  
  [1X79 [33X[0;0YCreating New Objects[133X[101X
  
  [33X[0;0YThis chapter is divided into three parts.[133X
  
  [33X[0;0YIn  the  first  part, it is explained how to create filters (see [14X79.1[114X, [14X79.2[114X,
  [14X79.3[114X,  [14X79.4[114X),  operations (see [14X79.5[114X), families (see [14X79.7[114X), types (see [14X79.8[114X),
  and objects with given type (see [14X79.9[114X).[133X
  
  [33X[0;0YIn  the  second part, first a few small examples are given, for dealing with
  the  usual  cases  of  component  objects (see [14X79.10[114X) and positional objects
  (see [14X79.11[114X),  and  for  the  implementation of new kinds of lists (see [14X79.12[114X
  and [14X79.15[114X).  Finally,  the  external representation of objects is introduced
  (see [14X79.16[114X), as a tool for representation independent access to an object.[133X
  
  [33X[0;0YThe  third part deals with some rules concerning the organization of the [5XGAP[105X
  library;  namely,  some commands for creating global variables are explained
  (see [14X79.18[114X)  that  correspond to the ones discussed in the first part of the
  chapter,  and the idea of distinguishing declaration and implementation part
  of [5XGAP[105X packages is outlined (see [14X79.19[114X).[133X
  
  [33X[0;0YSee  also  Chapter [14X81[114X for examples how the functions from the first part are
  used,  and why it is useful to have a declaration part and an implementation
  part.[133X
  
  
  [1X79.1 [33X[0;0YCreating Categories[133X[101X
  
  [1X79.1-1 NewCategory[101X
  
  [33X[1;0Y[29X[2XNewCategory[102X( [3Xname[103X, [3Xsuper[103X[, [3Xrank[103X] ) [32X function[133X
  
  [33X[0;0Y[2XNewCategory[102X  returns  a  new  category  [3Xcat[103X  that  has  the name [3Xname[103X and is
  contained in the filter [3Xsuper[103X, see [14X13.2[114X. This means that every object in [3Xcat[103X
  lies  automatically  also  in  [3Xsuper[103X.  We  say also that [3Xsuper[103X is an implied
  filter of [3Xcat[103X.[133X
  
  [33X[0;0YFor  example, if one wants to create a category of group elements then [3Xsuper[103X
  should  be [2XIsMultiplicativeElementWithInverse[102X ([14X31.14-13[114X) or a subcategory of
  it.  If  no  specific  supercategory  of [3Xcat[103X is known, [3Xsuper[103X may be [2XIsObject[102X
  ([14X12.1-1[114X).[133X
  
  [33X[0;0YThe  optional third argument [3Xrank[103X denotes the incremental rank (see [14X13.2[114X) of
  [3Xcat[103X, the default value is 1.[133X
  
  [1X79.1-2 CategoryFamily[101X
  
  [33X[1;0Y[29X[2XCategoryFamily[102X( [3Xcat[103X ) [32X function[133X
  
  [33X[0;0YFor  a category [3Xcat[103X, [2XCategoryFamily[102X returns the [13Xfamily category[113X of [3Xcat[103X. This
  is  a  category in which all families lie that know from their creation that
  all their elements are in the category [3Xcat[103X, see [14X79.7[114X.[133X
  
  [33X[0;0YFor   example,   a   family   of   associative  words  is  in  the  category
  [10XCategoryFamily(  IsAssocWord  )[110X,  and one can distinguish such a family from
  others by this category. So it is possible to install methods for operations
  that require one argument to be a family of associative words.[133X
  
  [33X[0;0Y[2XCategoryFamily[102X is quite technical, and in fact of minor importance.[133X
  
  [33X[0;0YSee also [2XCategoryCollections[102X ([14X30.2-4[114X).[133X
  
  
  [1X79.2 [33X[0;0YCreating Representations[133X[101X
  
  [1X79.2-1 NewRepresentation[101X
  
  [33X[1;0Y[29X[2XNewRepresentation[102X( [3Xname[103X, [3Xsuper[103X, [3Xslots[103X[, [3Xreq[103X] ) [32X function[133X
  
  [33X[0;0Y[2XNewRepresentation[102X  returns  a  new representation [3Xrep[103X that has the name [3Xname[103X
  and  is  a  subrepresentation  of  the representation [3Xsuper[103X. This means that
  every object in [3Xrep[103X lies automatically also in [3Xsuper[103X. We say also that [3Xsuper[103X
  is an implied filter of [3Xrep[103X.[133X
  
  [33X[0;0YEach representation in [5XGAP[105X is a subrepresentation of exactly one of the four
  representations    [10XIsInternalRep[110X,   [10XIsDataObjectRep[110X,   [10XIsComponentObjectRep[110X,
  [10XIsPositionalObjectRep[110X.  The data describing objects in the former two can be
  accessed  only  via [5XGAP[105X kernel functions, the data describing objects in the
  latter  two is accessible also in library functions, see [14X79.10[114X and [14X79.11[114X for
  the details.[133X
  
  [33X[0;0YThe  third argument [3Xslots[103X is a list either of integers or of strings. In the
  former case, [3Xrep[103X must be [10XIsPositionalObjectRep[110X or a subrepresentation of it,
  and  [3Xslots[103X tells what positions of the objects in the representation [3Xrep[103X may
  be  bound.  In  the  latter  case,  [3Xrep[103X  must  be  [10XIsComponentObjectRep[110X or a
  subrepresentation  of,  and  [3Xslots[103X  lists the admissible names of components
  that  objects  in  the representation [3Xrep[103X may have. The admissible positions
  resp. component names of [3Xsuper[103X need not be be listed in [3Xslots[103X.[133X
  
  [33X[0;0YThe incremental rank (see [14X13.2[114X) of [3Xrep[103X is 1.[133X
  
  [33X[0;0YNote  that  for  objects  in  the  representation [3Xrep[103X, of course some of the
  component names and positions reserved via [3Xslots[103X may be unbound.[133X
  
  [33X[0;0YExamples  for the use of [2XNewRepresentation[102X can be found in [14X79.10[114X, [14X79.11[114X, and
  also in [14X81.3[114X.[133X
  
  
  [1X79.3 [33X[0;0YCreating Attributes and Properties[133X[101X
  
  [33X[0;0YEach   method  that  is  installed  for  an  attribute  or  a  property  via
  [2XInstallMethod[102X  ([14X78.2-1[114X) must require exactly one argument, and this must lie
  in  the  filter  [3Xfilter[103X  that was entered as second argument of [2XNewAttribute[102X
  ([14X79.3-1[114X) resp. [2XNewProperty[102X ([14X79.3-2[114X).[133X
  
  [33X[0;0YAs  for  any  operation  (see [14X79.5[114X),  for  attributes and properties one can
  install  a  method  taking  an  argument  that  does  not  lie  in  [3Xfilt[103X via
  [2XInstallOtherMethod[102X  ([14X78.2-2[114X), or a method for more than one argument; in the
  latter case, clearly the result value is [13Xnot[113X stored in any of the arguments.[133X
  
  [1X79.3-1 NewAttribute[101X
  
  [33X[1;0Y[29X[2XNewAttribute[102X( [3Xname[103X, [3Xfilter[103X[, [3X"mutable"[103X][, [3Xrank[103X] ) [32X function[133X
  
  [33X[0;0Y[2XNewAttribute[102X  returns  a  new  attribute  getter  with  name  [3Xname[103X  that  is
  applicable to objects with the property [3Xfilter[103X.[133X
  
  [33X[0;0YContrary to the situation with categories and representations, the tester of
  the  new  attribute  does  [13Xnot[113X  imply [3Xfilter[103X. This is exactly because of the
  possibility to install methods that do not require [3Xfilter[103X.[133X
  
  [33X[0;0YFor  example, the attribute [2XSize[102X ([14X30.4-6[114X) was created with second argument a
  list  or  a collection, but there is also a method for [2XSize[102X ([14X30.4-6[114X) that is
  applicable to a character table, which is neither a list nor a collection.[133X
  
  [33X[0;0YIf  the  optional  third argument is given then there are two possibilities.
  Either it is an integer [3Xrank[103X, then the attribute tester has this incremental
  rank  (see [14X13.2[114X).  Or  it  is  the  string [10X"mutable"[110X, then the values of the
  attribute  shall  be mutable; more precisely, when a value of such a mutable
  attribute  is set then this value itself is stored, not an immutable copy of
  it.  (So  it  is  the user's responsibility to set an object that is in fact
  mutable.)  This  is  useful  for  an  attribute  whose value is some partial
  information  that may be completed later. For example, there is an attribute
  [10XComputedSylowSubgroups[110X for the list holding those Sylow subgroups of a group
  that have been computed already by the function [2XSylowSubgroup[102X ([14X39.13-1[114X), and
  this  list  is  mutable because one may want to enter groups into it as they
  are computed.[133X
  
  [33X[0;0YIf no third argument is given then the rank of the tester is 1.[133X
  
  [33X[0;0YEach  method for the new attribute that does [13Xnot[113X require its argument to lie
  in [3Xfilter[103X must be installed using [2XInstallOtherMethod[102X ([14X78.2-2[114X).[133X
  
  [1X79.3-2 NewProperty[101X
  
  [33X[1;0Y[29X[2XNewProperty[102X( [3Xname[103X, [3Xfilter[103X[, [3Xrank[103X] ) [32X function[133X
  
  [33X[0;0Y[2XNewProperty[102X  returns a new property [3Xprop[103X with name [3Xname[103X (see also [14X13.7[114X). The
  filter  [3Xfilter[103X  describes  the  involved  filters of [3Xprop[103X. As in the case of
  attributes, [3Xfilter[103X is not implied by [3Xprop[103X.[133X
  
  [33X[0;0YThe  optional third argument [3Xrank[103X denotes the incremental rank (see [14X13.2[114X) of
  the property [3Xprop[103X itself, i.e. [13Xnot[113X of its tester; the default value is 1.[133X
  
  
  [1X79.4 [33X[0;0YCreating Other Filters[133X[101X
  
  [33X[0;0YIn  order to change the value of [3Xfilt[103X for an object [3Xobj[103X, one can use logical
  implications (see [14X78.7[114X) or [2XSetFilterObj[102X ([14X79.4-2[114X), [2XResetFilterObj[102X ([14X79.4-3[114X).[133X
  
