Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Bib Ind
[Top of Book] [Contents] [Previous Chapter] [Next Chapter]

[MathJax off]

79 Creating New Objects
79.1 Creating Categories
79.1-1 NewCategory
79.1-2 CategoryFamily
79.2 Creating Representations
79.2-1 NewRepresentation
79.3 Creating Attributes and Properties
79.3-1 NewAttribute
79.3-2 NewProperty
79.4 Creating Other Filters
79.4-1 NewFilter
79.4-2 SetFilterObj
79.4-3 ResetFilterObj
79.5 Creating Operations
79.5-1 NewOperation
79.6 Creating Constructors
79.6-1 NewConstructor
79.7 Creating Families
79.7-1 NewFamily
79.8 Creating Types
79.8-1 NewType
79.9 Creating Objects
79.9-1 Objectify
79.9-2 ObjectifyWithAttributes
79.10 Component Objects
79.10-1 NamesOfComponents
79.11 Positional Objects
79.12 Implementing New List Objects
79.13 Example – Constructing Enumerators
79.14 Example – Constructing Iterators
79.15 Arithmetic Issues in the Implementation of New Kinds of Lists
79.16 External Representation
79.16-1 ExtRepOfObj
79.17 Mutability and Copying
79.18 Global Variables in the Library
79.18-1 DeclareCategory
79.18-2 DeclareRepresentation
79.18-3 DeclareAttribute
79.18-4 DeclareProperty
79.18-5 DeclareFilter
79.18-6 DeclareOperation
79.18-7 DeclareGlobalFunction
79.18-8 DeclareGlobalVariable
79.18-9 InstallValue
79.18-10 DeclareSynonym
79.18-11 FlushCaches
79.19 Declaration and Implementation Part

### 79 Creating New Objects

This chapter is divided into three parts.

In the first part, it is explained how to create filters (see 79.1, 79.2, 79.3, 79.4), operations (see 79.5), families (see 79.7), types (see 79.8), and objects with given type (see 79.9).

In the second part, first a few small examples are given, for dealing with the usual cases of component objects (see 79.10) and positional objects (see 79.11), and for the implementation of new kinds of lists (see 79.12 and 79.15). Finally, the external representation of objects is introduced (see 79.16), as a tool for representation independent access to an object.

The third part deals with some rules concerning the organization of the GAP library; namely, some commands for creating global variables are explained (see 79.18) that correspond to the ones discussed in the first part of the chapter, and the idea of distinguishing declaration and implementation part of GAP packages is outlined (see 79.19).

See also Chapter 81 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.

#### 79.1 Creating Categories

##### 79.1-1 NewCategory

`‣ NewCategory`( name, super[, rank] ) | ( function )

`NewCategory` returns a new category cat that has the name name and is contained in the filter super, see 13.2. This means that every object in cat lies automatically also in super. We say also that super is an implied filter of cat.

For example, if one wants to create a category of group elements then super should be `IsMultiplicativeElementWithInverse` (31.14-13) or a subcategory of it. If no specific supercategory of cat is known, super may be `IsObject` (12.1-1).

The optional third argument rank denotes the incremental rank (see 13.2) of cat, the default value is 1.

##### 79.1-2 CategoryFamily

`‣ CategoryFamily`( cat ) | ( function )

For a category cat, `CategoryFamily` returns the family category of cat. This is a category in which all families lie that know from their creation that all their elements are in the category cat, see 79.7.

For example, a family of associative words is in the category `CategoryFamily( IsAssocWord )`, 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.

`CategoryFamily` is quite technical, and in fact of minor importance.

See also `CategoryCollections` (30.2-4).

#### 79.2 Creating Representations

##### 79.2-1 NewRepresentation

`‣ NewRepresentation`( name, super, slots[, req] ) | ( function )

`NewRepresentation` returns a new representation rep that has the name name and is a subrepresentation of the representation super. This means that every object in rep lies automatically also in super. We say also that super is an implied filter of rep.

Each representation in GAP is a subrepresentation of exactly one of the four representations `IsInternalRep`, `IsDataObjectRep`, `IsComponentObjectRep`, `IsPositionalObjectRep`. The data describing objects in the former two can be accessed only via GAP kernel functions, the data describing objects in the latter two is accessible also in library functions, see 79.10 and 79.11 for the details.

The third argument slots is a list either of integers or of strings. In the former case, rep must be `IsPositionalObjectRep` or a subrepresentation of it, and slots tells what positions of the objects in the representation rep may be bound. In the latter case, rep must be `IsComponentObjectRep` or a subrepresentation of, and slots lists the admissible names of components that objects in the representation rep may have. The admissible positions resp. component names of super need not be be listed in slots.

The incremental rank (see 13.2) of rep is 1.

Note that for objects in the representation rep, of course some of the component names and positions reserved via slots may be unbound.

Examples for the use of `NewRepresentation` can be found in 79.10, 79.11, and also in 81.3.

#### 79.3 Creating Attributes and Properties

Each method that is installed for an attribute or a property via `InstallMethod` (78.2-1) must require exactly one argument, and this must lie in the filter filter that was entered as second argument of `NewAttribute` (79.3-1) resp. `NewProperty` (79.3-2).

As for any operation (see 79.5), for attributes and properties one can install a method taking an argument that does not lie in filt via `InstallOtherMethod` (78.2-2), or a method for more than one argument; in the latter case, clearly the result value is not stored in any of the arguments.

