.3-5[114X) method is applicable to our elements, we
  need only a [2XPrintObj[102X ([14X6.3-5[114X) method.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( PrintObj,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x )[127X[104X
    [4X[25X>[125X [27X   Print( "( ", x![1], " mod ", FamilyObj(x)!.modulus, " )" );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  installed  a  method  for  the  operation  [2XPrintObj[102X  ([14X6.3-5[114X)  (first
  argument),  and we gave it a suitable information message (second argument),
  see [14X7.2-1[114X  and [14X7.3[114X  for  applications  of this information string. The third
  argument tells [5XGAP[105X that the method is applicable for objects that lie in the
  category  [10XIsMyZmodnZObj[110X  and  in  the representation [10XIsMyModulusRep[110X. and the
  fourth  argument  is  the  method  itself.  More details about [2XInstallMethod[102X
  ([14X78.2-1[114X) can be found in [14X78.2[114X.[133X
  
  [33X[0;0YNote  that  the  requirement  [10XIsMyModulusRep[110X for the argument [10Xx[110X allows us to
  access the residue as [10Xx![1][110X. Since the family of [10Xx[110X has the component [10Xmodulus[110X
  bound  if  it  is  constructed by [10XMyZmodnZ[110X, we may access this component. We
  check whether the method installation has some effect.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgens;[127X[104X
    [4X[28X[ ( 1 mod 4 ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we  install  methods  for  the comparison operations. Note that we can
  assume that the residues in the representation chosen are normalized.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \=,[127X[104X
    [4X[25X>[125X [27X   "for two elements in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [IsMyZmodnZObj and IsMyModulusRep, IsMyZmodnZObj and IsMyModulusRep],[127X[104X
    [4X[25X>[125X [27X   function( x, y ) return x![1] = y![1]; end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \<,[127X[104X
    [4X[25X>[125X [27X   "for two elements in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [IsMyZmodnZObj and IsMyModulusRep, IsMyZmodnZObj and IsMyModulusRep],[127X[104X
    [4X[25X>[125X [27X   function( x, y ) return x![1] < y![1]; end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  third  argument  used  in  these  installations  specifies the required
  relation  between the families of the arguments (see [14X13.1[114X). This argument of
  a  method  installation,  if present, is a function that shall be applied to
  the  families  of  the  arguments.  [2XIsIdenticalObj[102X  ([14X12.5-1[114X)  means that the
  methods  are  applicable  only if both arguments lie in the same family. (In
  installations  for unary methods, obviously no relation is required, so this
  argument is left out there.)[133X
  
  [33X[0;0YUp  to  now,  we  see  no  advantage  of  the  new  approach over the one in
  Section [14X81.1[114X.  For  a  residue  class represented as [10X[ [3Xk[103X[10X, [3Xn[103X[10X ][110X, the way it is
  printed  on  the  screen is sufficient, and equality and comparison of lists
  are  good  enough  to  define  equality and comparison of residue classes if
  needed.  But  this  is  not the case in other situations. For example, if we
  would  have  decided that the residue [3Xk[103X need not be normalized then we would
  have  needed  functions  in  Section [14X81.1[114X  that  compute whether two residue
  classes  are  equal,  and which of two residue classes is regarded as larger
  than  another. Note that we are free to define what [21Xlarger[121X means for objects
  that are newly introduced.[133X
  
  [33X[0;0YNext  we  install  methods  for  the  arithmetic  operations,  first for the
  additive structure.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \+,[127X[104X
    [4X[25X>[125X [27X   "for two elements in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [IsMyZmodnZObj and IsMyModulusRep, IsMyZmodnZObj and IsMyModulusRep],[127X[104X
    [4X[25X>[125X [27X   function( x, y )[127X[104X
    [4X[25X>[125X [27X   return MyZmodnZObj( FamilyObj( x ), x![1] + y![1] );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( ZeroOp,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObj ],[127X[104X
    [4X[25X>[125X [27X   x -> MyZmodnZObj( FamilyObj( x ), 0 ) );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( AdditiveInverseOp,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   x -> MyZmodnZObj( FamilyObj( x ), AdditiveInverse( x![1] ) ) );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YHere  the  new  approach  starts to pay off. The method for the operation [2X\+[102X
  ([14X31.12-1[114X)  allows  us  to  use the infix operator [10X+[110X for residue classes. The
  method  for  [2XZeroOp[102X  ([14X31.10-3[114X)  is  used  when we call this operation or the
  attribute  [2XZero[102X  ([14X31.10-3[114X)  explicitly, and [2XZeroOp[102X ([14X31.10-3[114X) it is also used
  when we ask for [10X0 * [3Xrescl[103X[10X[110X, where [3Xrescl[103X is a residue class.[133X
  
  [33X[0;0Y(Note that [2XZero[102X ([14X31.10-3[114X) and [2XZeroOp[102X ([14X31.10-3[114X) are distinguished because [10X0 *
  [3Xobj[103X[10X[110X  is  guaranteed to return a [13Xmutable[113X result whenever a mutable version of
  this  result  exists  in  [5XGAP[105X  –for example if [3Xobj[103X is a matrix– whereas [2XZero[102X
  ([14X31.10-3[114X)  is  an attribute and therefore returns [13Ximmutable[113X results; for our
  example  there  is  no  difference  since  the  residue  classes  are always
  immutable,  nevertheless we have to install the method for [2XZeroOp[102X ([14X31.10-3[114X).
  The  same  holds  for  [2XAdditiveInverse[102X ([14X31.10-9[114X), [2XOne[102X ([14X31.10-2[114X), and [2XInverse[102X
  ([14X31.10-8[114X).)[133X
  
  [33X[0;0YSimilarly,  [2XAdditiveInverseOp[102X ([14X31.10-9[114X) can be either called directly or via
  the  unary [10X-[110X operator; so we can compute the additive inverse of the residue
  class [3Xrescl[103X as [10X-[3Xrescl[103X[10X[110X.[133X
  