  [1X79.4-1 NewFilter[101X
  
  [33X[1;0Y[29X[2XNewFilter[102X( [3Xname[103X[, [3Xrank[103X] ) [32X function[133X
  
  [33X[0;0Y[2XNewFilter[102X  returns  a  simple filter with name [3Xname[103X (see [14X13.8[114X). The optional
  second  argument [3Xrank[103X denotes the incremental rank (see [14X13.2[114X) of the filter,
  the default value is 1.[133X
  
  [33X[0;0YThe default value of the new simple filter for each object is [9Xfalse[109X.[133X
  
  [1X79.4-2 SetFilterObj[101X
  
  [33X[1;0Y[29X[2XSetFilterObj[102X( [3Xobj[103X, [3Xfilter[103X ) [32X function[133X
  
  [33X[0;0Y[2XSetFilterObj[102X sets the value of [3Xfilter[103X (and of all filters implied by [3Xfilter[103X)
  for [3Xobj[103X to [9Xtrue[109X,[133X
  
  [1X79.4-3 ResetFilterObj[101X
  
  [33X[1;0Y[29X[2XResetFilterObj[102X( [3Xobj[103X, [3Xfilter[103X ) [32X function[133X
  
  [33X[0;0Y[2XResetFilterObj[102X  sets  the value of [3Xfilter[103X for [3Xobj[103X to [9Xfalse[109X. (Implied filters
  of  [3Xfilt[103X  are  not  touched.  This  might  create inconsistent situations if
  applied carelessly).[133X
  
  
  [1X79.5 [33X[0;0YCreating Operations[133X[101X
  
  [1X79.5-1 NewOperation[101X
  
  [33X[1;0Y[29X[2XNewOperation[102X( [3Xname[103X, [3Xargs-filts[103X ) [32X function[133X
  
  [33X[0;0Y[2XNewOperation[102X  returns  an  operation [3Xopr[103X with name [3Xname[103X. The list [3Xargs-filts[103X
  describes  requirements  about  the  arguments  of [3Xopr[103X, namely the number of
  arguments  must  be equal to the length of [3Xargs-filts[103X, and the [22Xi[122X-th argument
  must lie in the filter [3Xargs-filts[103X[22X[i][122X.[133X
  
  [33X[0;0YEach  method  that  is  installed  for  [3Xopr[103X  via [2XInstallMethod[102X ([14X78.2-1[114X) must
  require that the [22Xi[122X-th argument lies in the filter [3Xargs-filts[103X[22X[i][122X.[133X
  
  [33X[0;0YOne  can  install  methods for other arguments tuples via [2XInstallOtherMethod[102X
  ([14X78.2-2[114X),  this  way  it is also possible to install methods for a different
  number of arguments than the length of [3Xargs-filts[103X.[133X
  
  
  [1X79.6 [33X[0;0YCreating Constructors[133X[101X
  
  [1X79.6-1 NewConstructor[101X
  
  [33X[1;0Y[29X[2XNewConstructor[102X( [3Xname[103X, [3Xargs-filts[103X ) [32X function[133X
  
  [33X[0;0Y[2XNewConstructor[102X   returns  a  constructor  [3Xcons[103X  with  name  [3Xname[103X.  The  list
  [3Xargs-filts[103X  describes requirements about the arguments of [3Xconstr[103X, namely the
  number  of arguments must be equal to the length of [3Xargs-filts[103X, and the [22Xi[122X-th
  argument  must  lie  in the filter [3Xargs-filts[103X[22X[i][122X. Additionally a constructor
  expects  the  first  argument  to  be  a  filter  to then select a method to
  construct an object.[133X
  
  [33X[0;0YEach  method  that  is  installed  for  [3Xcons[103X via [2XInstallMethod[102X ([14X78.2-1[114X) must
  require that the [22Xi[122X-th argument lies in the filter [3Xargs-filts[103X[22X[i][122X.[133X
  
  [33X[0;0YOne  can  install  methods for other arguments tuples via [2XInstallOtherMethod[102X
  ([14X78.2-2[114X),  this  way  it is also possible to install methods for a different
  number of arguments than the length of [3Xargs-filts[103X.[133X
  
  
  [1X79.7 [33X[0;0YCreating Families[133X[101X
  
  [33X[0;0YFamilies are probably the least obvious part of the [5XGAP[105X type system, so some
  remarks  about  the  role of families are necessary. When one uses [5XGAP[105X as it
  is,  one  will (better: should) not meet families at all. The two situations
  where families come into play are the following.[133X
  
  [33X[0;0YFirst,  since  families  are used to describe relations between arguments of
  operations  in  the  method  selection  mechanism  (see Chapter [14X78[114X, and also
  Chapter [14X13[114X),   one   has  to  prescribe  such  a  relation  in  each  method
  installation (see [14X78.2[114X); usual relations are [2XReturnTrue[102X ([14X5.4-1[114X) (which means
  that  any  relation  of  the actual arguments is admissible), [2XIsIdenticalObj[102X
  ([14X12.5-1[114X)  (which  means  that  there  are two arguments that lie in the same
  family),  and  [10XIsCollsElms[110X  (which  means  that there are two arguments, the
  first  being  a  collection  of  elements that lie in the same family as the
  second argument).[133X
  
  [33X[0;0YSecond  –and  this is the more complicated situation– whenever one creates a
  new  kind of objects, one has to decide what its family shall be. If the new
  object  shall  be  equal  to  existing  objects,  for  example if it is just
  represented  in a different way, there is no choice: The new object must lie
  in  the same family as all objects that shall be equal to it. So only if the
  new  object is different (w.r.t. the equality [21X[10X=[110X[121X) from all other [5XGAP[105X objects,
  we are likely to create a new family for it. Note that enlarging an existing
  family  by  such new objects may be problematic because of implications that
  have been installed for all objects of the family in question. The choice of
  families  depends  on  the applications one has in mind. For example, if the
  new  objects  in  question  are not likely to be arguments of operations for
  which   family   relations  are  relevant  (for  example  binary  arithmetic
  operations), one could create one family for all such objects, and regard it
  as [21Xthe family of all those [5XGAP[105X objects that would in fact not need a family[121X.
  On the other extreme, if one wants to create domains of the new objects then
  one  has  to choose the family in such a way that all intended elements of a
  domain  do  in  fact  lie  in  the same family. (Remember that a domain is a
  collection,  see Chapter [14X12.4[114X, and that a collection consists of elements in
  the same family, see Chapter [14X30[114X and Section [14X13.1[114X.)[133X
  
  [33X[0;0YLet  us  look  at  an example. Suppose that no permutations are available in
  [5XGAP[105X,  and that we want to implement permutations. Clearly we want to support
  permutation  groups,  but it is not a priori clear how to distribute the new
  permutations  into  families.  We  can put all permutations into one family;
  this  is  how in fact permutations are implemented in [5XGAP[105X. But it would also
  be possible to put all permutations of a given degree into a family of their
  own;  this  would  for  example  mean  that  for each degree, there would be
  distinguished  trivial  permutations, and that the stabilizer of the point [10X5[110X
  in  the  symmetric group on the points [10X1[110X, [10X2[110X, [22X...[122X, [10X5[110X is not regarded as equal
  to  the  symmetric  group on [10X1[110X, [10X2[110X, [10X3[110X, [10X4[110X. Note that the latter approach would
  have  the  advantage  that  it  is  no problem to construct permutations and
  permutation  groups  acting  on  arbitrary  (finite)  sets,  for  example by
  constructing  first  the  symmetric group on the set and then generating any
  desired permutation group as a subgroup of this symmetric group.[133X
  
  [33X[0;0YSo  one  aspect  concerning  a  reasonable choice of families is to make the
  families large enough for being able to form interesting domains of elements
  in  the  family.  But on the other hand, it is useful to choose the families
  small  enough  for  admitting  meaningful  relations  between  objects.  For
  example,  the  elements  of  different  free  groups in [5XGAP[105X lie in different
  families;  the  multiplication  of free group elements is installed only for
  the  case that the two operands lie in the same family, with the effect that
  one  cannot  erroneously  form  the  product of elements from different free
  groups.  In  this  case,  families  appear  as  a  tool for providing useful
  restrictions.[133X
  
  [33X[0;0YAs  another  example,  note that an element and a collection containing this
  element  never  lie  in  the  same  family, by the general implementation of
  collections;  namely,  the  family of a collection of elements in the family
  [3XFam[103X  is the collections family of [3XFam[103X (see [2XCollectionsFamily[102X ([14X30.2-1[114X)). This
  means  that  for  a collection, we need not (because we cannot) decide about
  its family.[133X
  
  [33X[0;0YA  few  functions in [5XGAP[105X return families, see [2XCollectionsFamily[102X ([14X30.2-1[114X) and
  [2XElementsFamily[102X ([14X30.2-3[114X).[133X
  
  [1X79.7-1 NewFamily[101X
  
  [33X[1;0Y[29X[2XNewFamily[102X( [3Xname[103X[, [3Xreq[103X[, [3Ximp[103X[, [3Xfamfilter[103X]]] ) [32X function[133X
  
  [33X[0;0Y[2XNewFamily[102X  returns  a  new  family  [3Xfam[103X with name [3Xname[103X. The argument [3Xreq[103X, if
  present,  is a filter of which [3Xfam[103X shall be a subset. If one tries to create
  an  object  in  [3Xfam[103X that does not lie in the filter [3Xreq[103X, an error message is
  printed.  Also  the argument [3Ximp[103X, if present, is a filter of which [3Xfam[103X shall
  be  a  subset.  Any  object  that  is  created  in  the  family [3Xfam[103X will lie
  automatically in the filter [3Ximp[103X.[133X
  
  [33X[0;0YThe  filter  [3Xfamfilter[103X,  if given, specifies a filter that will hold for the
  family [3Xfam[103X (not for objects in [3Xfam[103X).[133X
  
  [33X[0;0YFamilies are always represented as component objects (see [14X79.10[114X). This means
  that components can be used to store and access useful information about the
  family.[133X
  
  
  [1X79.8 [33X[0;0YCreating Types[133X[101X
  
  [1X79.8-1 NewType[101X
  
  [33X[1;0Y[29X[2XNewType[102X( [3Xfamily[103X, [3Xfilter[103X[, [3Xdata[103X] ) [32X function[133X
  
  [33X[0;0Y[2XNewType[102X  returns  the type given by the family [3Xfamily[103X and the filter [3Xfilter[103X.
  The optional third argument [3Xdata[103X is any object that denotes defining data of
  the desired type.[133X
  
