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<div class="ChapSects"><a href="chap79_mj.html#X83548994805AD1C9">79 Creating New Objects</a>
<div class="ContSect"> <a href="chap79_mj.html#X82E86CF37B123FD4">79.1 Creating Objects</a>

<div class="ContSSBlock">
 <a href="chap79_mj.html#X7CB5C12E813F512B">79.1-1 Objectify</a>
 <a href="chap79_mj.html#X85377AC07E775066">79.1-2 ObjectifyWithAttributes</a>
</div></div>
<div class="ContSect"> <a href="chap79_mj.html#X866E223484649E5A">79.2 Component Objects</a>

<div class="ContSSBlock">
 <a href="chap79_mj.html#X823965BF7DFDACC9">79.2-1 NamesOfComponents</a>
</div></div>
<div class="ContSect"> <a href="chap79_mj.html#X834893D07FAA6FD2">79.3 Positional Objects</a>

</div>
<div class="ContSect"> <a href="chap79_mj.html#X82309B3F81DD2237">79.4 Implementing New List Objects</a>

</div>
<div class="ContSect"> <a href="chap79_mj.html#X849D8BC278649EA5">79.5 Example – Constructing Enumerators</a>

</div>
<div class="ContSect"> <a href="chap79_mj.html#X7F6BF6CE7AD04EFC">79.6 Example – Constructing Iterators</a>

</div>
<div class="ContSect"> <a href="chap79_mj.html#X829629E87E30090C">79.7 Arithmetic Issues in the Implementation of New Kinds of Lists</a>

</div>
<div class="ContSect"> <a href="chap79_mj.html#X7EBB961E7FE1B0EB">79.8 External Representation</a>

<div class="ContSSBlock">
 <a href="chap79_mj.html#X8542B32A8206118C">79.8-1 ExtRepOfObj</a>
</div></div>
<div class="ContSect"> <a href="chap79_mj.html#X8090219A7C76AF55">79.9 Mutability and Copying</a>

</div>
<div class="ContSect"> <a href="chap79_mj.html#X87E29BA57C8208A4">79.10 Global Variables in the Library</a>

<div class="ContSSBlock">
 <a href="chap79_mj.html#X828E14ED7EE39522">79.10-1 DeclareGlobalName</a>
 <a href="chap79_mj.html#X8324B5DE8300E0F2">79.10-2 DeclareGlobalVariable</a>
 <a href="chap79_mj.html#X7A23F09886E936D2">79.10-3 InstallValue</a>
 <a href="chap79_mj.html#X87A4316C818B3DE3">79.10-4 FlushCaches</a>
 <a href="chap79_mj.html#X834A8CC587A609BE">79.10-5 DeclareGlobalFunction</a>
 <a href="chap79_mj.html#X851654DA87616207">79.10-6 DeclareSynonym</a>
</div></div>
<div class="ContSect"> <a href="chap79_mj.html#X7837CA9A83D93B38">79.11 Declaration and Implementation Part</a>

</div>
</div>

<h3>79 Creating New Objects</h3>

This chapter is divided into three parts.

In the first part, it is explained how to create objects with given type (see <a href="chap79_mj.html#X82E86CF37B123FD4">79.1</a>).

In the second part, first a few small examples are given, for dealing with the usual cases of component objects (see <a href="chap79_mj.html#X866E223484649E5A">79.2</a>) and positional objects (see <a href="chap79_mj.html#X834893D07FAA6FD2">79.3</a>), and for the implementation of new kinds of lists (see <a href="chap79_mj.html#X82309B3F81DD2237">79.4</a> and <a href="chap79_mj.html#X829629E87E30090C">79.7</a>). Finally, the external representation of objects is introduced (see <a href="chap79_mj.html#X7EBB961E7FE1B0EB">79.8</a>), 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 <a href="chap79_mj.html#X87E29BA57C8208A4">79.10</a>) 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 <a href="chap79_mj.html#X7837CA9A83D93B38">79.11</a>).

See also Chapter <a href="chap81_mj.html#X8125CC6A87409887">81</a> 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.