  [33X[0;0YIt  is  not  necessary  to  install  methods  for subtraction, since this is
  handled  via  addition  of the additive inverse of the second argument if no
  other method is installed.[133X
  
  [33X[0;0YLet us try what we can do with the methods that are available now.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xx:= gens[1];  y:= x + x;[127X[104X
    [4X[28X( 1 mod 4 )[128X[104X
    [4X[28X( 2 mod 4 )[128X[104X
    [4X[25Xgap>[125X [27X0 * x;  -x;[127X[104X
    [4X[28X( 0 mod 4 )[128X[104X
    [4X[28X( 3 mod 4 )[128X[104X
    [4X[25Xgap>[125X [27Xy = -y;  x = y;  x < y;  -x < y;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  might  want  to admit the addition of integers and elements in rings [22Xℤ /
  nℤ[122X,  where an integer is implicitly identified with its residue modulo [22Xn[122X. To
  achieve  this,  we  install  methods  to  add  an  integer  to  an object in
  [10XIsMyZmodnZObj[110X from the left and from the right.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \+,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/nZ (ModulusRep) and integer",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObj and IsMyModulusRep, IsInt ],[127X[104X
    [4X[25X>[125X [27X   function( x, y )[127X[104X
    [4X[25X>[125X [27X   return MyZmodnZObj( FamilyObj( x ), x![1] + y );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \+,[127X[104X
    [4X[25X>[125X [27X   "for integer and element in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   [ IsInt, IsMyZmodnZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y )[127X[104X
    [4X[25X>[125X [27X   return MyZmodnZObj( FamilyObj( y ), x + y![1] );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we can do also the following.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X2 + x;  7 - x;  y - 2;[127X[104X
    [4X[28X( 3 mod 4 )[128X[104X
    [4X[28X( 2 mod 4 )[128X[104X
    [4X[28X( 0 mod 4 )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSimilarly  we install the methods dealing with the multiplicative structure.
  We  need methods to multiply two of our objects, and to compute identity and
  inverse.  The  operation  [2XOneOp[102X ([14X31.10-2[114X) is called when we ask for [10X[3Xrescl[103X[10X^0[110X,
  and  [2XInverseOp[102X  ([14X31.10-8[114X)  is called when we ask for [10X[3Xrescl[103X[10X^-1[110X. Note that the
  method  for  [2XInverseOp[102X  ([14X31.10-8[114X)  returns  [9Xfail[109X  if  the  argument  is  not
  invertible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \*,[127X[104X
    [4X[25X>[125X [27X   "for two elements in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [IsMyZmodnZObj and IsMyModulusRep, IsMyZmodnZObj and IsMyModulusRep],[127X[104X
    [4X[25X>[125X [27X   function( x, y )[127X[104X
    [4X[25X>[125X [27X   return MyZmodnZObj( FamilyObj( x ), x![1] * y![1] );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( OneOp,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObj ],[127X[104X
    [4X[25X>[125X [27X   elm -> MyZmodnZObj( FamilyObj( elm ), 1 ) );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( InverseOp,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( elm )[127X[104X
    [4X[25X>[125X [27X   local residue;[127X[104X
    [4X[25X>[125X [27X   residue:= QuotientMod( 1, elm![1], FamilyObj( elm )!.modulus );[127X[104X
    [4X[25X>[125X [27X   if residue <> fail then[127X[104X
    [4X[25X>[125X [27X     residue:= MyZmodnZObj( FamilyObj( elm ), residue );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X   return residue;[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YTo  be  able to multiply our objects with integers, we need not (but we may,
  and  we should if we are going for efficiency) install special methods. This
  is  because  in general, [5XGAP[105X interprets the multiplication of an integer and
  an additive object as abbreviation of successive additions, and there is one
  generic  method  for  such a multiplication that uses only additions and –in
  the  case  of  a negative integer– taking the additive inverse. Analogously,
  there  is  a  generic  method  for  powering  by  integers  that  uses  only
  multiplications and taking the multiplicative inverse.[133X
  
  [33X[0;0YNote  that  we  could also interpret the multiplication with an integer as a
  shorthand  for  the  multiplication with the corresponding residue class. We
  are  lucky  that  this  interpretation  is  compatible  with the one that is
  already available. If this would not be the case then of course we would get
  into  trouble  by  installing  a  concurrent  multiplication  that  computes
  something  different  from the multiplication that is already defined, since
  [5XGAP[105X  does  not  guarantee which of the applicable methods is actually chosen
  (see [14X78.3[114X).[133X
  
  [33X[0;0YNow  we  have  implemented  methods  for  the  arithmetic operations for our
  elements, and the following calculations work.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xy:= 2 * x;  z:= (-5) * x;[127X[104X
    [4X[28X( 2 mod 4 )[128X[104X
    [4X[28X( 3 mod 4 )[128X[104X
    [4X[25Xgap>[125X [27Xy * z;  y * y;[127X[104X
    [4X[28X( 2 mod 4 )[128X[104X
    [4X[28X( 0 mod 4 )[128X[104X
    [4X[25Xgap>[125X [27Xy^-1;  y^0;[127X[104X
    [4X[28Xfail[128X[104X
    [4X[28X( 1 mod 4 )[128X[104X
    [4X[25Xgap>[125X [27Xz^-1;[127X[104X
    [4X[28X( 3 mod 4 )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere  are  some  other  operations  in  [5XGAP[105X  that we may want to accept our
  elements  as  arguments.  An  example  is  the  operation  [2XInt[102X ([14X14.2-3[114X) that
  returns,  e.g.,  the  integral  part  of  a  rational  number or the integer
  corresponding to an element in a finite prime field. For our objects, we may
  define that [2XInt[102X ([14X14.2-3[114X) returns the normalized residue.[133X
  