  [33X[0;0YFor  examples  where  [2XNewType[102X  is used, see [14X79.10[114X, [14X79.11[114X, and the example in
  Chapter [14X81[114X.[133X
  
  
  [1X79.9 [33X[0;0YCreating Objects[133X[101X
  
  [1X79.9-1 Objectify[101X
  
  [33X[1;0Y[29X[2XObjectify[102X( [3Xtype[103X, [3Xdata[103X ) [32X function[133X
  
  [33X[0;0YNew  objects  are created by [2XObjectify[102X. [3Xdata[103X is a list or a record, and [3Xtype[103X
  is the type that the desired object shall have. [2XObjectify[102X turns [3Xdata[103X into an
  object  with type [3Xtype[103X. That is, [3Xdata[103X is changed, and afterwards it will not
  be a list or a record unless [3Xtype[103X is of type list resp. record.[133X
  
  [33X[0;0YIf  [3Xdata[103X is a list then [2XObjectify[102X turns it into a positional object, if [3Xdata[103X
  is  a  record then [2XObjectify[102X turns it into a component object (for examples,
  see [14X79.10[114X and [14X79.11[114X).[133X
  
  [33X[0;0Y[2XObjectify[102X does also return the object that it made out of [3Xdata[103X.[133X
  
  [33X[0;0YFor  examples  where [2XObjectify[102X is used, see [14X79.10[114X, [14X79.11[114X, and the example in
  Chapter [14X81[114X.[133X
  
  [1X79.9-2 ObjectifyWithAttributes[101X
  
  [33X[1;0Y[29X[2XObjectifyWithAttributes[102X( [3Xobj[103X, [3Xtype[103X, [3Xattr1[103X, [3Xval1[103X, [3Xattr2[103X, [3Xval2[103X, [3X...[103X ) [32X function[133X
  
  [33X[0;0YAttribute  assignments will change the type of an object. If you create many
  objects, code of the form[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28Xo:=Objectify(type,rec());[128X[104X
    [4X[28XSetMyAttribute(o,value);[128X[104X
  [4X[32X[104X
  
  [33X[0;0Ywill  take a lot of time for type changes. You can avoid this by setting the
  attributes   immediately   while   the   object   is  created,  as  follows.
  [2XObjectifyWithAttributes[102X changes the type of object [3Xobj[103X to type [3Xtype[103X and sets
  attribute [3Xattr1[103X to [3Xval1[103X, sets attribute [3Xattr2[103X to [3Xval2[103X and so forth.[133X
  
  [33X[0;0YIf  the  filter list of [3Xtype[103X includes that these attributes are set (and the
  properties  also  include values of the properties) and if no special setter
  methods  are  installed for any of the involved attributes then they are set
  simultaneously without type changes. This can produce a substantial speedup.[133X
  
  [33X[0;0YIf  the  conditions  of  the  last  sentence  are not fulfilled, an ordinary
  [2XObjectify[102X  ([14X79.9-1[114X)  with  subsequent  setter  calls  for  the attributes is
  performed instead.[133X
  
  
  [1X79.10 [33X[0;0YComponent Objects[133X[101X
  
  [33X[0;0YA  [13Xcomponent  object[113X is an object in the representation [10XIsComponentObjectRep[110X
  or  a  subrepresentation of it. Such an object [3Xcobj[103X is built from subobjects
  that can be accessed via [10X[3Xcobj[103X[10X!.[3Xname[103X[10X[110X, similar to components of a record. Also
  analogously  to  records,  values  can be assigned to components of [3Xcobj[103X via
  [10X[3Xcobj[103X[10X!.[3Xname[103X[10X:=  [3Xval[103X[10X[110X. For the creation of component objects, see [14X79.9[114X. One must
  be  [13Xvery  careful[113X  when  using  the  [10X!.[110X  operator, in order to interpret the
  component  in the right way, and even more careful when using the assignment
  to  components  using  [10X!.[110X,  in  order to keep the information stored in [3Xcobj[103X
  consistent.[133X
  
  [33X[0;0YFirst  of  all,  in  the access or assignment to a component as shown above,
  [3Xname[103X  must be among the admissible component names for the representation of
  [3Xcobj[103X,  see [14X79.2[114X.  Second, preferably only few low level functions should use
  [10X!.[110X, whereas this operator should not occur in [21Xuser interactions[121X.[133X
  
  [33X[0;0YNote  that even if [3Xcobj[103X claims that it is immutable, i.e., if [3Xcobj[103X is not in
  the category [2XIsMutable[102X ([14X12.6-2[114X), access and assignment via [10X!.[110X and [10X!.:=[110X work.
  This  is  necessary  for being able to store newly discovered information in
  immutable objects.[133X
  
  [33X[0;0YThe following example shows the implementation of an iterator (see [14X30.8[114X) for
  the  domain of integers, which is represented as component object. See [14X79.11[114X
  for  an  implementation  using  positional  objects.  (In  practice, such an
  iterator   can  be  implemented  more  elegantly  using  [2XIteratorByFunctions[102X
  ([14X30.8-8[114X), see [14X79.14[114X.)[133X
  
  [33X[0;0YThe used succession of integers is [22X0, 1, -1, 2, -2, 3, -3, ...[122X, that is, [22Xa_n
  = n/2[122X if [22Xn[122X is even, and [22Xa_n = (1-n)/2[122X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XIsIntegersIteratorCompRep := NewRepresentation( "IsIntegersIteratorRep",[128X[104X
    [4X[28X    IsComponentObjectRep, [ "counter" ] );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above  command  creates  a  new  representation  (see [2XNewRepresentation[102X
  ([14X79.2-1[114X))    [10XIsIntegersIteratorCompRep[110X,    as    a    subrepresentation   of
  [10XIsComponentObjectRep[110X, and with one admissible component [10Xcounter[110X. So no other
  components than [10Xcounter[110X will be needed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XInstallMethod( Iterator,[128X[104X
    [4X[28X    "method for `Integers'",[128X[104X
    [4X[28X    [ IsIntegers ],[128X[104X
    [4X[28X    function( Integers )[128X[104X
    [4X[28X    return Objectify( NewType( IteratorsFamily,[128X[104X
    [4X[28X                                   IsIterator[128X[104X
    [4X[28X                               and IsIntegersIteratorCompRep ),[128X[104X
    [4X[28X                      rec( counter := 0 ) );[128X[104X
    [4X[28X    end );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAfter  the  above  method  installation,  one  can already ask for [10XIterator(
  Integers  )[110X.  Note  that  exactly the domain of integers is described by the
  filter [2XIsIntegers[102X ([14X14.1-2[114X).[133X
  
  [33X[0;0YBy  the  call  to  [2XNewType[102X  ([14X79.8-1[114X), the returned object lies in the family
  containing  all iterators, which is [10XIteratorsFamily[110X, it lies in the category
  [2XIsIterator[102X  ([14X30.8-3[114X)  and  in  the representation [10XIsIntegersIteratorCompRep[110X;
  furthermore, it has the component [10Xcounter[110X with value [10X0[110X.[133X
  
  [33X[0;0YWhat  is  missing now are methods for the two basic operations of iterators,
  namely  [2XIsDoneIterator[102X  ([14X30.8-4[114X)  and [2XNextIterator[102X ([14X30.8-5[114X). The former must
  always  return  [9Xfalse[109X,  since there are infinitely many integers. The latter
  must  return  the  next integer in the iteration, and update the information
  stored  in  the  iterator,  that  is,  increase  the  value of the component
  [10Xcounter[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XInstallMethod( IsDoneIterator,[128X[104X
    [4X[28X    "method for iterator of `Integers'",[128X[104X
    [4X[28X    [ IsIterator and IsIntegersIteratorCompRep ],[128X[104X
    [4X[28X    ReturnFalse );[128X[104X
    [4X[28X[128X[104X
    [4X[28XInstallMethod( NextIterator,[128X[104X
    [4X[28X    "method for iterator of `Integers'",[128X[104X
    [4X[28X    [ IsIntegersIteratorCompRep ],[128X[104X
    [4X[28X    function( iter )[128X[104X
    [4X[28X    iter!.counter:= iter!.counter + 1;[128X[104X
    [4X[28X    if iter!.counter mod 2 = 0 then[128X[104X
    [4X[28X      return iter!.counter / 2;[128X[104X
    [4X[28X    else[128X[104X
    [4X[28X      return ( 1 - iter!.counter ) / 2;[128X[104X
    [4X[28X    fi;[128X[104X
    [4X[28X    end );[128X[104X
  [4X[32X[104X
  
  [1X79.10-1 NamesOfComponents[101X
  
  [33X[1;0Y[29X[2XNamesOfComponents[102X( [3Xcomobj[103X ) [32X function[133X
  
  [33X[0;0YFor  a component object [3Xcomobj[103X, [2XNamesOfComponents[102X returns a list of strings,
  which are the names of components currently bound in [3Xcomobj[103X.[133X
  
  [33X[0;0YFor  a  record  [3Xcomobj[103X,  [2XNamesOfComponents[102X  returns  the  result of [2XRecNames[102X
  ([14X29.1-2[114X).[133X
  
  
  [1X79.11 [33X[0;0YPositional Objects[133X[101X
  
  [33X[0;0YA [13Xpositional object[113X is an object in the representation [10XIsPositionalObjectRep[110X
  or  a  subrepresentation of it. Such an object [3Xpobj[103X is built from subobjects
  that  can  be  accessed via [10X[3Xpobj[103X[10X![[3Xpos[103X[10X][110X, similar to positions in a list. Also
  analogously  to  lists,  values  can  be  assigned  to positions of [3Xpobj[103X via
  [10X[3Xpobj[103X[10X![[3Xpos[103X[10X]:= [3Xval[103X[10X[110X. For the creation of positional objects, see [14X79.9[114X.[133X
  
  [33X[0;0YOne  must be [13Xvery careful[113X when using the [10X![][110X operator, in order to interpret
  the  position  in  the  right  way,  and  even  more  careful when using the
  assignment  to  positions using [10X![][110X, in order to keep the information stored
  in [3Xpobj[103X consistent.[133X
  
  [33X[0;0YFirst  of all, in the access or assignment to a position as shown above, [3Xpos[103X
  must  be  among  the  admissible  positions  for the representation of [3Xpobj[103X,
  see [14X79.2[114X.  Second,  preferably  only few low level functions should use [10X![][110X,
  whereas this operator should not occur in [21Xuser interactions[121X.[133X
  