##### 79.3-1 NewAttribute

`‣ NewAttribute`( name, filter[, "mutable"][, rank] ) | ( function )

`NewAttribute` returns a new attribute getter with name name that is applicable to objects with the property filter.

Contrary to the situation with categories and representations, the tester of the new attribute does not imply filter. This is exactly because of the possibility to install methods that do not require filter.

For example, the attribute `Size` (30.4-6) was created with second argument a list or a collection, but there is also a method for `Size` (30.4-6) that is applicable to a character table, which is neither a list nor a collection.

If the optional third argument is given then there are two possibilities. Either it is an integer rank, then the attribute tester has this incremental rank (see 13.2). Or it is the string `"mutable"`, 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 `ComputedSylowSubgroups` for the list holding those Sylow subgroups of a group that have been computed already by the function `SylowSubgroup` (39.13-1), and this list is mutable because one may want to enter groups into it as they are computed.

If no third argument is given then the rank of the tester is 1.

Each method for the new attribute that does not require its argument to lie in filter must be installed using `InstallOtherMethod` (78.2-2).

##### 79.3-2 NewProperty

`‣ NewProperty`( name, filter[, rank] ) | ( function )

`NewProperty` returns a new property prop with name name (see also 13.7). The filter filter describes the involved filters of prop. As in the case of attributes, filter is not implied by prop.

The optional third argument rank denotes the incremental rank (see 13.2) of the property prop itself, i.e. not of its tester; the default value is 1.

#### 79.4 Creating Other Filters

In order to change the value of filt for an object obj, one can use logical implications (see 78.7) or `SetFilterObj` (79.4-2), `ResetFilterObj` (79.4-3).

##### 79.4-1 NewFilter

`‣ NewFilter`( name[, rank] ) | ( function )

`NewFilter` returns a simple filter with name name (see 13.8). The optional second argument rank denotes the incremental rank (see 13.2) of the filter, the default value is 1.

The default value of the new simple filter for each object is `false`.

##### 79.4-2 SetFilterObj

`‣ SetFilterObj`( obj, filter ) | ( function )

`SetFilterObj` sets the value of filter (and of all filters implied by filter) for obj to `true`,

##### 79.4-3 ResetFilterObj

`‣ ResetFilterObj`( obj, filter ) | ( function )

`ResetFilterObj` sets the value of filter for obj to `false`. (Implied filters of filt are not touched. This might create inconsistent situations if applied carelessly).

#### 79.5 Creating Operations

##### 79.5-1 NewOperation

`‣ NewOperation`( name, args-filts ) | ( function )

`NewOperation` returns an operation opr with name name. The list args-filts describes requirements about the arguments of opr, namely the number of arguments must be equal to the length of args-filts, and the \(i\)-th argument must lie in the filter args-filts\([i]\).

Each method that is installed for opr via `InstallMethod` (78.2-1) must require that the \(i\)-th argument lies in the filter args-filts\([i]\).

One can install methods for other arguments tuples via `InstallOtherMethod` (78.2-2), this way it is also possible to install methods for a different number of arguments than the length of args-filts.

#### 79.6 Creating Constructors

##### 79.6-1 NewConstructor

`‣ NewConstructor`( name, args-filts ) | ( function )

`NewConstructor` returns a constructor cons with name name. The list args-filts describes requirements about the arguments of constr, namely the number of arguments must be equal to the length of args-filts, and the \(i\)-th argument must lie in the filter args-filts\([i]\). Additionally a constructor expects the first argument to be a filter to then select a method to construct an object.

Each method that is installed for cons via `InstallMethod` (78.2-1) must require that the \(i\)-th argument lies in the filter args-filts\([i]\).

One can install methods for other arguments tuples via `InstallOtherMethod` (78.2-2), this way it is also possible to install methods for a different number of arguments than the length of args-filts.

#### 79.7 Creating Families

Families are probably the least obvious part of the GAP type system, so some remarks about the role of families are necessary. When one uses GAP as it is, one will (better: should) not meet families at all. The two situations where families come into play are the following.

First, since families are used to describe relations between arguments of operations in the method selection mechanism (see Chapter 78, and also Chapter 13), one has to prescribe such a relation in each method installation (see 78.2); usual relations are `ReturnTrue` (5.4-1) (which means that any relation of the actual arguments is admissible), `IsIdenticalObj` (12.5-1) (which means that there are two arguments that lie in the same family), and `IsCollsElms` (which means that there are two arguments, the first being a collection of elements that lie in the same family as the second argument).

Second –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 "`=`") from all other GAP 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 "the family of all those GAP objects that would in fact not need a family". 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 12.4, and that a collection consists of elements in the same family, see Chapter 30 and Section 13.1.)

Let us look at an example. Suppose that no permutations are available in GAP, 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 GAP. 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 `5` in the symmetric group on the points `1`, `2`, \(\ldots\), `5` is not regarded as equal to the symmetric group on `1`, `2`, `3`, `4`. 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.

So 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 GAP 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.

As 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 Fam is the collections family of Fam (see `CollectionsFamily` (30.2-1)). This means that for a collection, we need not (because we cannot) decide about its family.

A few functions in GAP return families, see `CollectionsFamily` (30.2-1) and `ElementsFamily` (30.2-3).

##### 79.7-1 NewFamily

`‣ NewFamily`( name[, req[, imp[, famfilter]]] ) | ( function )

`NewFamily` returns a new family fam with name name. The argument req, if present, is a filter of which fam shall be a subset. If one tries to create an object in fam that does not lie in the filter req, an error message is printed. Also the argument imp, if present, is a filter of which fam shall be a subset. Any object that is created in the family fam will lie automatically in the filter imp.