  [33X[0;0YNote  that  we  [13Xdefine[113X  this  behaviour for elements but we [13Ximplement[113X it for
  objects  in  the  representation  [10XIsMyModulusRep[110X. This means that if someone
  implements  another  representation of residue classes then this person must
  be  careful  to  implement  [2XInt[102X  ([14X14.2-3[114X)  methods  for  objects in this new
  representation compatibly with our definition, i.e., such that the result is
  independent of the representation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( Int,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   z -> z![1] );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAnother  example of an operation for which we might want to install a method
  is  [2X\mod[102X  ([14X31.12-1[114X).  We make the ring print itself as [2XIntegers[102X ([14X14[114X) mod the
  modulus,  and  then it is reasonable to allow a construction this way, which
  makes the [2XPrintObj[102X ([14X6.3-5[114X) output of the ring [5XGAP[105X readable.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( PrintObj,[127X[104X
    [4X[25X>[125X [27X   "for full collection Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [ CategoryCollections( IsMyZmodnZObj ) and IsWholeFamily ],[127X[104X
    [4X[25X>[125X [27X   function( R )[127X[104X
    [4X[25X>[125X [27X   Print( "(Integers mod ",[127X[104X
    [4X[25X>[125X [27X          ElementsFamily( FamilyObj(R) )!.modulus, ")" );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \mod,[127X[104X
    [4X[25X>[125X [27X   "for `Integers', and a positive integer",[127X[104X
    [4X[25X>[125X [27X   [ IsIntegers, IsPosRat and IsInt ],[127X[104X
    [4X[25X>[125X [27X   function( Integers, n ) return MyZmodnZ( n ); end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YLet us try this.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInt( y );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XIntegers mod 1789;[127X[104X
    [4X[28X(Integers mod 1789)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YProbably  it  is  not  necessary  to  emphasize  that  with  the approach of
  Section [14X81.1[114X,  installing  methods  for  existing  operations is usually not
  possible  or  at least not recommended. For example, installing the function
  [10Xresclass_sum[110X defined in Section [14X81.1[114X as a [2X\+[102X ([14X31.12-1[114X) method for adding two
  lists  of length two (with integer entries) would not be compatible with the
  general definition of the addition of two lists of same length. Installing a
  method for the operation [2XInt[102X ([14X14.2-3[114X) that takes a list [10X[ [3Xk[103X[10X, [3Xn[103X[10X ][110X and returns
  [3Xk[103X  would in principle be possible, since there is no [2XInt[102X ([14X14.2-3[114X) method for
  lists  yet,  but  it is not sensible to do so because one can think of other
  interpretations of such a list where different [2XInt[102X ([14X14.2-3[114X) methods could be
  installed with the same right.[133X
  
  [33X[0;0YAs mentioned in Section [14X81.2[114X, one advantage of the new approach is that with
  the implementation we have up to now, automatically also matrices of residue
  classes can be treated.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr:= Integers mod 16;[127X[104X
    [4X[28X(Integers mod 16)[128X[104X
    [4X[25Xgap>[125X [27Xx:= One( r );[127X[104X
    [4X[28X( 1 mod 16 )[128X[104X
    [4X[25Xgap>[125X [27Xmat:= IdentityMat( 2 ) * x;[127X[104X
    [4X[28X[ [ ( 1 mod 16 ), ( 0 mod 16 ) ], [ ( 0 mod 16 ), ( 1 mod 16 ) ] ][128X[104X
    [4X[25Xgap>[125X [27Xmat[1][2]:= x;;[127X[104X
    [4X[25Xgap>[125X [27Xmat;[127X[104X
    [4X[28X[ [ ( 1 mod 16 ), ( 1 mod 16 ) ], [ ( 0 mod 16 ), ( 1 mod 16 ) ] ][128X[104X
    [4X[25Xgap>[125X [27XOrder( mat );[127X[104X
    [4X[28X16[128X[104X
    [4X[25Xgap>[125X [27Xmat + mat;[127X[104X
    [4X[28X[ [ ( 2 mod 16 ), ( 2 mod 16 ) ], [ ( 0 mod 16 ), ( 2 mod 16 ) ] ][128X[104X
    [4X[25Xgap>[125X [27Xlast^4;[127X[104X
    [4X[28X[ [ ( 0 mod 16 ), ( 0 mod 16 ) ], [ ( 0 mod 16 ), ( 0 mod 16 ) ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSuch  matrices,  if  they  are  invertible, are valid as group elements. One
  technical  problem  is that the default algorithm for inverting matrices may
  give  up  since  Gaussian  elimination  need  not  be  successful over rings
  containing  zero  divisors.  Therefore  we  install a simpleminded inversion
  method that inverts an integer matrix.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( InverseOp,[127X[104X
    [4X[25X>[125X [27X   "for an ordinary matrix over a ring Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [ IsMatrix and IsOrdinaryMatrix[127X[104X
    [4X[25X>[125X [27X     and CategoryCollections( CategoryCollections( IsMyZmodnZObj ) ) ],[127X[104X
    [4X[25X>[125X [27X   function( mat )[127X[104X
    [4X[25X>[125X [27X   local one, modulus;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   one:= One( mat[1][1] );[127X[104X
    [4X[25X>[125X [27X   modulus:= FamilyObj( one )!.modulus;[127X[104X
    [4X[25X>[125X [27X   mat:= InverseOp( List( mat, row -> List( row, Int ) ) );[127X[104X
    [4X[25X>[125X [27X   if mat <> fail then[127X[104X
    [4X[25X>[125X [27X     mat:= ( mat mod modulus ) * one;[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X   if not IsMatrix( mat ) then[127X[104X
    [4X[25X>[125X [27X     mat:= fail;[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X   return mat;[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAdditionally  we  install  a  method  for finding a domain that contains the
  matrix entries; this is used by some [5XGAP[105X library functions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( DefaultFieldOfMatrixGroup,[127X[104X
    [4X[25X>[125X [27X    "for a matrix group over a ring Z/nZ",[127X[104X
    [4X[25X>[125X [27X    [ IsMatrixGroup and CategoryCollections( CategoryCollections([127X[104X
    [4X[25X>[125X [27X          CategoryCollections( IsMyZmodnZObj ) ) ) ],[127X[104X
    [4X[25X>[125X [27X    G -> RingWithOneByGenerators([ One( Representative( G )[1][1] ) ]));[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we can deal with matrix groups over residue class rings.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmat2:= IdentityMat( 2 ) * x;;[127X[104X
    [4X[25Xgap>[125X [27Xmat2[2][1]:= x;;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Group( mat, mat2 );;[127X[104X
    [4X[25Xgap>[125X [27XSize( g );[127X[104X
    [4X[28X3072[128X[104X
    [4X[25Xgap>[125X [27XFactors( last );[127X[104X
    [4X[28X[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xsyl3:= SylowSubgroup( g, 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= GeneratorsOfGroup( syl3 );[127X[104X
    [4X[28X[ [ [ ( 1 mod 16 ), ( 7 mod 16 ) ], [ ( 11 mod 16 ), ( 14 mod 16 ) ] ] ][128X[104X
    [4X[25Xgap>[125X [27XOrder( gens[1] );[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt should be noted that this way more involved methods for matrix groups may
  not  be  available.  For example, many questions about a finite matrix group
  can  be  delegated  to  an isomorphic permutation group via a so-called [21Xnice
  monomorphism[121X;     this     can     be     controlled     by    the    filter
  [2XIsHandledByNiceMonomorphism[102X ([14X40.5-1[114X).[133X
  