  [33X[0;0YNote  that even if [3Xpobj[103X claims that it is immutable, i.e., if [3Xpobj[103X is not in
  the category [2XIsMutable[102X ([14X12.6-2[114X), access and assignment via [10X![][110X work. This is
  necessary  for being able to store newly discovered information in immutable
  objects.[133X
  
  [33X[0;0YThe following example shows the implementation of an iterator (see [14X30.8[114X) for
  the domain of integers, which is represented as positional object. See [14X79.10[114X
  for an implementation using component objects, and more details.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XIsIntegersIteratorPosRep := NewRepresentation( "IsIntegersIteratorRep",[128X[104X
    [4X[28X    IsPositionalObjectRep, [ 1 ] );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above  command  creates  a  new  representation  (see [2XNewRepresentation[102X
  ([14X79.2-1[114X))    [10XIsIntegersIteratorPosRep[110X,    as    a    subrepresentation    of
  [10XIsComponentObjectRep[110X,  and with only the first position being admissible for
  storing data.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XInstallMethod( Iterator,[128X[104X
    [4X[28X    "method for `Integers'",[128X[104X
    [4X[28X    [ IsIntegers ],[128X[104X
    [4X[28X    function( Integers )[128X[104X
    [4X[28X    return Objectify( NewType( IteratorsFamily,[128X[104X
    [4X[28X                                   IsIterator[128X[104X
    [4X[28X                               and IsIntegersIteratorRep ),[128X[104X
    [4X[28X                      [ 0 ] );[128X[104X
    [4X[28X    end );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAfter  the  above  method  installation,  one  can already ask for [10XIterator(
  Integers  )[110X.  Note  that  exactly the domain of integers is described by the
  filter [2XIsIntegers[102X ([14X14.1-2[114X).[133X
  
  [33X[0;0YBy  the  call  to  [2XNewType[102X  ([14X79.8-1[114X), the returned object lies in the family
  containing  all iterators, which is [10XIteratorsFamily[110X, it lies in the category
  [2XIsIterator[102X  ([14X30.8-3[114X)  and  in  the  representation [10XIsIntegersIteratorPosRep[110X;
  furthermore, the first position has value [10X0[110X.[133X
  
  [33X[0;0YWhat  is  missing now are methods for the two basic operations of iterators,
  namely  [2XIsDoneIterator[102X  ([14X30.8-4[114X)  and [2XNextIterator[102X ([14X30.8-5[114X). The former must
  always  return  [9Xfalse[109X,  since there are infinitely many integers. The latter
  must  return  the  next integer in the iteration, and update the information
  stored  in  the  iterator,  that  is, increase the value stored in the first
  position.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XInstallMethod( IsDoneIterator,[128X[104X
    [4X[28X    "method for iterator of `Integers'",[128X[104X
    [4X[28X    [ IsIterator and IsIntegersIteratorPosRep ],[128X[104X
    [4X[28X    ReturnFalse );[128X[104X
    [4X[28X[128X[104X
    [4X[28XInstallMethod( NextIterator,[128X[104X
    [4X[28X    "method for iterator of `Integers'",[128X[104X
    [4X[28X    [ IsIntegersIteratorPosRep ],[128X[104X
    [4X[28X    function( iter )[128X[104X
    [4X[28X    iter![1]:= iter![1] + 1;[128X[104X
    [4X[28X    if iter![1] mod 2 = 0 then[128X[104X
    [4X[28X      return iter![1] / 2;[128X[104X
    [4X[28X    else[128X[104X
    [4X[28X      return ( 1 - iter![1] ) / 2;[128X[104X
    [4X[28X    fi;[128X[104X
    [4X[28X    end );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  should be noted that one can of course install both the methods shown in
  Section [14X79.10[114X and [14X79.11[114X. The call [10XIterator( Integers )[110X will cause one of the
  methods  to  be selected, and for the returned iterator, which will have one
  of  the  representations  we  constructed,  the  right [2XNextIterator[102X ([14X30.8-5[114X)
  method will be chosen.[133X
  
  
  [1X79.12 [33X[0;0YImplementing New List Objects[133X[101X
  
  [33X[0;0YThis  section  gives some hints for the quite usual situation that one wants
  to implement new objects that are lists. More precisely, one either wants to
  deal  with  lists  that  have  additional  features,  or one wants that some
  objects also behave as lists. An example can be found in [14X79.13[114X.[133X
  
  [33X[0;0YA [13Xlist[113X in [5XGAP[105X is an object in the category [2XIsList[102X ([14X21.1-1[114X). Basic operations
  for  lists  are  [2XLength[102X  ([14X21.17-5[114X),  [2X\[\][102X ([14X21.2-1[114X), and [2XIsBound\[\][102X ([14X21.2-1[114X)
  (see [14X21.2[114X).[133X
  
  [33X[0;0YNote  that  the access to the position [3Xpos[103X in the list [3Xlist[103X via [10X[3Xlist[103X[10X[[3Xpos[103X[10X][110X is
  handled  by  the  call  [10X\[\]( [3Xlist[103X[10X, [3Xpos[103X[10X )[110X to the operation [2X\[\][102X ([14X21.2-1[114X). To
  explain  the  somewhat  strange  name  [10X\[\][110X  of  this  operation,  note that
  non-alphanumeric  characters  like  [10X[[110X  and [10X][110X may occur in [5XGAP[105X variable names
  only if they are escaped by a [10X\[110X character.[133X
  
  [33X[0;0YAnalogously,  the check [10XIsBound( [3Xlist[103X[10X[[3Xpos[103X[10X] )[110X whether the position [3Xpos[103X of the
  list  [3Xlist[103X  is  bound is handled by the call [10XIsBound\[\]( [3Xlist[103X[10X, [3Xpos[103X[10X )[110X to the
  operation [2XIsBound\[\][102X ([14X21.2-1[114X).[133X
  
  [33X[0;0YFor  mutable  lists, also assignment to positions and unbinding of positions
  via  the  operations  [2X\[\]\:\=[102X  ([14X21.2-1[114X)  and  [2XUnbind\[\][102X ([14X21.2-1[114X) are basic
  operations.  The assignment [10X[3Xlist[103X[10X[[3Xpos[103X[10X]:= [3Xval[103X[10X[110X is handled by the call [10X\[\]\:\=(
  [3Xlist[103X[10X, [3Xpos[103X[10X, [3Xval[103X[10X )[110X, and [10XUnbind( [3Xlist[103X[10X[[3Xpos[103X[10X] )[110X is handled by the call [10XUnbind\[\](
  [3Xlist[103X[10X, [3Xpos[103X[10X )[110X.[133X
  
  [33X[0;0YAll  other  operations  for  lists, e.g., [2XAdd[102X ([14X21.4-2[114X), [2XAppend[102X ([14X21.4-5[114X), [2XSum[102X
  ([14X21.20-26[114X),  are based on these operations. This means that it is sufficient
  to install methods for the new list objects only for the basic operations.[133X
  
  [33X[0;0YSo  if  one  wants  to  implement  new list objects then one creates them as
  objects  in  the  category  [2XIsList[102X ([14X21.1-1[114X), and installs methods for [2XLength[102X
  ([14X21.17-5[114X),  [2X\[\][102X  ([14X21.2-1[114X), and [2XIsBound\[\][102X ([14X21.2-1[114X). If the new lists shall
  be  mutable,  one  needs  to  install also methods for [2X\[\]\:\=[102X ([14X21.2-1[114X) and
  [2XUnbind\[\][102X ([14X21.2-1[114X).[133X
  
  [33X[0;0YOne  application  for this is the implementation of [13Xenumerators[113X for domains.
  An  enumerator  for  the  domain  [22XD[122X  is  a  dense  list whose entries are in
  bijection  with  the  elements  of [22XD[122X. If [22XD[122X is large then it is not useful to
  write  down  all  elements.  Instead  one  can  implement  such  a bijection
  implicitly. This works also for infinite domains.[133X
  
  [33X[0;0YIn this situation, one implements a new representation of the lists that are
  already  available  in  [5XGAP[105X,  in particular the family of such a list is the
  same as the family of the domain [22XD[122X.[133X
  
  [33X[0;0YBut  it  is  also  possible  to implement new kinds of lists that lie in new
  families, and thus are not equal to lists that were available in [5XGAP[105X before.
  An  example  for this is the implementation of matrices whose multiplication
  via [21X[10X*[110X[121X is the Lie product of matrices.[133X
  
  [33X[0;0YIn  this  situation, it makes no sense to put the new matrices into the same
  family  as  the original matrices. Note that the product of two Lie matrices
  shall be defined but not the product of an ordinary matrix and a Lie matrix.
  So  it is possible to have two lists that have the same entries but that are
  not equal w.r.t. [21X[10X=[110X[121X because they lie in different families.[133X
  
  
  [1X79.13 [33X[0;0YExample – Constructing Enumerators[133X[101X
  
  [33X[0;0YWhen  dealing  with  countable sets, a usual task is to define enumerations,
  i.e.,  bijections  to the positive integers. In [5XGAP[105X, this can be implemented
  via  [13Xenumerators[113X  (see [14X21.23[114X).  These are lists containing the elements in a
  specified  ordering,  and  the operations [2XPosition[102X ([14X21.16-1[114X) and list access
  via  [2X\[\][102X  ([14X21.2-1[114X)  define  the desired bijection. For implementing such an
  enumerator,  one mainly needs to install the appropriate functions for these
  operations.[133X
  
  [33X[0;0YA  general  setup  for creating such lists is given by [2XEnumeratorByFunctions[102X
  ([14X30.3-4[114X).[133X
  