The filter famfilter, if given, specifies a filter that will hold for the family fam (not for objects in fam).

Families are always represented as component objects (see 79.10). This means that components can be used to store and access useful information about the family.

#### 79.8 Creating Types

##### 79.8-1 NewType

`‣ NewType`( family, filter[, data] ) | ( function )

`NewType` returns the type given by the family family and the filter filter. The optional third argument data is any object that denotes defining data of the desired type.

For examples where `NewType` is used, see 79.10, 79.11, and the example in Chapter 81.

#### 79.9 Creating Objects

##### 79.9-1 Objectify

`‣ Objectify`( type, data ) | ( function )

New objects are created by `Objectify`. data is a list or a record, and type is the type that the desired object shall have. `Objectify` turns data into an object with type type. That is, data is changed, and afterwards it will not be a list or a record unless type is of type list resp. record.

If data is a list then `Objectify` turns it into a positional object, if data is a record then `Objectify` turns it into a component object (for examples, see 79.10 and 79.11).

`Objectify` does also return the object that it made out of data.

For examples where `Objectify` is used, see 79.10, 79.11, and the example in Chapter 81.

##### 79.9-2 ObjectifyWithAttributes

`‣ ObjectifyWithAttributes`( obj, type, attr1, val1, attr2, val2, ... ) | ( function )

Attribute assignments will change the type of an object. If you create many objects, code of the form

    o:=Objectify(type,rec());
    SetMyAttribute(o,value);

will take a lot of time for type changes. You can avoid this by setting the attributes immediately while the object is created, as follows. `ObjectifyWithAttributes` changes the type of object obj to type type and sets attribute attr1 to val1, sets attribute attr2 to val2 and so forth.

If the filter list of type 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.

If the conditions of the last sentence are not fulfilled, an ordinary `Objectify` (79.9-1) with subsequent setter calls for the attributes is performed instead.

#### 79.10 Component Objects

A component object is an object in the representation `IsComponentObjectRep` or a subrepresentation of it. Such an object cobj is built from subobjects that can be accessed via `cobj!.name`, similar to components of a record. Also analogously to records, values can be assigned to components of cobj via `cobj!.name:= val`. For the creation of component objects, see 79.9. One must be very careful when using the `!.` operator, in order to interpret the component in the right way, and even more careful when using the assignment to components using `!.`, in order to keep the information stored in cobj consistent.

First of all, in the access or assignment to a component as shown above, name must be among the admissible component names for the representation of cobj, see 79.2. Second, preferably only few low level functions should use `!.`, whereas this operator should not occur in "user interactions".

Note that even if cobj claims that it is immutable, i.e., if cobj is not in the category `IsMutable` (12.6-2), access and assignment via `!.` and `!.:=` work. This is necessary for being able to store newly discovered information in immutable objects.

The following example shows the implementation of an iterator (see 30.8) for the domain of integers, which is represented as component object. See 79.11 for an implementation using positional objects. (In practice, such an iterator can be implemented more elegantly using `IteratorByFunctions` (30.8-8), see 79.14.)

The used succession of integers is \(0, 1, -1, 2, -2, 3, -3, \ldots\), that is, \(a_n = n/2\) if \(n\) is even, and \(a_n = (1-n)/2\) otherwise.

    IsIntegersIteratorCompRep := NewRepresentation( "IsIntegersIteratorRep",
        IsComponentObjectRep, [ "counter" ] );

The above command creates a new representation (see `NewRepresentation` (79.2-1)) `IsIntegersIteratorCompRep`, as a subrepresentation of `IsComponentObjectRep`, and with one admissible component `counter`. So no other components than `counter` will be needed.

    InstallMethod( Iterator,
        "method for `Integers'",
        [ IsIntegers ],
        function( Integers )
        return Objectify( NewType( IteratorsFamily,
                                       IsIterator
                                   and IsIntegersIteratorCompRep ),
                          rec( counter := 0 ) );
        end );

After the above method installation, one can already ask for `Iterator( Integers )`. Note that exactly the domain of integers is described by the filter `IsIntegers` (14.1-2).

By the call to `NewType` (79.8-1), the returned object lies in the family containing all iterators, which is `IteratorsFamily`, it lies in the category `IsIterator` (30.8-3) and in the representation `IsIntegersIteratorCompRep`; furthermore, it has the component `counter` with value `0`.

What is missing now are methods for the two basic operations of iterators, namely `IsDoneIterator` (30.8-4) and `NextIterator` (30.8-5). The former must always return `false`, 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 `counter`.

    InstallMethod( IsDoneIterator,
        "method for iterator of `Integers'",
        [ IsIterator and IsIntegersIteratorCompRep ],
        ReturnFalse );

    InstallMethod( NextIterator,
        "method for iterator of `Integers'",
        [ IsIntegersIteratorCompRep ],
        function( iter )
        iter!.counter:= iter!.counter + 1;
        if iter!.counter mod 2 = 0 then
          return iter!.counter / 2;
        else
          return ( 1 - iter!.counter ) / 2;
        fi;
        end );

##### 79.10-1 NamesOfComponents

`‣ NamesOfComponents`( comobj ) | ( function )

For a component object comobj, `NamesOfComponents` returns a list of strings, which are the names of components currently bound in comobj.