  [33X[0;0YBy  the  way, also groups of (invertible) residue classes can be formed, but
  this may be of minor interest.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= Group( x );;  Size( g );[127X[104X
    [4X[28X#I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for [128X[104X
    [4X[28X[ ( 1 mod 16 ) ][128X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xg:= Group( 3*x );;  Size( g );[127X[104X
    [4X[28X#I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for [128X[104X
    [4X[28X[ ( 3 mod 16 ) ][128X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(The  messages  above  tell  that  [5XGAP[105X  does  not know a method for deciding
  whether  the  given  elements  are  valid  group  elements.  We could add an
  appropriate [10XIsGeneratorsOfMagmaWithInverses[110X method if we would want.)[133X
  
  [33X[0;0YHaving  done  enough  for the elements, we may install some more methods for
  the  rings  if we want to use them as arguments. These rings are finite, and
  there  are  many  generic methods that will work if they are able to compute
  the list of elements of the ring, so we install a method for this.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( Enumerator,[127X[104X
    [4X[25X>[125X [27X   "for full collection Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [ CategoryCollections( IsMyZmodnZObj ) and IsWholeFamily ],[127X[104X
    [4X[25X>[125X [27X   function( R )[127X[104X
    [4X[25X>[125X [27X   local F;[127X[104X
    [4X[25X>[125X [27X   F:= ElementsFamily( FamilyObj(R) );[127X[104X
    [4X[25X>[125X [27X   return List( [ 0 .. Size( R ) - 1 ], x -> MyZmodnZObj( F, x ) );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  this  method is applicable only to full rings [22Xℤ / nℤ[122X, for proper
  subrings  it  would  return  a wrong result. Furthermore, it is not required
  that  the  argument is a ring; in fact this method is applicable also to the
  additive  group formed by all elements in the family, provided that it knows
  to contain the whole family.[133X
  
  [33X[0;0YAnalogously,  we  install methods to compute the size, a random element, and
  the units of full rings [22Xℤ / nℤ[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( Random,[127X[104X
    [4X[25X>[125X [27X   "for full collection Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [ CategoryCollections( IsMyZmodnZObj ) and IsWholeFamily ],[127X[104X
    [4X[25X>[125X [27X   R -> MyZmodnZObj( ElementsFamily( FamilyObj(R) ),[127X[104X
    [4X[25X>[125X [27X                   Random( [ 0 .. Size( R ) - 1 ] ) ) );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( Size,[127X[104X
    [4X[25X>[125X [27X   "for full ring Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [ CategoryCollections( IsMyZmodnZObj ) and IsWholeFamily ],[127X[104X
    [4X[25X>[125X [27X   R -> ElementsFamily( FamilyObj(R) )!.modulus );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( Units,[127X[104X
    [4X[25X>[125X [27X   "for full ring Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [     CategoryCollections( IsMyZmodnZObj )[127X[104X
    [4X[25X>[125X [27X     and IsWholeFamily and IsRing ],[127X[104X
    [4X[25X>[125X [27X   function( R )[127X[104X
    [4X[25X>[125X [27X   local F;[127X[104X
    [4X[25X>[125X [27X   F:= ElementsFamily( FamilyObj( R ) );[127X[104X
    [4X[25X>[125X [27X   return List( PrimeResidues( Size(R) ), x -> MyZmodnZObj( F, x ) );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  [2XUnits[102X  ([14X56.5-2[114X) method has the disadvantage that the result is returned
  as  a list (in fact this list is also strictly sorted). We could improve the
  implementation  by returning the units as a group; if we do not want to take
  the   full  list  of  elements  as  generators,  we  can  use  the  function
  [2XGeneratorsPrimeResidues[102X ([14X15.2-4[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( Units,[127X[104X
    [4X[25X>[125X [27X   "for full ring Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [     CategoryCollections( IsMyZmodnZObj )[127X[104X
    [4X[25X>[125X [27X     and IsWholeFamily and IsRing ],[127X[104X
    [4X[25X>[125X [27X   function( R )[127X[104X
    [4X[25X>[125X [27X   local G, gens;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   gens:= GeneratorsPrimeResidues( Size( R ) ).generators;[127X[104X
    [4X[25X>[125X [27X   if not IsEmpty( gens ) and gens[ 1 ] = 1 then[127X[104X
    [4X[25X>[125X [27X     gens:= gens{ [ 2 .. Length( gens ) ] };[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X   gens:= Flat( gens ) * One( R );[127X[104X
    [4X[25X>[125X [27X   return GroupByGenerators( gens, One( R ) );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YEach  ring [22Xℤ / nℤ[122X is finite, and we could install a method that returns [9Xtrue[109X
  when [2XIsFinite[102X ([14X30.4-2[114X) is called with [22Xℤ / nℤ[122X as argument. But we can do this
  more elegantly via installing a [13Xlogical implication[113X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallTrueMethod( IsFinite,[127X[104X
    [4X[25X>[125X [27X   CategoryCollections( IsMyZmodnZObj ) and IsDomain );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  effect,  every  domain  that  consists of elements in [10XIsMyZmodnZObj[110X will
  automatically  store  that  it  is  finite, even if [2XIsFinite[102X ([14X30.4-2[114X) is not
  called for it.[133X
  