  [33X[0;0YIf  the  set  in  question is a domain [3XD[103X for which a [2XSize[102X ([14X30.4-6[114X) method is
  available  then  all  one  has  to  do  is  to  write down the functions for
  computing  the  [22Xn[122X-th element of the list and for computing the position of a
  given  [5XGAP[105X object in the list, to put them into the components [10XElementNumber[110X
  and  [10XNumberElement[110X  of  a record, and to call [2XEnumeratorByFunctions[102X ([14X30.3-4[114X)
  with  the  domain [3XD[103X and this record as arguments. For example, the following
  lines  of  code install an [2XEnumerator[102X ([14X30.3-2[114X) method for the case that [3XD[103X is
  the  domain of rational integers. (Note that [2XIsIntegers[102X ([14X14.1-2[114X) is a filter
  that describes exactly the domain of rational integers.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XInstallMethod( Enumerator,[128X[104X
    [4X[28X    "for integers",[128X[104X
    [4X[28X    [ IsIntegers ],[128X[104X
    [4X[28X    Integers -> EnumeratorByFunctions( Integers, rec([128X[104X
    [4X[28X                    ElementNumber := function( e, n ) ... end,[128X[104X
    [4X[28X                    NumberElement := function( e, x ) ... end ) ) );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  bodies  of the functions have been omitted above; here is the code that
  is actually used in [5XGAP[105X. (The ordering coincides with that for the iterators
  for  the  domain  of  rational  integers  that  have been discussed in [14X79.10[114X
  and [14X79.11[114X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xenum:= Enumerator( Integers );[127X[104X
    [4X[28X<enumerator of Integers>[128X[104X
    [4X[25Xgap>[125X [27XPrint( enum!.NumberElement, "\n" );[127X[104X
    [4X[28Xfunction ( e, x )[128X[104X
    [4X[28X    local  pos;[128X[104X
    [4X[28X    if not IsInt( x )  then[128X[104X
    [4X[28X        return fail;[128X[104X
    [4X[28X    elif 0 < x  then[128X[104X
    [4X[28X        pos := 2 * x;[128X[104X
    [4X[28X    else[128X[104X
    [4X[28X        pos := -2 * x + 1;[128X[104X
    [4X[28X    fi;[128X[104X
    [4X[28X    return pos;[128X[104X
    [4X[28Xend[128X[104X
    [4X[25Xgap>[125X [27XPrint( enum!.ElementNumber, "\n" );[127X[104X
    [4X[28Xfunction ( e, n )[128X[104X
    [4X[28X    if n mod 2 = 0  then[128X[104X
    [4X[28X        return n / 2;[128X[104X
    [4X[28X    else[128X[104X
    [4X[28X        return (1 - n) / 2;[128X[104X
    [4X[28X    fi;[128X[104X
    [4X[28X    return;[128X[104X
    [4X[28Xend[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  situation becomes slightly more complicated if the set [22XS[122X in question is
  not  a  domain.  This is because one must provide also at least a method for
  computing  the  length  of  the  list,  and because one has to determine the
  family  in  which  it  lies  (see [14X79.9[114X).  The latter should usually not be a
  problem  since  either  [22XS[122X  is  nonempty and all its elements lie in the same
  family  –in  this case one takes the collections family of any element in [22XS[122X–
  or the family of the enumerator must be [10XListsFamily[110X.[133X
  
  [33X[0;0YAn  example in the [5XGAP[105X library is an enumerator for the set of [22Xk[122X-tuples over
  a finite set; the function is called [2XEnumeratorOfTuples[102X ([14X16.2-9[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrint( EnumeratorOfTuples, "\n" );[127X[104X
    [4X[28Xfunction ( set, k )[128X[104X
    [4X[28X    local  enum;[128X[104X
    [4X[28X    if k = 0  then[128X[104X
    [4X[28X        return Immutable( [ [  ] ] );[128X[104X
    [4X[28X    elif IsEmpty( set )  then[128X[104X
    [4X[28X        return Immutable( [  ] );[128X[104X
    [4X[28X    fi;[128X[104X
    [4X[28X    enum [128X[104X
    [4X[28X     := EnumeratorByFunctions( CollectionsFamily( FamilyObj( set ) ), [128X[104X
    [4X[28X       rec([128X[104X
    [4X[28X          ElementNumber := function ( enum, n )[128X[104X
    [4X[28X                local  nn, t, i;[128X[104X
    [4X[28X                nn := n - 1;[128X[104X
    [4X[28X                t := [  ];[128X[104X
    [4X[28X                for i  in [ 1 .. enum!.k ]  do[128X[104X
    [4X[28X                    t[i] := RemInt( nn, Length( enum!.set ) ) + 1;[128X[104X
    [4X[28X                    nn := QuoInt( nn, Length( enum!.set ) );[128X[104X
    [4X[28X                od;[128X[104X
    [4X[28X                if nn <> 0  then[128X[104X
    [4X[28X                    Error( "<enum>[", n, [128X[104X
    [4X[28X                     "] must have an assigned value" );[128X[104X
    [4X[28X                fi;[128X[104X
    [4X[28X                nn := enum!.set{Reversed( t )};[128X[104X
    [4X[28X                MakeImmutable( nn );[128X[104X
    [4X[28X                return nn;[128X[104X
    [4X[28X            end,[128X[104X
    [4X[28X          NumberElement := function ( enum, elm )[128X[104X
    [4X[28X                local  n, i;[128X[104X
    [4X[28X                if not IsList( elm )  then[128X[104X
    [4X[28X                    return fail;[128X[104X
    [4X[28X                fi;[128X[104X
    [4X[28X                elm := List( elm, function ( x )[128X[104X
    [4X[28X                        return Position( enum!.set, x );[128X[104X
    [4X[28X                    end );[128X[104X
    [4X[28X                if fail in elm or Length( elm ) <> enum!.k  then[128X[104X
    [4X[28X                    return fail;[128X[104X
    [4X[28X                fi;[128X[104X
    [4X[28X                n := 0;[128X[104X
    [4X[28X                for i  in [ 1 .. enum!.k ]  do[128X[104X
    [4X[28X                    n := Length( enum!.set ) * n + elm[i] - 1;[128X[104X
    [4X[28X                od;[128X[104X
    [4X[28X                return n + 1;[128X[104X
    [4X[28X            end,[128X[104X
    [4X[28X          Length := function ( enum )[128X[104X
    [4X[28X                return Length( enum!.set ) ^ enum!.k;[128X[104X
    [4X[28X            end,[128X[104X
    [4X[28X          PrintObj := function ( enum )[128X[104X
    [4X[28X                Print( "EnumeratorOfTuples( ", enum!.set, ", ", [128X[104X
    [4X[28X                 enum!.k, " )" );[128X[104X
    [4X[28X                return;[128X[104X
    [4X[28X            end,[128X[104X
    [4X[28X          set := Set( set ),[128X[104X
    [4X[28X          k := k ) );[128X[104X
    [4X[28X    SetIsSSortedList( enum, true );[128X[104X
    [4X[28X    return enum;[128X[104X
    [4X[28Xend[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that  the  enumerator  is a homogeneous list that stores individual
  functions  [10XElementNumber[110X, [10XNumberElement[110X, [10XLength[110X, and [10XPrintObj[110X; besides that,
  the data components [22XS[122X and [22Xk[122X are contained.[133X
  
  
  [1X79.14 [33X[0;0YExample – Constructing Iterators[133X[101X
  
  [33X[0;0YIterators  are a kind of objects that is implemented for several collections
  in  the  [5XGAP[105X  library  and  which  might be interesting also in other cases,
  see [14X30.8[114X.  A  general  setup  for  implementing new iterators is provided by
  [2XIteratorByFunctions[102X ([14X30.8-8[114X).[133X
  
  [33X[0;0YAll  one has to do is to write down the functions for [2XNextIterator[102X ([14X30.8-5[114X),
  [2XIsDoneIterator[102X   ([14X30.8-4[114X),   and   [2XShallowCopy[102X   ([14X12.7-1[114X),   and   to   call
  [2XIteratorByFunctions[102X  ([14X30.8-8[114X) with this record as argument. For example, the
  following  lines  of  code  install an [2XIterator[102X ([14X30.8-1[114X) method for the case
  that the argument is the domain of rational integers.[133X
  
  [33X[0;0Y(Note that [2XIsIntegers[102X ([14X14.1-2[114X) is a filter that describes exactly the domain
  of rational integers.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XInstallMethod( Iterator,[128X[104X
    [4X[28X    "for integers",[128X[104X
    [4X[28X    [ IsIntegers ],[128X[104X
    [4X[28X    Integers -> IteratorByFunctions( rec([128X[104X
    [4X[28X                    NextIterator:= function( iter ) ... end,[128X[104X
    [4X[28X                    IsDoneIterator := ReturnFalse,[128X[104X
    [4X[28X                    ShallowCopy := function( iter ) ... end ) ) );[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe bodies of two of the functions have been omitted above; here is the code
  that  is  actually  used  in  [5XGAP[105X. (The ordering coincides with that for the
  iterators  for  the  domain  of  rational  integers that have been discussed
  in [14X79.10[114X and [14X79.11[114X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xiter:= Iterator( Integers );[127X[104X
    [4X[28X<iterator of Integers at 0>[128X[104X
    [4X[25Xgap>[125X [27XPrint( iter!.NextIterator, "\n" );[127X[104X
    [4X[28Xfunction ( iter )[128X[104X
    [4X[28X    iter!.counter := iter!.counter + 1;[128X[104X
    [4X[28X    if iter!.counter mod 2 = 0  then[128X[104X
    [4X[28X        return iter!.counter / 2;[128X[104X
    [4X[28X    else[128X[104X
    [4X[28X        return (1 - iter!.counter) / 2;[128X[104X
    [4X[28X    fi;[128X[104X
    [4X[28X    return;[128X[104X
    [4X[28Xend[128X[104X
    [4X[25Xgap>[125X [27XPrint( iter!.ShallowCopy, "\n" );   [127X[104X
    [4X[28Xfunction ( iter )[128X[104X
    [4X[28X    return rec([128X[104X
    [4X[28X        counter := iter!.counter );[128X[104X
    [4X[28Xend[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  the  [10XShallowCopy[110X component of the record must be a function that
  does not return an iterator but a record that can be used as the argument of
  [2XIteratorByFunctions[102X ([14X30.8-8[114X) in order to create the desired shallow copy.[133X
  
  
  [1X79.15 [33X[0;0YArithmetic Issues in the Implementation of New Kinds of Lists[133X[101X
  
  [33X[0;0YWhen  designing  a  new kind of list objects in [5XGAP[105X, defining the arithmetic
  behaviour of these objects is an issue.[133X
  