For a record comobj, `NamesOfComponents` returns the result of `RecNames` (29.1-2).

#### 79.11 Positional Objects

A positional object is an object in the representation `IsPositionalObjectRep` or a subrepresentation of it. Such an object pobj is built from subobjects that can be accessed via `pobj![pos]`, similar to positions in a list. Also analogously to lists, values can be assigned to positions of pobj via `pobj![pos]:= val`. For the creation of positional objects, see 79.9.

One must be very careful when using the `![]` operator, in order to interpret the position in the right way, and even more careful when using the assignment to positions using `![]`, in order to keep the information stored in pobj consistent.

First of all, in the access or assignment to a position as shown above, pos must be among the admissible positions for the representation of pobj, see 79.2. Second, preferably only few low level functions should use `![]`, whereas this operator should not occur in "user interactions".

Note that even if pobj claims that it is immutable, i.e., if pobj is not in the category `IsMutable` (12.6-2), access and assignment via `![]` work. This is necessary for being able to store newly discovered information in immutable objects.

The following example shows the implementation of an iterator (see 30.8) for the domain of integers, which is represented as positional object. See 79.10 for an implementation using component objects, and more details.

    IsIntegersIteratorPosRep := NewRepresentation( "IsIntegersIteratorRep",
        IsPositionalObjectRep, [ 1 ] );

The above command creates a new representation (see `NewRepresentation` (79.2-1)) `IsIntegersIteratorPosRep`, as a subrepresentation of `IsComponentObjectRep`, and with only the first position being admissible for storing data.

    InstallMethod( Iterator,
        "method for `Integers'",
        [ IsIntegers ],
        function( Integers )
        return Objectify( NewType( IteratorsFamily,
                                       IsIterator
                                   and IsIntegersIteratorRep ),
                          [ 0 ] );
        end );

After the above method installation, one can already ask for `Iterator( Integers )`. Note that exactly the domain of integers is described by the filter `IsIntegers` (14.1-2).

By the call to `NewType` (79.8-1), the returned object lies in the family containing all iterators, which is `IteratorsFamily`, it lies in the category `IsIterator` (30.8-3) and in the representation `IsIntegersIteratorPosRep`; furthermore, the first position has value `0`.

What is missing now are methods for the two basic operations of iterators, namely `IsDoneIterator` (30.8-4) and `NextIterator` (30.8-5). The former must always return `false`, 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.

    InstallMethod( IsDoneIterator,
        "method for iterator of `Integers'",
        [ IsIterator and IsIntegersIteratorPosRep ],
        ReturnFalse );

    InstallMethod( NextIterator,
        "method for iterator of `Integers'",
        [ IsIntegersIteratorPosRep ],
        function( iter )
        iter![1]:= iter![1] + 1;
        if iter![1] mod 2 = 0 then
          return iter![1] / 2;
        else
          return ( 1 - iter![1] ) / 2;
        fi;
        end );

It should be noted that one can of course install both the methods shown in Section 79.10 and 79.11. The call `Iterator( Integers )` 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 `NextIterator` (30.8-5) method will be chosen.

#### 79.12 Implementing New List Objects

This 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 79.13.

A list in GAP is an object in the category `IsList` (21.1-1). Basic operations for lists are `Length` (21.17-5), `\[\]` (21.2-1), and `IsBound\[\]` (21.2-1) (see 21.2).

Note that the access to the position pos in the list list via `list[pos]` is handled by the call `\[\]( list, pos )` to the operation `\[\]` (21.2-1). To explain the somewhat strange name `\[\]` of this operation, note that non-alphanumeric characters like `[` and `]` may occur in GAP variable names only if they are escaped by a `\` character.

Analogously, the check `IsBound( list[pos] )` whether the position pos of the list list is bound is handled by the call `IsBound\[\]( list, pos )` to the operation `IsBound\[\]` (21.2-1).

For mutable lists, also assignment to positions and unbinding of positions via the operations `\[\]\:\=` (21.2-1) and `Unbind\[\]` (21.2-1) are basic operations. The assignment `list[pos]:= val` is handled by the call `\[\]\:\=( list, pos, val )`, and `Unbind( list[pos] )` is handled by the call `Unbind\[\]( list, pos )`.

All other operations for lists, e.g., `Add` (21.4-2), `Append` (21.4-5), `Sum` (21.20-26), are based on these operations. This means that it is sufficient to install methods for the new list objects only for the basic operations.

So if one wants to implement new list objects then one creates them as objects in the category `IsList` (21.1-1), and installs methods for `Length` (21.17-5), `\[\]` (21.2-1), and `IsBound\[\]` (21.2-1). If the new lists shall be mutable, one needs to install also methods for `\[\]\:\=` (21.2-1) and `Unbind\[\]` (21.2-1).

One application for this is the implementation of enumerators for domains. An enumerator for the domain \(D\) is a dense list whose entries are in bijection with the elements of \(D\). If \(D\) 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.

In this situation, one implements a new representation of the lists that are already available in GAP, in particular the family of such a list is the same as the family of the domain \(D\).

But 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 GAP before. An example for this is the implementation of matrices whose multiplication via "`*`" is the Lie product of matrices.

In 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. "`=`" because they lie in different families.

#### 79.13 Example – Constructing Enumerators

When dealing with countable sets, a usual task is to define enumerations, i.e., bijections to the positive integers. In GAP, this can be implemented via enumerators (see 21.23). These are lists containing the elements in a specified ordering, and the operations `Position` (21.16-1) and list access via `\[\]` (21.2-1) define the desired bijection. For implementing such an enumerator, one mainly needs to install the appropriate functions for these operations.