  
  [1X81.4 [33X[0;0YCompatibility of Residue Class Rings with Prime Fields[133X[101X
  
  [33X[0;0YThe  above  implementation of residue classes and residue class rings has at
  least  two  disadvantages. First, if [22Xp[122X is a prime then the ring [22Xℤ / pℤ[122X is in
  fact a field, but the return values of [10XMyZmodnZ[110X are never regarded as fields
  because  they are not in the category [2XIsMagmaWithInversesIfNonzero[102X ([14X35.1-3[114X).
  Second,  and  this  makes  the example really interesting, there are already
  elements  of  finite  prime  fields  implemented  in [5XGAP[105X, and we may want to
  identify them with elements in [22Xℤ / pℤ[122X.[133X
  
  [33X[0;0YTo  be  more  precise,  elements of finite fields in [5XGAP[105X lie in the category
  [2XIsFFE[102X  ([14X59.1-1[114X),  and  there  is already a representation, [10XIsInternalRep[110X, of
  these  elements  via discrete logarithms. The aim of this section is to make
  [10XIsMyModulusRep[110X  an  alternative  representation  of elements in finite prime
  fields.[133X
  
  [33X[0;0YNote  that  this is only one step towards the desired compatibility. Namely,
  after  having a second representation of elements in finite prime fields, we
  may  wish  that  the  function  [2XGF[102X  ([14X59.3-2[114X) (which is the usual function to
  create finite fields in [5XGAP[105X) is able to return [10XMyZmodnZ( [3Xp[103X[10X )[110X when [10XGF( [3Xp[103X[10X )[110X is
  called  for  a  prime  [3Xp[103X.  Moreover,  then we have to decide about a default
  representation  of  elements  in  [10XGF(  [3Xp[103X[10X  )[110X  for  primes  [3Xp[103X  for  which both
  representations are available. Of course we can force the new representation
  by  explicitly  calling [10XMyZmodnZ[110X and [10XMyZmodnZObj[110X whenever we want, but it is
  not a priori clear in which situation which representation is preferable.[133X
  
  [33X[0;0YThe same questions will occur when we want to implement a new representation
  for  non-prime  fields. The steps of this implementation will be the same as
  described  in  this  chapter, and we will have to achieve compatibility with
  both  the internal representation of elements in small finite fields and the
  representation [10XIsMyModulusRep[110X of elements in arbitrary prime fields.[133X
  
  [33X[0;0YBut  let  us  now turn back to the task of this section. We first adjust the
  setup  of  the declaration part of the previous section, and then repeat the
  installations with suitable modifications.[133X
  
  [33X[0;0Y(We  should  start  a  new [5XGAP[105X session for that, otherwise [5XGAP[105X will complain
  that the objects to be declared are already bound; additionally, the methods
  installed above may be not compatible with the ones we want.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDeclareCategory( "IsMyZmodnZObj", IsScalar );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XDeclareCategory( "IsMyZmodnZObjNonprime", IsMyZmodnZObj );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XDeclareSynonym( "IsMyZmodpZObj", IsMyZmodnZObj and IsFFE );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XDeclareRepresentation( "IsMyModulusRep", IsPositionalObjectRep, [ 1 ] );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XDeclareGlobalFunction( "MyZmodnZObj" );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XDeclareGlobalFunction( "MyZmodnZ" );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  in the previous section, all (newly introduced) elements of rings [22Xℤ / nℤ[122X
  lie  in  the category [10XIsMyZmodnZObj[110X. But now we introduce two subcategories,
  namely [10XIsMyZmodnZObjNonprime[110X for all elements in rings [22Xℤ / nℤ[122X where [22Xn[122X is not
  a  prime, and [10XIsMyZmodpZObj[110X for elements in finite prime fields. All objects
  in  the latter are automatically known to lie in the category [2XIsFFE[102X ([14X59.1-1[114X)
  of finite field elements.[133X
  
  [33X[0;0YIt  would be reasonable if also those internally represented elements in the
  category  [2XIsFFE[102X ([14X59.1-1[114X) that do in fact lie in a prime field would also lie
  in  the category [10XIsMyZmodnZObj[110X (and thus in fact in [10XIsMyZmodpZObj[110X). But this
  cannot  be  achieved because internally represented finite field elements do
  in general not store whether they lie in a prime field.[133X
  