  [33X[0;0YThere  are  situations  where  arithmetic  operations  of  list  objects are
  unimportant  in the sense that adding two such lists need not be represented
  in  a  special  way.  In  such cases it might be useful either to support no
  arithmetics  at  all  for the new lists, or to enable the default arithmetic
  methods.   The   former   can   be  achieved  by  not  setting  the  filters
  [2XIsGeneralizedRowVector[102X  ([14X21.12-1[114X)  and  [2XIsMultiplicativeGeneralizedRowVector[102X
  ([14X21.12-2[114X)  in  the types of the lists, the latter can be achieved by setting
  the filter [2XIsListDefault[102X ([14X21.12-3[114X). (for details, see [14X21.12[114X). An example for
  [21Xwrapped  lists[121X with default behaviour are vector space bases; they are lists
  with  additional  properties concerning the computation of coefficients, but
  arithmetic  properties  are  not  important.  So it is no loss to enable the
  default methods for these lists.[133X
  
  [33X[0;0YHowever,  often  the  arithmetic behaviour of new list objects is important,
  and  one  wants  to keep these lists away from default methods for addition,
  multiplication  etc.  For  example,  the sum and the product of (compatible)
  block  matrices  shall  be  represented  as  a  block matrix, so the default
  methods  for  sum  and product of matrices shall not be applicable, although
  the  results will be equal to those of the default methods in the sense that
  their entries at corresponding positions are equal.[133X
  
  [33X[0;0YSo  one  does  not set the filter [2XIsListDefault[102X ([14X21.12-3[114X) in such cases, and
  thus  one  can  implement  one's  own methods for arithmetic operations. (Of
  course  [21Xcan[121X  means on the other hand that one [13Xmust[113X implement such methods if
  one is interested in arithmetics of the new lists.)[133X
  
  [33X[0;0YThe  specific binary arithmetic methods for the new lists will usually cover
  the  case  that  both  arguments  are  of the new kind, and perhaps also the
  interaction  between a list of the new kind and certain other kinds of lists
  may be handled if this appears to be useful.[133X
  
  [33X[0;0YFor  the  last  situation, interaction between a new kind of lists and other
  kinds  of  lists,  [5XGAP[105X  provides  already  a  setup.  Namely,  there are the
  categories           [2XIsGeneralizedRowVector[102X           ([14X21.12-1[114X)          and
  [2XIsMultiplicativeGeneralizedRowVector[102X ([14X21.12-2[114X), which are concerned with the
  additive and the multiplicative behaviour, respectively, of lists. For lists
  in  these  filters, the structure of the results of arithmetic operations is
  prescribed (see [14X21.13[114X and [14X21.14[114X).[133X
  
  [33X[0;0YFor     example,     if     one     implements     block     matrices     in
  [2XIsMultiplicativeGeneralizedRowVector[102X   ([14X21.12-2[114X)   then   automatically  the
  product  of  such  a  block matrix and a (plain) list of such block matrices
  will be defined as the obvious list of matrix products, and a default method
  for plain lists will handle this multiplication. (Note that this method will
  rely  on  a  method  for computing the product of the block matrices, and of
  course  no  default  method is available for that.) Conversely, if the block
  matrices  are not in [2XIsMultiplicativeGeneralizedRowVector[102X ([14X21.12-2[114X) then the
  product  of  a  block  matrix  and  a  (plain) list of block matrices is not
  defined.  (There  is no default method for it, and one can define the result
  and provide a method for computing it.)[133X
  
  [33X[0;0YThus  if one decides to set the filters [2XIsGeneralizedRowVector[102X ([14X21.12-1[114X) and
  [2XIsMultiplicativeGeneralizedRowVector[102X ([14X21.12-2[114X) for the new lists, on the one
  hand  one  loses  freedom in defining arithmetic behaviour, but on the other
  hand one gains several default methods for a more or less natural behaviour.[133X
  
  [33X[0;0YIf    a    list    in    the    filter    [2XIsGeneralizedRowVector[102X   ([14X21.12-1[114X)
  ([2XIsMultiplicativeGeneralizedRowVector[102X        ([14X21.12-2[114X))        lies       in
  [10XIsAttributeStoringRep[110X, the values of additive (multiplicative) nesting depth
  is  stored  in  the  list  and  need  not  be calculated for each arithmetic
  operation.  One  can  then  store  the value(s) already upon creation of the
  lists,  with  the  effect that the default arithmetic operations will access
  elements of these lists only if this is unavoidable. For example, the sum of
  two  plain lists of [21Xwrapped matrices[121X with stored nesting depths are computed
  via  the method for adding two such wrapped lists, and without accessing any
  of  their  rows (which might be expensive). In this sense, the wrapped lists
  are treated as black boxes.[133X
  
  
  [1X79.16 [33X[0;0YExternal Representation[133X[101X
  
  [33X[0;0YAn  operation  is  defined for elements rather than for objects in the sense
  that  if  the  arguments  are  replaced by objects that are equal to the old
  arguments w.r.t. the equivalence relation [21X[10X=[110X[121X then the result must be equal to
  the old result w.r.t. [21X[10X=[110X[121X.[133X
  
  [33X[0;0YBut  the  implementation  of many methods is representation dependent in the
  sense that certain representation dependent subobjects are accessed.[133X
  
  [33X[0;0YFor example, a method that implements the addition of univariate polynomials
  may  access  coefficients  lists  of  its  arguments only if they are really
  stored,  while  in  the case of sparsely represented polynomials a different
  approach is needed.[133X
  
  [33X[0;0YIn  spite  of  this,  for  many operations one does not want to write an own
  method  for  each  possible  representations  of  each argument, for example
  because  none  of  the  methods could in fact take advantage of the actually
  given representations of the objects. Another reason could be that one wants
  to  install first a representation independent method, and then add specific
  methods as they are needed to gain more efficiency, by really exploiting the
  fact that the arguments have certain representations.[133X
  
  [33X[0;0YFor the purpose of admitting representation independent code, one can define
  an  [13Xexternal representation[113X of objects in a given family, install methods to
  compute this external representation for each representation of the objects,
  and  then  use  this  external  representation  of the objects whenever they
  occur.[133X
  
  [33X[0;0YWe  cannot  provide  conversion functions that allow us to first convert any
  object  in  question  to  one  particular  [21Xstandard representation[121X, and then
  access  the  data in the way defined for this representation, simply because
  it  may be impossible to choose such a [21Xstandard representation[121X uniformly for
  all objects in the given family.[133X
  
  [33X[0;0YSo  the  aim  of  an external representation of an object [3Xobj[103X is a different
  one,  namely to describe the data from which [3Xobj[103X is composed. In particular,
  the   external   representation  of  [3Xobj[103X  is  [13Xnot[113X  one  possible  ([21Xstandard[121X)
  representation  of  [3Xobj[103X,  in  fact  the external representation of [3Xobj[103X is in
  general  different  from  [3Xobj[103X  w.r.t. [21X[10X=[110X[121X,  first  of all because the external
  representation of [3Xobj[103X does in general not lie in the same family as [3Xobj[103X.[133X
  
  [33X[0;0YFor  example the external representation of a rational function is a list of
  length  two or three, the first entry being the zero coefficient, the second
  being a list describing the coefficients and monomials of the numerator, and
  the  third, if bound, being a list describing the coefficients and monomials
  of  the  denominator.  In  particular,  the  external  representation  of  a
  polynomial is a list and not a polynomial.[133X
  
  [33X[0;0YThe  other way round, the external representation of [3Xobj[103X encodes [3Xobj[103X in such
  a  way  that  from this data and the family of [3Xobj[103X, one can create an object
  that  is equal to [3Xobj[103X. Usually the external representation of an object is a
  list or a record.[133X
  
  [33X[0;0YAlthough  the external representation of [3Xobj[103X is by definition independent of
  the  actually  available  representations  for  [3Xobj[103X,  it  is  usual  that  a
  representation  of  [3Xobj[103X  exists  for  which  the computation of the external
  representation  is  obtained  by  just  [21Xunpacking[121X [3Xobj[103X, in the sense that the
  desired  data  is  stored  in  a component or a position of [3Xobj[103X, if [3Xobj[103X is a
  component object (see [14X79.10[114X) or a positional object (see [14X79.11[114X).[133X
  
  [33X[0;0YTo  implement  an  external  representation means to install methods for the
  following two operations.[133X
  
  [1X79.16-1 ExtRepOfObj[101X
  
  [33X[1;0Y[29X[2XExtRepOfObj[102X( [3Xobj[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XObjByExtRep[102X( [3Xfam[103X, [3Xdata[103X ) [32X operation[133X
  
  [33X[0;0Y[2XExtRepOfObj[102X  returns  the  external  representation  of  its  argument,  and
  [2XObjByExtRep[102X   returns  an  object  in  the  family  [3Xfam[103X  that  has  external
  representation [3Xdata[103X.[133X
  
  [33X[0;0YOf course, [10XObjByExtRep( FamilyObj( [3Xobj[103X[10X ), ExtRepOfObj( [3Xobj[103X[10X ) )[110X must be equal
  to  [3Xobj[103X w.r.t. the operation [2X\=[102X ([14X31.11-1[114X). But it is [13Xnot[113X required that equal
  objects have equal external representations.[133X
  
  [33X[0;0YNote  that  if  one  defines  a  new  representation of objects for which an
  external representation does already exist then one [13Xmust[113X install a method to
  compute   this   external   representation   for  the  objects  in  the  new
  representation.[133X
  
  
  [1X79.17 [33X[0;0YMutability and Copying[133X[101X
  
  [33X[0;0YAny  [5XGAP[105X  object is either mutable or immutable. This can be tested with the
  function  [2XIsMutable[102X  ([14X12.6-2[114X).  The  intended meaning of (im)mutability is a
  mathematical one: an immutable object should never change in such a way that
  it  represents  a  different  Element. Objects [13Xmay[113X change in other ways, for
  instance  to  store more information, or represent an element in a different
  way.[133X
  
  [33X[0;0YImmutability  is  enforced  in  different  ways  for  built-in objects (like
  records, or lists) and for external objects (made using [2XObjectify[102X ([14X79.9-1[114X)).[133X
  