A general setup for creating such lists is given by `EnumeratorByFunctions` (30.3-4).

If the set in question is a domain D for which a `Size` (30.4-6) method is available then all one has to do is to write down the functions for computing the \(n\)-th element of the list and for computing the position of a given GAP object in the list, to put them into the components `ElementNumber` and `NumberElement` of a record, and to call `EnumeratorByFunctions` (30.3-4) with the domain D and this record as arguments. For example, the following lines of code install an `Enumerator` (30.3-2) method for the case that D is the domain of rational integers. (Note that `IsIntegers` (14.1-2) is a filter that describes exactly the domain of rational integers.)

    InstallMethod( Enumerator,
        "for integers",
        [ IsIntegers ],
        Integers -> EnumeratorByFunctions( Integers, rec(
                        ElementNumber := function( e, n ) ... end,
                        NumberElement := function( e, x ) ... end ) ) );

The bodies of the functions have been omitted above; here is the code that is actually used in GAP. (The ordering coincides with that for the iterators for the domain of rational integers that have been discussed in 79.10 and 79.11.)

    gap> enum:= Enumerator( Integers );
    <enumerator of Integers>
    gap> Print( enum!.NumberElement, "\n" );
    function ( e, x )
        local  pos;
        if not IsInt( x )  then
            return fail;
        elif 0 < x  then
            pos := 2 * x;
        else
            pos := -2 * x + 1;
        fi;
        return pos;
    end
    gap> Print( enum!.ElementNumber, "\n" );
    function ( e, n )
        if n mod 2 = 0  then
            return n / 2;
        else
            return (1 - n) / 2;
        fi;
        return;
    end

The situation becomes slightly more complicated if the set \(S\) 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 79.9). The latter should usually not be a problem since either \(S\) is nonempty and all its elements lie in the same family –in this case one takes the collections family of any element in \(S\)– or the family of the enumerator must be `ListsFamily`.

An example in the GAP library is an enumerator for the set of \(k\)-tuples over a finite set; the function is called `EnumeratorOfTuples` (16.2-9).

    gap> Print( EnumeratorOfTuples, "\n" );
    function ( set, k )
        local  enum;
        if k = 0  then
            return Immutable( [ [  ] ] );
        elif IsEmpty( set )  then
            return Immutable( [  ] );
        fi;
        enum
         := EnumeratorByFunctions( CollectionsFamily( FamilyObj( set ) ),
           rec(
              ElementNumber := function ( enum, n )
                    local  nn, t, i;
                    nn := n - 1;
                    t := [  ];
                    for i  in [ 1 .. enum!.k ]  do
                        t[i] := RemInt( nn, Length( enum!.set ) ) + 1;
                        nn := QuoInt( nn, Length( enum!.set ) );
                    od;
                    if nn <> 0  then
                        Error( "<enum>[", n,
                         "] must have an assigned value" );
                    fi;
                    nn := enum!.set{Reversed( t )};
                    MakeImmutable( nn );
                    return nn;
                end,
              NumberElement := function ( enum, elm )
                    local  n, i;
                    if not IsList( elm )  then
                        return fail;
                    fi;
                    elm := List( elm, function ( x )
                            return Position( enum!.set, x );
                        end );
                    if fail in elm or Length( elm ) <> enum!.k  then
                        return fail;
                    fi;
                    n := 0;
                    for i  in [ 1 .. enum!.k ]  do
                        n := Length( enum!.set ) * n + elm[i] - 1;
                    od;
                    return n + 1;
                end,
              Length := function ( enum )
                    return Length( enum!.set ) ^ enum!.k;
                end,
              PrintObj := function ( enum )
                    Print( "EnumeratorOfTuples( ", enum!.set, ", ",
                     enum!.k, " )" );
                    return;
                end,
              set := Set( set ),
              k := k ) );
        SetIsSSortedList( enum, true );
        return enum;
    end

We see that the enumerator is a homogeneous list that stores individual functions `ElementNumber`, `NumberElement`, `Length`, and `PrintObj`; besides that, the data components \(S\) and \(k\) are contained.

#### 79.14 Example – Constructing Iterators

Iterators are a kind of objects that is implemented for several collections in the GAP library and which might be interesting also in other cases, see 30.8. A general setup for implementing new iterators is provided by `IteratorByFunctions` (30.8-8).

All one has to do is to write down the functions for `NextIterator` (30.8-5), `IsDoneIterator` (30.8-4), and `ShallowCopy` (12.7-1), and to call `IteratorByFunctions` (30.8-8) with this record as argument. For example, the following lines of code install an `Iterator` (30.8-1) method for the case that the argument is the domain of rational integers.

(Note that `IsIntegers` (14.1-2) is a filter that describes exactly the domain of rational integers.)