<a id="X82E86CF37B123FD4" name="X82E86CF37B123FD4"></a>

<h4>79.1 Creating Objects</h4>

<a id="X7CB5C12E813F512B" name="X7CB5C12E813F512B"></a>

<h5>79.1-1 Objectify</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Objectify</code>( <var class="Arg">type</var>, <var class="Arg">data</var> )</td><td class="tdright">( function )</td></tr></table></div>
New objects are created by <code class="func">Objectify</code>. <var class="Arg">data</var> must be a plain list or a plain record, and <var class="Arg">type</var> is the type that the desired object shall have. <code class="func">Objectify</code> turns <var class="Arg">data</var> into an object with type <var class="Arg">type</var>. That is, <var class="Arg">data</var> is changed, and afterwards it will not be a list or a record unless <var class="Arg">type</var> is of type list resp. record.

If <var class="Arg">data</var> is a list then <code class="func">Objectify</code> turns it into a positional object, if <var class="Arg">data</var> is a record then <code class="func">Objectify</code> turns it into a component object (for examples, see <a href="chap79_mj.html#X866E223484649E5A">79.2</a> and <a href="chap79_mj.html#X834893D07FAA6FD2">79.3</a>).

<code class="func">Objectify</code> does also return the object that it made out of <var class="Arg">data</var>.

For examples where <code class="func">Objectify</code> is used, see <a href="chap79_mj.html#X866E223484649E5A">79.2</a>, <a href="chap79_mj.html#X834893D07FAA6FD2">79.3</a>, and the example in Chapter <a href="chap81_mj.html#X8125CC6A87409887">81</a>.

<a id="X85377AC07E775066" name="X85377AC07E775066"></a>

<h5>79.1-2 ObjectifyWithAttributes</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ObjectifyWithAttributes</code>( <var class="Arg">obj</var>, <var class="Arg">type</var>, <var class="Arg">attr1</var>, <var class="Arg">val1</var>, <var class="Arg">attr2</var>, <var class="Arg">val2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
Attribute assignments will change the type of an object. If you create many objects, code of the form

<div class="example"><pre>
o:=Objectify(type,rec());
SetMyAttribute(o,value);
</pre></div>

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. <code class="func">ObjectifyWithAttributes</code> takes a plain list or record <var class="Arg">obj</var> and turns it an object just like <code class="func">Objectify</code> (<a href="chap79_mj.html#X7CB5C12E813F512B">79.1-1</a>) and sets attribute <var class="Arg">attr1</var> to <var class="Arg">val1</var>, sets attribute <var class="Arg">attr2</var> to <var class="Arg">val2</var> and so forth.

If the filter list of <var class="Arg">type</var> 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 <code class="func">Objectify</code> (<a href="chap79_mj.html#X7CB5C12E813F512B">79.1-1</a>) with subsequent setter calls for the attributes is performed instead.

<a id="X866E223484649E5A" name="X866E223484649E5A"></a>

<h4>79.2 Component Objects</h4>

A component object is an object in the representation <code class="func">IsComponentObjectRep</code> (<a href="chap13_mj.html#X805F1C3B7C730062">13.4-1</a>) or a subrepresentation of it. Such an object <var class="Arg">cobj</var> is built from subobjects that can be accessed via <code class="code"><var class="Arg">cobj</var>!.<var class="Arg">name</var></code>, similar to components of a record. Also analogously to records, values can be assigned to components of <var class="Arg">cobj</var> via <code class="code"><var class="Arg">cobj</var>!.<var class="Arg">name</var>:= <var class="Arg">val</var></code>. For the creation of component objects, see <a href="chap79_mj.html#X82E86CF37B123FD4">79.1</a>. One must be very careful when using the <code class="code">!.</code> operator, in order to interpret the component in the right way, and even more careful when using the assignment to components using <code class="code">!.</code>, in order to keep the information stored in <var class="Arg">cobj</var> consistent.

First of all, in the access or assignment to a component as shown above, <var class="Arg">name</var> must be among the admissible component names for the representation of <var class="Arg">cobj</var>, see <code class="func">NewRepresentation</code> (<a href="chap13_mj.html#X7CC8106F809E15CF">13.4-4</a>). Second, preferably only few low level functions should use <code class="code">!.</code>, whereas this operator should not occur in <q>user interactions</q>.