  [33X[0;0YAs  for  the implementation part, again let us start with the definitions of
  [10XMyZmodnZObj[110X and [10XMyZmodnZ[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallGlobalFunction( MyZmodnZObj, function( Fam, residue )[127X[104X
    [4X[25X>[125X [27X   if IsFFEFamily( Fam ) then[127X[104X
    [4X[25X>[125X [27X     return Objectify( NewType( Fam, IsMyZmodpZObj[127X[104X
    [4X[25X>[125X [27X                                 and IsMyModulusRep ),[127X[104X
    [4X[25X>[125X [27X                   [ residue mod Characteristic( Fam ) ] );[127X[104X
    [4X[25X>[125X [27X   else[127X[104X
    [4X[25X>[125X [27X     return Objectify( NewType( Fam, IsMyZmodnZObjNonprime[127X[104X
    [4X[25X>[125X [27X                                 and IsMyModulusRep ),[127X[104X
    [4X[25X>[125X [27X                   [ residue mod Fam!.modulus ] );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XInstallGlobalFunction( MyZmodnZ, function( n )[127X[104X
    [4X[25X>[125X [27X   local F, R;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   if not ( IsInt( n ) and IsPosRat( n ) ) then[127X[104X
    [4X[25X>[125X [27X     Error( "<n> must be a positive integer" );[127X[104X
    [4X[25X>[125X [27X   elif IsPrimeInt( n ) then[127X[104X
    [4X[25X>[125X [27X     # Construct the family of element objects of our field.[127X[104X
    [4X[25X>[125X [27X     F:= FFEFamily( n );[127X[104X
    [4X[25X>[125X [27X     # Make the domain.[127X[104X
    [4X[25X>[125X [27X     R:= FieldOverItselfByGenerators( [ MyZmodnZObj( F, 1 ) ] );[127X[104X
    [4X[25X>[125X [27X     SetIsPrimeField( R, true );[127X[104X
    [4X[25X>[125X [27X   else[127X[104X
    [4X[25X>[125X [27X     # Construct the family of element objects of our ring.[127X[104X
    [4X[25X>[125X [27X     F:= NewFamily( Concatenation( "MyZmod", String( n ), "Z" ),[127X[104X
    [4X[25X>[125X [27X                    IsMyZmodnZObjNonprime );[127X[104X
    [4X[25X>[125X [27X     # Install the data.[127X[104X
    [4X[25X>[125X [27X     F!.modulus:= n;[127X[104X
    [4X[25X>[125X [27X     # Make the domain.[127X[104X
    [4X[25X>[125X [27X     R:= RingWithOneByGenerators( [ MyZmodnZObj( F, 1 ) ] );[127X[104X
    [4X[25X>[125X [27X     SetIsWholeFamily( R, true );[127X[104X
    [4X[25X>[125X [27X     SetName( R, Concatenation( "(Integers mod ",String(n),")" ) );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   # Return the ring resp. field.[127X[104X
    [4X[25X>[125X [27X   return R;[127X[104X
    [4X[25X>[125X [27Xend );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  the  result of [10XMyZmodnZ[110X with a prime as argument is a field that
  does  not  contain  the whole family of its elements, since all finite field
  elements of a fixed characteristic lie in the same family. Further note that
  we  cannot  expect  a  family  of  finite field elements to have a component
  [10Xmodulus[110X,  so  we  use [2XCharacteristic[102X ([14X31.10-1[114X) to get the modulus. Requiring
  that  [10XFam!.modulus[110X  works  also  if [10XFam[110X is a family of finite field elements
  would  violate the rule that an extension of [5XGAP[105X should not force changes in
  existing  code,  in  this  case  code  dealing with families of finite field
  elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( PrintObj,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObjNonprime and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x )[127X[104X
    [4X[25X>[125X [27X   Print( "( ", x![1], " mod ", FamilyObj(x)!.modulus, " )" );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( PrintObj,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/pZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodpZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x )[127X[104X
    [4X[25X>[125X [27X   Print( "( ", x![1], " mod ", Characteristic(x), " )" );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \=,[127X[104X
    [4X[25X>[125X [27X   "for two elements in Z/nZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObj and IsMyModulusRep,[127X[104X
    [4X[25X>[125X [27X     IsMyZmodnZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y ) return x![1] = y![1]; end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above  method to check equality is independent of whether the arguments
  have  a  prime  or  nonprime  modulus,  so  we installed it for arguments in
  [10XIsMyZmodnZObj[110X.   Now   we   install  also  methods  to  compare  objects  in
  [10XIsMyZmodpZObj[110X with the [21Xold[121X finite field elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \=,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/pZ (ModulusRep) and internal FFE",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodpZObj and IsMyModulusRep, IsFFE and IsInternalRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y )[127X[104X
    [4X[25X>[125X [27X   return DegreeFFE( y ) = 1 and x![1] = IntFFE( y );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \=,[127X[104X
    [4X[25X>[125X [27X   "for internal FFE and element in Z/pZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsFFE and IsInternalRep, IsMyZmodpZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y )[127X[104X
    [4X[25X>[125X [27X   return DegreeFFE( x ) = 1 and IntFFE( x ) = y![1];[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  situation with the operation [10X<[110X is more difficult. Of course we are free
  to  define  the  comparison of objects in [10XIsMyZmodnZObjNonprime[110X, but for the
  finite field elements, the comparison must be compatible with the predefined
  comparison  of  the  [21Xold[121X  finite  field  elements.  The  definition of the [10X<[110X
  comparison  of  internally represented finite field elements can be found in
  Chapter [14X59[114X.  In  situations  where  the  documentation  does not provide the
  required  information,  one  has to look it up in the [5XGAP[105X code; for example,
  the  comparison in our case can be found in the appropriate source code file
  of the [5XGAP[105X kernel.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \<,[127X[104X
    [4X[25X>[125X [27X   "for two elements in Z/nZ (ModulusRep, nonprime)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObjNonprime and IsMyModulusRep,[127X[104X
    [4X[25X>[125X [27X     IsMyZmodnZObjNonprime and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y ) return x![1] < y![1]; end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \<,[127X[104X
    [4X[25X>[125X [27X   "for two elements in Z/pZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodpZObj and IsMyModulusRep,[127X[104X
    [4X[25X>[125X [27X     IsMyZmodpZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y )[127X[104X
    [4X[25X>[125X [27X   local p, r;      # characteristic and primitive root[127X[104X
    [4X[25X>[125X [27X   if x![1] = 0 then[127X[104X
    [4X[25X>[125X [27X     return y![1] <> 0;[127X[104X
    [4X[25X>[125X [27X   elif y![1] = 0 then[127X[104X
    [4X[25X>[125X [27X     return false;[127X[104X
    [4X[25X>[125X [27X   else[127X[104X
    [4X[25X>[125X [27X     p:= Characteristic( x );[127X[104X
    [4X[25X>[125X [27X     r:= PrimitiveRootMod( p );[127X[104X
    [4X[25X>[125X [27X     return LogMod( x![1], r, p ) < LogMod( y![1], r, p );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \<,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/pZ (ModulusRep) and internal FFE",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodpZObj and IsMyModulusRep, IsFFE and IsInternalRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y )[127X[104X
    [4X[25X>[125X [27X   return x![1] * One( y ) < y;[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \<,[127X[104X
    [4X[25X>[125X [27X   "for internal FFE and element in Z/pZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsFFE and IsInternalRep, IsMyZmodpZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y )[127X[104X
    [4X[25X>[125X [27X   return x < y![1] * One( x );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we install the same methods for the arithmetic operations [2X\+[102X ([14X31.12-1[114X),
  [2XZeroOp[102X  ([14X31.10-3[114X),  [2XAdditiveInverseOp[102X ([14X31.10-9[114X), [10X\-[110X, [2X\*[102X ([14X31.12-1[114X), and [2XOneOp[102X
  ([14X31.10-2[114X)  as  in the previous section, without listing them below. Also the
  same  [2XInt[102X  ([14X14.2-3[114X)  method  is installed for objects in [10XIsMyZmodnZObj[110X. Note
  that  it  is compatible with the definition of [2XInt[102X ([14X14.2-3[114X) for finite field
  elements. And of course the same method for [2X\mod[102X ([14X31.12-1[114X) is installed.[133X
  