    InstallMethod( Iterator,
        "for integers",
        [ IsIntegers ],
        Integers -> IteratorByFunctions( rec(
                        NextIterator:= function( iter ) ... end,
                        IsDoneIterator := ReturnFalse,
                        ShallowCopy := function( iter ) ... end ) ) );

The bodies of two of the functions have been omitted above; here is the code that is actually used in GAP. (The ordering coincides with that for the iterators for the domain of rational integers that have been discussed in 79.10 and 79.11.)

    gap> iter:= Iterator( Integers );
    <iterator of Integers at 0>
    gap> Print( iter!.NextIterator, "\n" );
    function ( iter )
        iter!.counter := iter!.counter + 1;
        if iter!.counter mod 2 = 0  then
            return iter!.counter / 2;
        else
            return (1 - iter!.counter) / 2;
        fi;
        return;
    end
    gap> Print( iter!.ShallowCopy, "\n" );
    function ( iter )
        return rec(
            counter := iter!.counter );
    end

Note that the `ShallowCopy` 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 `IteratorByFunctions` (30.8-8) in order to create the desired shallow copy.

#### 79.15 Arithmetic Issues in the Implementation of New Kinds of Lists

When designing a new kind of list objects in GAP, defining the arithmetic behaviour of these objects is an issue.

There 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 `IsGeneralizedRowVector` (21.12-1) and `IsMultiplicativeGeneralizedRowVector` (21.12-2) in the types of the lists, the latter can be achieved by setting the filter `IsListDefault` (21.12-3). (for details, see 21.12). An example for "wrapped lists" 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.

However, 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.

So one does not set the filter `IsListDefault` (21.12-3) in such cases, and thus one can implement one's own methods for arithmetic operations. (Of course "can" means on the other hand that one must implement such methods if one is interested in arithmetics of the new lists.)

The 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.

For the last situation, interaction between a new kind of lists and other kinds of lists, GAP provides already a setup. Namely, there are the categories `IsGeneralizedRowVector` (21.12-1) and `IsMultiplicativeGeneralizedRowVector` (21.12-2), 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 21.13 and 21.14).

For example, if one implements block matrices in `IsMultiplicativeGeneralizedRowVector` (21.12-2) 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 `IsMultiplicativeGeneralizedRowVector` (21.12-2) 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.)

Thus if one decides to set the filters `IsGeneralizedRowVector` (21.12-1) and `IsMultiplicativeGeneralizedRowVector` (21.12-2) 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.

If a list in the filter `IsGeneralizedRowVector` (21.12-1) (`IsMultiplicativeGeneralizedRowVector` (21.12-2)) lies in `IsAttributeStoringRep`, 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 "wrapped matrices" 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.

#### 79.16 External Representation

An 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 "`=`" then the result must be equal to the old result w.r.t. "`=`".

But the implementation of many methods is representation dependent in the sense that certain representation dependent subobjects are accessed.

For 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.

In 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.

For the purpose of admitting representation independent code, one can define an external representation 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.

We cannot provide conversion functions that allow us to first convert any object in question to one particular "standard representation", and then access the data in the way defined for this representation, simply because it may be impossible to choose such a "standard representation" uniformly for all objects in the given family.

So the aim of an external representation of an object obj is a different one, namely to describe the data from which obj is composed. In particular, the external representation of obj is not one possible ("standard") representation of obj, in fact the external representation of obj is in general different from obj w.r.t. "`=`", first of all because the external representation of obj does in general not lie in the same family as obj.

For 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.

The other way round, the external representation of obj encodes obj in such a way that from this data and the family of obj, one can create an object that is equal to obj. Usually the external representation of an object is a list or a record.

Although the external representation of obj is by definition independent of the actually available representations for obj, it is usual that a representation of obj exists for which the computation of the external representation is obtained by just "unpacking" obj, in the sense that the desired data is stored in a component or a position of obj, if obj is a component object (see 79.10) or a positional object (see 79.11).

To implement an external representation means to install methods for the following two operations.

##### 79.16-1 ExtRepOfObj

`‣ ExtRepOfObj`( obj ) | ( operation )

`‣ ObjByExtRep`( fam, data ) | ( operation )

`ExtRepOfObj` returns the external representation of its argument, and `ObjByExtRep` returns an object in the family fam that has external representation data.

Of course, `ObjByExtRep( FamilyObj( obj ), ExtRepOfObj( obj ) )` must be equal to obj w.r.t. the operation `\=` (31.11-1). But it is not required that equal objects have equal external representations.

Note that if one defines a new representation of objects for which an external representation does already exist then one must install a method to compute this external representation for the objects in the new representation.

#### 79.17 Mutability and Copying

Any GAP object is either mutable or immutable. This can be tested with the function `IsMutable` (12.6-2). 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 may change in other ways, for instance to store more information, or represent an element in a different way.

Immutability is enforced in different ways for built-in objects (like records, or lists) and for external objects (made using `Objectify` (79.9-1)).

For built-in objects which are immutable, the kernel will prevent you from changing them. Thus

    gap> l := [1,2,4];
    [ 1, 2, 4 ]
    gap> MakeImmutable(l);
    [ 1, 2, 4 ]
    gap> l[3] := 5;
    Error, Lists Assignment: <list> must be a mutable list

For external objects, the situation is different. An external object which claims to be immutable (i.e. its type does not contain `IsMutable` (12.6-2)) should not admit any methods which change the element it represents. The kernel does not prevent the use of `!.` and `![` to change the underlying data structure. This is used for instance by the code that stores attribute values for reuse. In general, these `!` operations should only be used in methods which depend on the representation of the object. Furthermore, we would not recommend users to install methods which depend on the representations of objects created by the library or by GAP packages, as there is certainly no guarantee of the representations being the same in future versions of GAP.