Note that even if <var class="Arg">cobj</var> claims that it is immutable, i.e., if <var class="Arg">cobj</var> is not in the category <code class="func">IsMutable</code> (<a href="chap12_mj.html#X7999AD1D7A4F1F46">12.6-2</a>), access and assignment via <code class="code">!.</code> and <code class="code">!.:=</code> 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 <a href="chap30_mj.html#X85A3F00985453F95">30.8</a>) for the domain of integers, which is represented as component object. See <a href="chap79_mj.html#X834893D07FAA6FD2">79.3</a> for an implementation using positional objects. (In practice, such an iterator can be implemented more elegantly using <code class="func">IteratorByFunctions</code> (<a href="chap30_mj.html#X82677D8F817D6701">30.8-8</a>), see <a href="chap79_mj.html#X7F6BF6CE7AD04EFC">79.6</a>.)

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.

<div class="example"><pre>
DeclareRepresentation( "IsIntegersIteratorCompRep",
 IsComponentObjectRep, [ "counter" ] );
</pre></div>

The above command creates a new representation (see <code class="func">NewRepresentation</code> (<a href="chap13_mj.html#X7CC8106F809E15CF">13.4-4</a>)) <code class="code">IsIntegersIteratorCompRep</code>, as a subrepresentation of <code class="func">IsComponentObjectRep</code> (<a href="chap13_mj.html#X805F1C3B7C730062">13.4-1</a>), and with one admissible component <code class="code">counter</code>. So no other components than <code class="code">counter</code> will be needed.

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

After the above method installation, one can already ask for <code class="code">Iterator( Integers )</code>. Note that exactly the domain of integers is described by the filter <code class="func">IsIntegers</code> (<a href="chap14_mj.html#X818683B17F8C97F3">14.1-2</a>).

By the call to <code class="func">NewType</code> (<a href="chap13_mj.html#X7CE39E9478AEC826">13.9-3</a>), the returned object lies in the family containing all iterators, which is <code class="code">IteratorsFamily</code>, it lies in the category <code class="func">IsIterator</code> (<a href="chap30_mj.html#X87168A827E5B28E4">30.8-3</a>) and in the representation <code class="code">IsIntegersIteratorCompRep</code>; furthermore, it has the component <code class="code">counter</code> with value <code class="code">0</code>.

What is missing now are methods for the two basic operations of iterators, namely <code class="func">IsDoneIterator</code> (<a href="chap30_mj.html#X8055FC557B5D899E">30.8-4</a>) and <code class="func">NextIterator</code> (<a href="chap30_mj.html#X879F62F77D1D1179">30.8-5</a>). The former must always return <code class="keyw">false</code>, 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 <code class="code">counter</code>.

<div class="example"><pre>
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 );
</pre></div>

<a id="X823965BF7DFDACC9" name="X823965BF7DFDACC9"></a>

<h5>79.2-1 NamesOfComponents</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NamesOfComponents</code>( <var class="Arg">comobj</var> )</td><td class="tdright">( function )</td></tr></table></div>
For a component object <var class="Arg">comobj</var>, <code class="func">NamesOfComponents</code> returns a list of strings, which are the names of components currently bound in <var class="Arg">comobj</var>.

For a record <var class="Arg">comobj</var> in internal representation, <code class="func">NamesOfComponents</code> returns the result of <code class="func">RecNames</code> (<a href="chap29_mj.html#X837F1E1F866FB1A0">29.1-2</a>).

<a id="X834893D07FAA6FD2" name="X834893D07FAA6FD2"></a>

<h4>79.3 Positional Objects</h4>

A positional object is an object in the representation <code class="func">IsPositionalObjectRep</code> (<a href="chap13_mj.html#X805F1C3B7C730062">13.4-1</a>) or a subrepresentation of it. Such an object <var class="Arg">pobj</var> is built from subobjects that can be accessed via <code class="code"><var class="Arg">pobj</var>![<var class="Arg">pos</var>]</code>, similar to positions in a list. Also analogously to lists, values can be assigned to positions of <var class="Arg">pobj</var> via <code class="code"><var class="Arg">pobj</var>![<var class="Arg">pos</var>]:= <var class="Arg">val</var></code>. For the creation of positional objects, see <a href="chap79_mj.html#X82E86CF37B123FD4">79.1</a>.