  [33X[0;0YFor  built-in  objects which are immutable, the kernel will prevent you from
  changing them. Thus[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xl := [1,2,4];[127X[104X
    [4X[28X[ 1, 2, 4 ][128X[104X
    [4X[25Xgap>[125X [27XMakeImmutable(l);[127X[104X
    [4X[28X[ 1, 2, 4 ][128X[104X
    [4X[25Xgap>[125X [27Xl[3] := 5;[127X[104X
    [4X[28XError, Lists Assignment: <list> must be a mutable list[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  external  objects, the situation is different. An external object which
  claims  to  be immutable (i.e. its type does not contain [2XIsMutable[102X ([14X12.6-2[114X))
  should  not  admit  any  methods which change the element it represents. The
  kernel  does  [13Xnot[113X prevent the use of [10X!.[110X and [10X![[110X to change the underlying data
  structure.  This  is  used  for  instance  by the code that stores attribute
  values  for  reuse.  In  general,  these [10X![110X operations should only be used in
  methods  which  depend  on the representation of the object. Furthermore, we
  would   [13Xnot[113X   recommend  users  to  install  methods  which  depend  on  the
  representations  of  objects  created  by the library or by [5XGAP[105X packages, as
  there  is  certainly  no  guarantee of the representations being the same in
  future versions of [5XGAP[105X.[133X
  
  [33X[0;0YHere  we  see  an  immutable  object (the group [22XS_4[122X), in which we improperly
  install a new component.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg := SymmetricGroup(IsPermGroup,4);[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsMutable(g);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XNamesOfComponents(g);[127X[104X
    [4X[28X[ "Size", "NrMovedPoints", "MovedPoints", [128X[104X
    [4X[28X  "GeneratorsOfMagmaWithInverses" ][128X[104X
    [4X[25Xgap>[125X [27Xg!.silly := "rubbish";[127X[104X
    [4X[28X"rubbish"[128X[104X
    [4X[25Xgap>[125X [27XNamesOfComponents(g);[127X[104X
    [4X[28X[ "Size", "NrMovedPoints", "MovedPoints", [128X[104X
    [4X[28X  "GeneratorsOfMagmaWithInverses", "silly" ][128X[104X
    [4X[25Xgap>[125X [27Xg!.silly;[127X[104X
    [4X[28X"rubbish"[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOn  the  other hand, if we form an immutable externally represented list, we
  find that [5XGAP[105X will not let us change the object.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xe := Enumerator(g);[127X[104X
    [4X[28X<enumerator of perm group>[128X[104X
    [4X[25Xgap>[125X [27XIsMutable(e);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsList(e);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xe[3];[127X[104X
    [4X[28X(1,2,4)[128X[104X
    [4X[25Xgap>[125X [27Xe[3] := false;[127X[104X
    [4X[28XError, The list you are trying to assign to is immutable[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWhen we consider copying objects, another filter [2XIsCopyable[102X ([14X12.6-1[114X), enters
  the  game  and we find that [2XShallowCopy[102X ([14X12.7-1[114X) and [2XStructuralCopy[102X ([14X12.7-2[114X)
  behave  quite  differently.  Objects  can  be  divided for this purpose into
  three:  mutable  objects,  immutable  but copyable objects, and non-copyable
  objects (called constants).[133X
  
  [33X[0;0YA  mutable  or  copyable  object  should  have  a  method  for the operation
  [2XShallowCopy[102X  ([14X12.7-1[114X),  which  should make a new mutable object, sharing its
  top-level  subobjects  with  the original. The exact definition of top-level
  subobject may be defined by the implementor for new kinds of object.[133X
  
  [33X[0;0Y[2XShallowCopy[102X ([14X12.7-1[114X) applied to a constant simply returns the constant.[133X
  
  [33X[0;0Y[2XStructuralCopy[102X  ([14X12.7-2[114X)  is  expected to be much less used than [2XShallowCopy[102X
  ([14X12.7-1[114X). Applied to a mutable object, it returns a new mutable object which
  shares no mutable sub-objects with the input. Applied to an immutable object
  (even  a  copyable  one), it just returns the object. It is not an operation
  (indeed, it's a rather special kernel function).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xe1 := StructuralCopy(e);[127X[104X
    [4X[28X<enumerator of perm group>[128X[104X
    [4X[25Xgap>[125X [27XIsMutable(e1);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xe2 := ShallowCopy(e);[127X[104X
    [4X[28X[ (), (1,4), (1,2,4), (1,3,4), (2,4), (1,4,2), (1,2), (1,3,4,2), [128X[104X
    [4X[28X  (2,3,4), (1,4,2,3), (1,2,3), (1,3)(2,4), (3,4), (1,4,3), (1,2,4,3), [128X[104X
    [4X[28X  (1,3), (2,4,3), (1,4,3,2), (1,2)(3,4), (1,3,2), (2,3), (1,4)(2,3), [128X[104X
    [4X[28X  (1,2,3,4), (1,3,2,4) ][128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThere are two other related functions: [2XImmutable[102X ([14X12.6-3[114X), which makes a new
  immutable  object  which  shares  no  mutable  subobjects with its input and
  [2XMakeImmutable[102X ([14X12.6-4[114X) which changes an object and its mutable subobjects [13Xin
  place[113X  to  be immutable. It should only be used on [21Xnew[121X objects that you have
  just created, and which cannot share mutable subobjects with anything else.[133X
  
  [33X[0;0YBoth  [2XImmutable[102X ([14X12.6-3[114X) and [2XMakeImmutable[102X ([14X12.6-4[114X) work on external objects
  by  just  resetting the [2XIsMutable[102X ([14X12.6-2[114X) filter in the object's type. This
  should  make  ineligible  any  methods  that  might  change the object. As a
  consequence, you must allow for the possibility of immutable versions of any
  objects you create.[133X
  
  [33X[0;0YSo,  if  you are implementing your own external objects. The rules amount to
  the following:[133X
  
  [31X1[131X   [33X[0;6YYou decide if your objects should be mutable or copyable or constants,
        by fixing whether their type includes [2XIsMutable[102X ([14X12.6-2[114X) or [2XIsCopyable[102X
        ([14X12.6-1[114X).[133X
  
  [31X2[131X   [33X[0;6YYou install methods for your objects respecting that decision:[133X
  
        [8Xfor constants:[108X
              [33X[0;12Yno methods change the underlying elements;[133X
  
        [8Xfor copyables:[108X
              [33X[0;12Yyou provide a method for [2XShallowCopy[102X ([14X12.7-1[114X);[133X
  
        [8Xfor mutables:[108X
              [33X[0;12Yyou  may  have  methods  that change the underlying elements and
              these should explicitly require [2XIsMutable[102X ([14X12.6-2[114X).[133X
  
  
  [1X79.18 [33X[0;0YGlobal Variables in the Library[133X[101X
  
  [33X[0;0YGlobal  variables in the [5XGAP[105X library are usually read-only in order to avoid
  their being overwritten accidentally. See also Section [14X4.9[114X.[133X
  
  [1X79.18-1 DeclareCategory[101X
  
  [33X[1;0Y[29X[2XDeclareCategory[102X( [3Xname[103X, [3Xsuper[103X[, [3Xrank[103X] ) [32X function[133X
  
  [33X[0;0Ydoes  the  same  as [2XNewCategory[102X ([14X79.1-1[114X) and additionally makes the variable
  [3Xname[103X read-only.[133X
  
  [1X79.18-2 DeclareRepresentation[101X
  
  [33X[1;0Y[29X[2XDeclareRepresentation[102X( [3Xname[103X, [3Xsuper[103X, [3Xslots[103X[, [3Xreq[103X] ) [32X function[133X
  
  [33X[0;0Ydoes  the  same  as  [2XNewRepresentation[102X  ([14X79.2-1[114X)  and additionally makes the
  variable [3Xname[103X read-only.[133X
  
  [1X79.18-3 DeclareAttribute[101X
  
  [33X[1;0Y[29X[2XDeclareAttribute[102X( [3Xname[103X, [3Xfilter[103X[, [3X"mutable"[103X][, [3Xrank[103X] ) [32X function[133X
  
  [33X[0;0Ydoes the same as [2XNewAttribute[102X ([14X79.3-1[114X), additionally makes the variable [3Xname[103X
  read-only  and  also binds read-only global variables with names [10XHas[3Xname[103X[10X[110X and
  [10XSet[3Xname[103X[10X[110X for the tester and setter of the attribute (see Section [14X13.6[114X).[133X
  
  [1X79.18-4 DeclareProperty[101X
  
  [33X[1;0Y[29X[2XDeclareProperty[102X( [3Xname[103X, [3Xfilter[103X[, [3Xrank[103X] ) [32X function[133X
  
  [33X[0;0Ydoes  the same as [2XNewProperty[102X ([14X79.3-2[114X), additionally makes the variable [3Xname[103X
  read-only  and  also binds read-only global variables with names [10XHas[3Xname[103X[10X[110X and
  [10XSet[3Xname[103X[10X[110X for the tester and setter of the property (see Section [14X13.6[114X).[133X
  
  [1X79.18-5 DeclareFilter[101X
  
  [33X[1;0Y[29X[2XDeclareFilter[102X( [3Xname[103X[, [3Xrank[103X] ) [32X function[133X
  
  [33X[0;0Ydoes the same as [2XNewFilter[102X ([14X79.4-1[114X) and additionally makes the variable [3Xname[103X
  read-only.[133X
  
  [1X79.18-6 DeclareOperation[101X
  
  [33X[1;0Y[29X[2XDeclareOperation[102X( [3Xname[103X, [3Xfilters[103X ) [32X function[133X
  
  [33X[0;0Ydoes  the  same as [2XNewOperation[102X ([14X79.5-1[114X) and additionally makes the variable
  [3Xname[103X read-only.[133X
  
  [1X79.18-7 DeclareGlobalFunction[101X
  
  [33X[1;0Y[29X[2XDeclareGlobalFunction[102X( [3Xname[103X, [3Xinfo[103X ) [32X function[133X
  [33X[1;0Y[29X[2XInstallGlobalFunction[102X( [3Xoper[103X, [3Xfunc[103X ) [32X function[133X
  
  [33X[0;0Y[2XDeclareGlobalFunction[102X  [5XGAP[105X  functions  that  are not operations and that are
  intended  to be called by users should be notified to [5XGAP[105X in the declaration
  part     of    the    respective    package    (see    Section [14X79.19[114X)    via
  [2XDeclareGlobalFunction[102X,  which  returns  a  function  that  serves as a place
  holder for the function that will be installed later, and that will print an
  error message if it is called. See also [2XDeclareSynonym[102X ([14X79.18-10[114X).[133X
  
  [33X[0;0YA global function declared with [2XDeclareGlobalFunction[102X can be given its value
  [3Xfunc[103X  via  [2XInstallGlobalFunction[102X;  [3Xgvar[103X  is the global variable (or a string
  denoting   its   name)   named  with  the  [3Xname[103X  argument  of  the  call  to
  [2XDeclareGlobalFunction[102X. For example, a declaration like[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XDeclareGlobalFunction( "SumOfTwoCubes" );[128X[104X
  [4X[32X[104X
  