Here we see an immutable object (the group \(S_4\)), in which we improperly install a new component.

    gap> g := SymmetricGroup(IsPermGroup,4);
    Sym( [ 1 .. 4 ] )
    gap> IsMutable(g);
    false
    gap> NamesOfComponents(g);
    [ "Size", "NrMovedPoints", "MovedPoints",
      "GeneratorsOfMagmaWithInverses" ]
    gap> g!.silly := "rubbish";
    "rubbish"
    gap> NamesOfComponents(g);
    [ "Size", "NrMovedPoints", "MovedPoints",
      "GeneratorsOfMagmaWithInverses", "silly" ]
    gap> g!.silly;
    "rubbish"

On the other hand, if we form an immutable externally represented list, we find that GAP will not let us change the object.

    gap> e := Enumerator(g);
    <enumerator of perm group>
    gap> IsMutable(e);
    false
    gap> IsList(e);
    true
    gap> e[3];
    (1,2,4)
    gap> e[3] := false;
    Error, The list you are trying to assign to is immutable

When we consider copying objects, another filter `IsCopyable` (12.6-1), enters the game and we find that `ShallowCopy` (12.7-1) and `StructuralCopy` (12.7-2) behave quite differently. Objects can be divided for this purpose into three: mutable objects, immutable but copyable objects, and non-copyable objects (called constants).

A mutable or copyable object should have a method for the operation `ShallowCopy` (12.7-1), 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.

`ShallowCopy` (12.7-1) applied to a constant simply returns the constant.

`StructuralCopy` (12.7-2) is expected to be much less used than `ShallowCopy` (12.7-1). 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).

    gap> e1 := StructuralCopy(e);
    <enumerator of perm group>
    gap> IsMutable(e1);
    false
    gap> e2 := ShallowCopy(e);
    [ (), (1,4), (1,2,4), (1,3,4), (2,4), (1,4,2), (1,2), (1,3,4,2),
      (2,3,4), (1,4,2,3), (1,2,3), (1,3)(2,4), (3,4), (1,4,3), (1,2,4,3),
      (1,3), (2,4,3), (1,4,3,2), (1,2)(3,4), (1,3,2), (2,3), (1,4)(2,3),
      (1,2,3,4), (1,3,2,4) ]
    gap>

There are two other related functions: `Immutable` (12.6-3), which makes a new immutable object which shares no mutable subobjects with its input and `MakeImmutable` (12.6-4) which changes an object and its mutable subobjects in place to be immutable. It should only be used on "new" objects that you have just created, and which cannot share mutable subobjects with anything else.

Both `Immutable` (12.6-3) and `MakeImmutable` (12.6-4) work on external objects by just resetting the `IsMutable` (12.6-2) 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.

So, if you are implementing your own external objects. The rules amount to the following:

1. You decide if your objects should be mutable or copyable or constants, by fixing whether their type includes `IsMutable` (12.6-2) or `IsCopyable` (12.6-1).

2. You install methods for your objects respecting that decision:

   for constants:

     no methods change the underlying elements;

   for copyables:

     you provide a method for `ShallowCopy` (12.7-1);

   for mutables:

     you may have methods that change the underlying elements and these should explicitly require `IsMutable` (12.6-2).

#### 79.18 Global Variables in the Library

Global variables in the GAP library are usually read-only in order to avoid their being overwritten accidentally. See also Section 4.9.

##### 79.18-1 DeclareCategory

`‣ DeclareCategory`( name, super[, rank] ) | ( function )

does the same as `NewCategory` (79.1-1) and additionally makes the variable name read-only.

##### 79.18-2 DeclareRepresentation

`‣ DeclareRepresentation`( name, super, slots[, req] ) | ( function )

does the same as `NewRepresentation` (79.2-1) and additionally makes the variable name read-only.

##### 79.18-3 DeclareAttribute

`‣ DeclareAttribute`( name, filter[, "mutable"][, rank] ) | ( function )

does the same as `NewAttribute` (79.3-1), additionally makes the variable name read-only and also binds read-only global variables with names `Hasname` and `Setname` for the tester and setter of the attribute (see Section 13.6).

##### 79.18-4 DeclareProperty

`‣ DeclareProperty`( name, filter[, rank] ) | ( function )

does the same as `NewProperty` (79.3-2), additionally makes the variable name read-only and also binds read-only global variables with names `Hasname` and `Setname` for the tester and setter of the property (see Section 13.6).

##### 79.18-5 DeclareFilter

`‣ DeclareFilter`( name[, rank] ) | ( function )

does the same as `NewFilter` (79.4-1) and additionally makes the variable name read-only.

##### 79.18-6 DeclareOperation

`‣ DeclareOperation`( name, filters ) | ( function )

does the same as `NewOperation` (79.5-1) and additionally makes the variable name read-only.

##### 79.18-7 DeclareGlobalFunction

`‣ DeclareGlobalFunction`( name, info ) | ( function )

`‣ InstallGlobalFunction`( oper, func ) | ( function )

`DeclareGlobalFunction` GAP functions that are not operations and that are intended to be called by users should be notified to GAP in the declaration part of the respective package (see Section 79.19) via `DeclareGlobalFunction`, 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 `DeclareSynonym` (79.18-10).