  [33X[0;0YWe have to be careful, however, with the methods for [2XInverseOp[102X ([14X31.10-8[114X), [2X\/[102X
  ([14X31.12-1[114X),  and  [2X\^[102X  ([14X31.12-1[114X).  These  methods  and the missing methods for
  arithmetic  operations  with one argument in [10XIsMyModulusRep[110X and the other in
  [10XIsInternalRep[110X are given below.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \+,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/pZ (ModulusRep) and internal FFE",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodpZObj and IsMyModulusRep, IsFFE and IsInternalRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y ) return x![1] + y; end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \+,[127X[104X
    [4X[25X>[125X [27X   "for internal FFE and element in Z/pZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsFFE and IsInternalRep, IsMyZmodpZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y ) return x + y![1]; end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \*,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/pZ (ModulusRep) and internal FFE",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodpZObj and IsMyModulusRep, IsFFE and IsInternalRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y ) return x![1] * y; end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( \*,[127X[104X
    [4X[25X>[125X [27X   "for internal FFE and element in Z/pZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   IsIdenticalObj,[127X[104X
    [4X[25X>[125X [27X   [ IsFFE and IsInternalRep, IsMyZmodpZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x, y ) return x * y![1]; end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( InverseOp,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/nZ (ModulusRep, nonprime)",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodnZObjNonprime and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x )[127X[104X
    [4X[25X>[125X [27X   local residue;[127X[104X
    [4X[25X>[125X [27X   residue:= QuotientMod( 1, x![1], FamilyObj(x)!.modulus );[127X[104X
    [4X[25X>[125X [27X   if residue <> fail then[127X[104X
    [4X[25X>[125X [27X     residue:= MyZmodnZObj( FamilyObj(x), residue );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X   return residue;[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( InverseOp,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/pZ (ModulusRep)",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodpZObj and IsMyModulusRep ],[127X[104X
    [4X[25X>[125X [27X   function( x )[127X[104X
    [4X[25X>[125X [27X   local residue;[127X[104X
    [4X[25X>[125X [27X   residue:= QuotientMod( 1, x![1], Characteristic( FamilyObj(x) ) );[127X[104X
    [4X[25X>[125X [27X   if residue <> fail then[127X[104X
    [4X[25X>[125X [27X     residue:= MyZmodnZObj( FamilyObj(x), residue );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X   return residue;[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  operation  [2XDegreeFFE[102X  ([14X59.2-1[114X) is defined for finite field elements, we
  need  a  method  for objects in [10XIsMyZmodpZObj[110X. Note that we need not require
  [10XIsMyModulusRep[110X since no access to representation dependent data occurs.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( DegreeFFE,[127X[104X
    [4X[25X>[125X [27X   "for element in Z/pZ",[127X[104X
    [4X[25X>[125X [27X   [ IsMyZmodpZObj ],[127X[104X
    [4X[25X>[125X [27X   z -> 1 );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  methods  for  [2XEnumerator[102X  ([14X30.3-2[114X), [2XRandom[102X ([14X30.7-1[114X), [2XSize[102X ([14X30.4-6[114X), and
  [2XUnits[102X  ([14X56.5-2[114X),  that  we  had  installed  in  the previous section had all
  assumed  that  their  argument contains the whole family of its elements. So
  these  methods  make  sense  only for the nonprime case. For the prime case,
  there are already methods for these operations with argument a field.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInstallMethod( Enumerator,[127X[104X
    [4X[25X>[125X [27X   "for full ring Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [ CategoryCollections( IsMyZmodnZObjNonprime ) and IsWholeFamily ],[127X[104X
    [4X[25X>[125X [27X   function( R )[127X[104X
    [4X[25X>[125X [27X   local F;[127X[104X
    [4X[25X>[125X [27X   F:= ElementsFamily( FamilyObj( R ) );[127X[104X
    [4X[25X>[125X [27X   return List( [ 0 .. Size( R ) - 1 ], x -> MyZmodnZObj( F, x ) );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( Random,[127X[104X
    [4X[25X>[125X [27X   "for full ring Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [ CategoryCollections( IsMyZmodnZObjNonprime ) and IsWholeFamily ],[127X[104X
    [4X[25X>[125X [27X   R -> MyZmodnZObj( ElementsFamily( FamilyObj( R ) ),[127X[104X
    [4X[25X>[125X [27X                   Random( [ 0 .. Size( R ) - 1 ] ) ) );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( Size,[127X[104X
    [4X[25X>[125X [27X   "for full ring Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [ CategoryCollections( IsMyZmodnZObjNonprime ) and IsWholeFamily ],[127X[104X
    [4X[25X>[125X [27X   R -> ElementsFamily( FamilyObj( R ) )!.modulus );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallMethod( Units,[127X[104X
    [4X[25X>[125X [27X   "for full ring Z/nZ",[127X[104X
    [4X[25X>[125X [27X   [     CategoryCollections( IsMyZmodnZObjNonprime )[127X[104X
    [4X[25X>[125X [27X     and IsWholeFamily and IsRing ],[127X[104X
    [4X[25X>[125X [27X   function( R )[127X[104X
    [4X[25X>[125X [27X   local G, gens;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   gens:= GeneratorsPrimeResidues( Size( R ) ).generators;[127X[104X
    [4X[25X>[125X [27X   if not IsEmpty( gens ) and gens[ 1 ] = 1 then[127X[104X
    [4X[25X>[125X [27X     gens:= gens{ [ 2 .. Length( gens ) ] };[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X   gens:= Flat( gens ) * One( R );[127X[104X
    [4X[25X>[125X [27X   return GroupByGenerators( gens, One( R ) );[127X[104X
    [4X[25X>[125X [27X   end );[127X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XInstallTrueMethod( IsFinite,[127X[104X
    [4X[25X>[125X [27X   CategoryCollections( IsMyZmodnZObjNonprime ) and IsDomain );[127X[104X
  [4X[32X[104X
  