  [33X[0;0Yin  the  [21Xdeclaration  part[121X  (see  Section [14X79.19[114X)  might have a corresponding
  [21Ximplementation part[121X of:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XInstallGlobalFunction( SumOfTwoCubes, function(x, y) return x^3 + y^3; end);[128X[104X
  [4X[32X[104X
  
  [1X79.18-8 DeclareGlobalVariable[101X
  
  [33X[1;0Y[29X[2XDeclareGlobalVariable[102X( [3Xname[103X[, [3Xdescription[103X] ) [32X function[133X
  
  [33X[0;0YFor  global  variables  that  are [13Xnot[113X functions, instead of using [2XBindGlobal[102X
  ([14X4.9-7[114X)  one  can also declare the variable with [2XDeclareGlobalVariable[102X which
  creates  a  new  global  variable  named  by  the string [3Xname[103X. If the second
  argument  [3Xdescription[103X  is  entered then this must be a string that describes
  the  meaning  of the global variable. [2XDeclareGlobalVariable[102X shall be used in
  the  declaration part of the respective package (see [14X79.19[114X), values can then
  be   assigned   to   the   new   variable   with   [2XInstallValue[102X   ([14X79.18-9[114X),
  [2XInstallFlushableValue[102X    ([14X79.18-9[114X)    or   [2XInstallFlushableValueFromFunction[102X
  ([14X79.18-9[114X), in the implementation part (again, see [14X79.19[114X).[133X
  
  [1X79.18-9 InstallValue[101X
  
  [33X[1;0Y[29X[2XInstallValue[102X( [3Xgvar[103X, [3Xvalue[103X ) [32X function[133X
  [33X[1;0Y[29X[2XInstallFlushableValue[102X( [3Xgvar[103X, [3Xvalue[103X ) [32X function[133X
  [33X[1;0Y[29X[2XInstallFlushableValueFromFunction[102X( [3Xgvar[103X, [3Xfunc[103X ) [32X function[133X
  
  [33X[0;0Y[2XInstallValue[102X   assigns   the  value  [3Xvalue[103X  to  the  global  variable  [3Xgvar[103X.
  [2XInstallFlushableValue[102X does the same but additionally provides that each call
  of  [2XFlushCaches[102X  ([14X79.18-11[114X)  will assign a structural copy of [3Xvalue[103X to [3Xgvar[103X.
  [2XInstallFlushableValueFromFunction[102X instead assigns the result of [3Xfunc[103X to [3Xgvar[103X
  ([3Xfunc[103X is re-evaluated for each invocation of [2XFlushCaches[102X ([14X79.18-11[114X)[133X
  
  [33X[0;0Y[2XInstallValue[102X  does  [13Xnot[113X  work  if  [3Xvalue[103X  is  an  [21Ximmediate object[121X, i.e., an
  internally  represented small integer or finite field element. It also fails
  for  booleans.  Furthermore,  [2XInstallFlushableValue[102X works only if [3Xvalue[103X is a
  list  or  a  record.  (Note  that [2XInstallFlushableValue[102X makes sense only for
  [13Xmutable[113X global variables.)[133X
  
  [1X79.18-10 DeclareSynonym[101X
  
  [33X[1;0Y[29X[2XDeclareSynonym[102X( [3Xname[103X, [3Xvalue[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDeclareSynonymAttr[102X( [3Xname[103X, [3Xvalue[103X ) [32X function[133X
  
  [33X[0;0Y[2XDeclareSynonym[102X assigns the string [3Xname[103X to a global variable as a synonym for
  [3Xvalue[103X. Two typical intended usages are to declare an [21Xand-filter[121X, e.g.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XDeclareSynonym( "IsGroup", IsMagmaWithInverses and IsAssociative );[128X[104X
  [4X[32X[104X
  
  [33X[0;0Yand  to  provide  a  previously declared global function with an alternative
  name, e.g.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XDeclareGlobalFunction( "SizeOfSomething" );[128X[104X
    [4X[28XDeclareSynonym( "OrderOfSomething", SizeOfSomething );[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[13XNote:[113X  Before  using  [2XDeclareSynonym[102X  in the way of this second example, one
  should  determine  whether  the  synonym  is really needed. Perhaps an extra
  index entry in the documentation would be sufficient.[133X
  
  [33X[0;0YWhen  [3Xvalue[103X is actually an attribute then [2XDeclareSynonymAttr[102X should be used;
  this  binds  also  globals  variables [10XSet[110X[3Xname[103X and [10XHas[110X[3Xname[103X for its setter and
  tester, respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XDeclareSynonymAttr( "IsField", IsDivisionRing and IsCommutative );[128X[104X
    [4X[28XDeclareAttribute( "GeneratorsOfDivisionRing", IsDivisionRing );[128X[104X
    [4X[28XDeclareSynonymAttr( "GeneratorsOfField", GeneratorsOfDivisionRing );[128X[104X
  [4X[32X[104X
  
  [1X79.18-11 FlushCaches[101X
  
  [33X[1;0Y[29X[2XFlushCaches[102X(  ) [32X operation[133X
  
  [33X[0;0Y[2XFlushCaches[102X  resets the value of each global variable that has been declared
  with  [2XDeclareGlobalVariable[102X  ([14X79.18-8[114X)  and  for which the initial value has
  been       set       with       [2XInstallFlushableValue[102X      ([14X79.18-9[114X)      or
  [2XInstallFlushableValueFromFunction[102X ([14X79.18-9[114X) to this initial value.[133X
  
  [33X[0;0Y[2XFlushCaches[102X  should  be used only for debugging purposes, since the involved
  global  variables  include  for  example  lists that store finite fields and
  cyclotomic  fields  used  in the current [5XGAP[105X session, in order to avoid that
  these  fields  are  constructed  anew  in  each  call  to [2XGF[102X ([14X59.3-2[114X) and [2XCF[102X
  ([14X60.1-1[114X).[133X
  
  
  [1X79.19 [33X[0;0YDeclaration and Implementation Part[133X[101X
  
  [33X[0;0YEach  package  of  [5XGAP[105X code consists of two parts, the [13Xdeclaration part[113X that
  defines  the new categories and operations for the objects the package deals
  with,  and  the  [13Ximplementation  part[113X  where  the  corresponding methods are
  installed.  The  declaration  part  should  be  representation  independent,
  representation   dependent   information   should   be  dealt  with  in  the
  implementation part.[133X
  
  [33X[0;0Y[5XGAP[105X  functions that are not operations and that are intended to be called by
  users   should   be   notified   to   [5XGAP[105X   in   the  declaration  part  via
  [2XDeclareGlobalFunction[102X ([14X79.18-7[114X). Values for these functions can be installed
  in the implementation part via [2XInstallGlobalFunction[102X ([14X79.18-7[114X).[133X
  
  [33X[0;0YCalls to the following functions belong to the declaration part.[133X
  
  [33X[0;0Y[2XDeclareAttribute[102X ([14X79.18-3[114X),[133X
  
  [33X[0;0Y[2XDeclareCategory[102X ([14X79.18-1[114X),[133X
  
  [33X[0;0Y[2XDeclareFilter[102X ([14X79.18-5[114X),[133X
  
  [33X[0;0Y[2XDeclareOperation[102X ([14X79.18-6[114X),[133X
  
  [33X[0;0Y[2XDeclareGlobalFunction[102X ([14X79.18-7[114X),[133X
  
  [33X[0;0Y[2XDeclareSynonym[102X ([14X79.18-10[114X),[133X
  
  [33X[0;0Y[2XDeclareSynonymAttr[102X ([14X79.18-10[114X),[133X
  
  [33X[0;0Y[2XDeclareProperty[102X ([14X79.18-4[114X),[133X
  
  [33X[0;0Y[2XInstallTrueMethod[102X ([14X78.7-1[114X).[133X
  
  [33X[0;0YCalls to the following functions belong to the implementation part.[133X
  
  [33X[0;0Y[2XDeclareRepresentation[102X ([14X79.18-2[114X),[133X
  
  [33X[0;0Y[2XInstallGlobalFunction[102X ([14X79.18-7[114X),[133X
  
  [33X[0;0Y[2XInstallMethod[102X ([14X78.2-1[114X),[133X
  
  [33X[0;0Y[2XInstallImmediateMethod[102X ([14X78.6-1[114X),[133X
  
  [33X[0;0Y[2XInstallOtherMethod[102X ([14X78.2-2[114X),[133X
  
  [33X[0;0Y[2XNewFamily[102X ([14X79.7-1[114X),[133X
  
  [33X[0;0Y[2XNewType[102X ([14X79.8-1[114X),[133X
  
  [33X[0;0Y[2XObjectify[102X ([14X79.9-1[114X).[133X
  
  [33X[0;0YWhenever  both  a  [10XNew[110X[3XSomething[103X and a [10XDeclare[110X[3XSomething[103X variant of a function
  exist  (see [14X79.18[114X),  the use of [10XDeclare[110X[3XSomething[103X is recommended because this
  protects  the  variables in question from being overwritten. Note that there
  are  [13Xno[113X functions [10XDeclareFamily[110X and [10XDeclareType[110X since families and types are
  created  dynamically,  hence  usually  no global variables are associated to
  them.  Further  note  that  [2XDeclareRepresentation[102X  ([14X79.18-2[114X)  is regarded as
  belonging  to  the  implementation  part, because usually representations of
  objects  are  accessed only in very few places, and all code that involves a
  particular   representation   is   contained   in  one  file;  additionally,
  representations  of objects are often not interesting for the user, so there
  is   no   need   to   provide   a  user  interface  or  documentation  about
  representations.[133X
  
  [33X[0;0YIt should be emphasized that [21Xdeclaration[121X means only an explicit notification
  of  mathematical  or  technical  terms  or  of concepts to [5XGAP[105X. For example,
  declaring  a category or property with name [10XIsInteresting[110X does of course not
  tell   [5XGAP[105X   what  this  shall  mean,  and  it  is  necessary  to  implement
  possibilities  to  create  objects  that  know  already  that  they  lie  in
  [10XIsInteresting[110X  in the case that it is a category, or to install implications
  or  methods  in order to compute for a given object whether [10XIsInteresting[110X is
  [9Xtrue[109X or [9Xfalse[109X for it in the case that [10XIsInteresting[110X is a property.[133X
  