A global function declared with `DeclareGlobalFunction` can be given its value func via `InstallGlobalFunction`; gvar is the global variable (or a string denoting its name) named with the name argument of the call to `DeclareGlobalFunction`. For example, a declaration like

    DeclareGlobalFunction( "SumOfTwoCubes" );

in the "declaration part" (see Section 79.19) might have a corresponding "implementation part" of:

    InstallGlobalFunction( SumOfTwoCubes, function(x, y) return x^3 + y^3; end);

##### 79.18-8 DeclareGlobalVariable

`‣ DeclareGlobalVariable`( name[, description] ) | ( function )

For global variables that are not functions, instead of using `BindGlobal` (4.9-7) one can also declare the variable with `DeclareGlobalVariable` which creates a new global variable named by the string name. If the second argument description is entered then this must be a string that describes the meaning of the global variable. `DeclareGlobalVariable` shall be used in the declaration part of the respective package (see 79.19), values can then be assigned to the new variable with `InstallValue` (79.18-9), `InstallFlushableValue` (79.18-9) or `InstallFlushableValueFromFunction` (79.18-9), in the implementation part (again, see 79.19).

##### 79.18-9 InstallValue

`‣ InstallValue`( gvar, value ) | ( function )

`‣ InstallFlushableValue`( gvar, value ) | ( function )

`‣ InstallFlushableValueFromFunction`( gvar, func ) | ( function )

`InstallValue` assigns the value value to the global variable gvar. `InstallFlushableValue` does the same but additionally provides that each call of `FlushCaches` (79.18-11) will assign a structural copy of value to gvar. `InstallFlushableValueFromFunction` instead assigns the result of func to gvar (func is re-evaluated for each invocation of `FlushCaches` (79.18-11)

`InstallValue` does not work if value is an "immediate object", i.e., an internally represented small integer or finite field element. It also fails for booleans. Furthermore, `InstallFlushableValue` works only if value is a list or a record. (Note that `InstallFlushableValue` makes sense only for mutable global variables.)

##### 79.18-10 DeclareSynonym

`‣ DeclareSynonym`( name, value ) | ( function )

`‣ DeclareSynonymAttr`( name, value ) | ( function )

`DeclareSynonym` assigns the string name to a global variable as a synonym for value. Two typical intended usages are to declare an "and-filter", e.g.

    DeclareSynonym( "IsGroup", IsMagmaWithInverses and IsAssociative );

and to provide a previously declared global function with an alternative name, e.g.

    DeclareGlobalFunction( "SizeOfSomething" );
    DeclareSynonym( "OrderOfSomething", SizeOfSomething );

Note: Before using `DeclareSynonym` 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.

When value is actually an attribute then `DeclareSynonymAttr` should be used; this binds also globals variables `Set` name and `Has` name for its setter and tester, respectively.

    DeclareSynonymAttr( "IsField", IsDivisionRing and IsCommutative );
    DeclareAttribute( "GeneratorsOfDivisionRing", IsDivisionRing );
    DeclareSynonymAttr( "GeneratorsOfField", GeneratorsOfDivisionRing );

##### 79.18-11 FlushCaches

`‣ FlushCaches`( ) | ( operation )

`FlushCaches` resets the value of each global variable that has been declared with `DeclareGlobalVariable` (79.18-8) and for which the initial value has been set with `InstallFlushableValue` (79.18-9) or `InstallFlushableValueFromFunction` (79.18-9) to this initial value.

`FlushCaches` 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 GAP session, in order to avoid that these fields are constructed anew in each call to `GF` (59.3-2) and `CF` (60.1-1).

#### 79.19 Declaration and Implementation Part

Each package of GAP code consists of two parts, the declaration part that defines the new categories and operations for the objects the package deals with, and the implementation part where the corresponding methods are installed. The declaration part should be representation independent, representation dependent information should be dealt with in the implementation part.

GAP functions that are not operations and that are intended to be called by users should be notified to GAP in the declaration part via `DeclareGlobalFunction` (79.18-7). Values for these functions can be installed in the implementation part via `InstallGlobalFunction` (79.18-7).

Calls to the following functions belong to the declaration part.

`DeclareAttribute` (79.18-3),

`DeclareCategory` (79.18-1),

`DeclareFilter` (79.18-5),

`DeclareOperation` (79.18-6),

`DeclareGlobalFunction` (79.18-7),

`DeclareSynonym` (79.18-10),

`DeclareSynonymAttr` (79.18-10),

`DeclareProperty` (79.18-4),

`InstallTrueMethod` (78.7-1).

Calls to the following functions belong to the implementation part.

`DeclareRepresentation` (79.18-2),

`InstallGlobalFunction` (79.18-7),

`InstallMethod` (78.2-1),

`InstallImmediateMethod` (78.6-1),

`InstallOtherMethod` (78.2-2),

`NewFamily` (79.7-1),

`NewType` (79.8-1),

`Objectify` (79.9-1).

Whenever both a `New` Something and a `Declare` Something variant of a function exist (see 79.18), the use of `Declare` Something is recommended because this protects the variables in question from being overwritten. Note that there are no functions `DeclareFamily` and `DeclareType` since families and types are created dynamically, hence usually no global variables are associated to them. Further note that `DeclareRepresentation` (79.18-2) 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.

It should be emphasized that "declaration" means only an explicit notification of mathematical or technical terms or of concepts to GAP. For example, declaring a category or property with name `IsInteresting` does of course not tell GAP what this shall mean, and it is necessary to implement possibilities to create objects that know already that they lie in `IsInteresting` in the case that it is a category, or to install implications or methods in order to compute for a given object whether `IsInteresting` is `true` or `false` for it in the case that `IsInteresting` is a property.

[Top of Book] [Contents] [Previous Chapter] [Next Chapter]
Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Bib Ind

generated by GAPDoc2HTML