  
  [1X81.5 [33X[0;0YFurther Improvements in Implementing Residue Class Rings[133X[101X
  
  [33X[0;0YThere are of course many possibilities to improve the implementation.[133X
  
  [33X[0;0YWith  the  setup as described above, subsequent calls [10XMyZmodnZ( [3Xn[103X[10X )[110X with the
  same  [3Xn[103X  yield  incompatible  rings  in  the sense that elements of one ring
  cannot  be  added to elements of an other one. The solution for this problem
  is  to  keep  a  global  list  of all results of [10XMyZmodnZ[110X in the current [5XGAP[105X
  session,  and  to return the stored values whenever possible. Note that this
  approach  would  admit  [2XPrintObj[102X  ([14X6.3-5[114X)  methods that produce [5XGAP[105X readable
  output.[133X
  
  [33X[0;0YOne  can  improve  the [2XUnits[102X ([14X56.5-2[114X) method for the full ring in such a way
  that  a  group  is  returned  and  not only a list of its elements; then the
  result  of  [2XUnits[102X  ([14X56.5-2[114X)  can  be used, e. g., as input for the operation
  [2XSylowSubgroup[102X ([14X39.13-1[114X).[133X
  
  [33X[0;0YTo  make  computations  more  efficient,  one can install methods for [10X\-[110X, [2X\/[102X
  ([14X31.12-1[114X), and [2X\^[102X ([14X31.12-1[114X); one reason for doing so may be that this avoids
  the  unnecessary  construction of the additive or multiplicative inverse, or
  of intermediate powers.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XInstallMethod( \-, "two elements in Z/nZ (ModulusRep)", ... );[128X[104X
    [4X[28XInstallMethod( \-, "Z/nZ-obj. (ModulusRep) and integer", ... );[128X[104X
    [4X[28XInstallMethod( \-, "integer and Z/nZ-obj. (ModulusRep)", ... );[128X[104X
    [4X[28XInstallMethod( \-, "Z/pZ-obj. (ModulusRep) and internal FFE", ... );[128X[104X
    [4X[28XInstallMethod( \-, "internal FFE and Z/pZ-obj. (ModulusRep)", ... );[128X[104X
    [4X[28XInstallMethod( \*, "Z/nZ-obj. (ModulusRep) and integer", ... );[128X[104X
    [4X[28XInstallMethod( \*, "integer and Z/nZ-obj. (ModulusRep)", ... );[128X[104X
    [4X[28XInstallMethod( \/, "two Z/nZ-objs. (ModulusRep, nonprime)", ... );[128X[104X
    [4X[28XInstallMethod( \/, "two Z/pZ-objs. (ModulusRep)", ... );[128X[104X
    [4X[28XInstallMethod( \/, "Z/nZ-obj. (ModulusRep) and integer", ... );[128X[104X
    [4X[28XInstallMethod( \/, "integer and Z/nZ-obj. (ModulusRep)", ... );[128X[104X
    [4X[28XInstallMethod( \/, "Z/pZ-obj. (ModulusRep) and internal FFE", ... );[128X[104X
    [4X[28XInstallMethod( \/, "internal FFE and Z/pZ-obj. (ModulusRep)", ... );[128X[104X
    [4X[28XInstallMethod( \^, "Z/nZ-obj. (ModulusRep, nonprime) & int.", ... );[128X[104X
    [4X[28XInstallMethod( \^, "Z/pZ-obj. (ModulusRep), and integer", ... );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  call  to  [2XNewType[102X ([14X79.8-1[114X) in [10XMyZmodnZObj[110X can be avoided by storing the
  required  type,  e.g.,  in the family. But note that it is [13Xnot[113X admissible to
  take the type of an existing object as first argument of [2XObjectify[102X ([14X79.9-1[114X).
  For  example,  suppose  two objects in [10XIsMyZmodnZObj[110X shall be added. Then we
  must  not  use  the  type  of  one  of  the arguments in a call of [2XObjectify[102X
  ([14X79.9-1[114X),  because  the  argument may have knowledge that is not correct for
  the  result  of  the addition. One may think of the property [2XIsOne[102X ([14X31.10-5[114X)
  that may hold for both arguments but certainly not for their sum.[133X
  
  [33X[0;0YFor  comparing two objects in [10XIsMyZmodpZObj[110X via [21X[10X<[110X[121X, we had to install a quite
  expensive  method because of the compatibility with the comparison of finite
  field  elements  that  did already exist. In fact [5XGAP[105X supports finite fields
  with  elements  represented via discrete logarithms only up to a given size.
  So in principle we have the freedom to define a cheaper comparison via [21X[10X<[110X[121X for
  objects in [10XIsMyZmodpZObj[110X if the modulus is large enough. This is possible by
  introducing  two categories [10XIsMyZmodpZObjSmall[110X and [10XIsMyZmodpZObjLarge[110X, which
  are  subcategories  of  [10XIsMyZmodpZObj[110X, and to install different [2X\<[102X ([14X31.11-1[114X)
  methods for pairs of objects in these categories.[133X
  