One must be very careful when using the <code class="code">![]</code> operator, in order to interpret the position in the right way, and even more careful when using the assignment to positions using <code class="code">![]</code>, in order to keep the information stored in <var class="Arg">pobj</var> consistent.

First of all, in the access or assignment to a position as shown above, <var class="Arg">pos</var> must be among the admissible positions for the representation of <var class="Arg">pobj</var>, see <code class="func">NewRepresentation</code> (<a href="chap13_mj.html#X7CC8106F809E15CF">13.4-4</a>). Second, preferably only few low level functions should use <code class="code">![]</code>, whereas this operator should not occur in <q>user interactions</q>.

Note that even if <var class="Arg">pobj</var> claims that it is immutable, i.e., if <var class="Arg">pobj</var> is not in the category <code class="func">IsMutable</code> (<a href="chap12_mj.html#X7999AD1D7A4F1F46">12.6-2</a>), access and assignment via <code class="code">![]</code> 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 <a href="chap30_mj.html#X85A3F00985453F95">30.8</a>) for the domain of integers, which is represented as positional object. See <a href="chap79_mj.html#X866E223484649E5A">79.2</a> for an implementation using component objects, and more details.

<div class="example"><pre>
DeclareRepresentation( "IsIntegersIteratorPosRep",
 IsPositionalObjectRep, [ 1 ] );
</pre></div>

The above command creates a new representation (see <code class="func">NewRepresentation</code> (<a href="chap13_mj.html#X7CC8106F809E15CF">13.4-4</a>)) <code class="code">IsIntegersIteratorPosRep</code>, as a subrepresentation of <code class="func">IsPositionalObjectRep</code> (<a href="chap13_mj.html#X805F1C3B7C730062">13.4-1</a>), and with only the first position being admissible for storing data.

<div class="example"><pre>
InstallMethod( Iterator,
 "method for `Integers'",
 [ IsIntegers ],
 function( Integers )
 return Objectify( NewType( IteratorsFamily,
 IsIterator
 and IsIntegersIteratorPosRep ),
 [ 0 ] );
 end );
</pre></div>

After the above method installation, one can already ask for <code class="code">Iterator( Integers )</code>. Note that exactly the domain of integers is described by the filter <code class="func">IsIntegers</code> (<a href="chap14_mj.html#X818683B17F8C97F3">14.1-2</a>).

By the call to <code class="func">NewType</code> (<a href="chap13_mj.html#X7CE39E9478AEC826">13.9-3</a>), the returned object lies in the family containing all iterators, which is <code class="code">IteratorsFamily</code>, it lies in the category <code class="func">IsIterator</code> (<a href="chap30_mj.html#X87168A827E5B28E4">30.8-3</a>) and in the representation <code class="code">IsIntegersIteratorPosRep</code>; furthermore, the first position has value <code class="code">0</code>.

What is missing now are methods for the two basic operations of iterators, namely <code class="func">IsDoneIterator</code> (<a href="chap30_mj.html#X8055FC557B5D899E">30.8-4</a>) and <code class="func">NextIterator</code> (<a href="chap30_mj.html#X879F62F77D1D1179">30.8-5</a>). The former must always return <code class="keyw">false</code>, 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.

<div class="example"><pre>
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 );
</pre></div>

It should be noted that one can of course install both the methods shown in Section <a href="chap79_mj.html#X866E223484649E5A">79.2</a> and <a href="chap79_mj.html#X834893D07FAA6FD2">79.3</a>. The call <code class="code">Iterator( Integers )</code> 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 <code class="func">NextIterator</code> (<a href="chap30_mj.html#X879F62F77D1D1179">30.8-5</a>) method will be chosen.

<a id="X82309B3F81DD2237" name="X82309B3F81DD2237"></a>

<h4>79.4 Implementing New List Objects</h4>

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 <a href="chap79_mj.html#X849D8BC278649EA5">79.5</a>.

A list in GAP is an object in the category <code class="func">IsList</code> (<a href="chap21_mj.html#X7C4CC4EA8299701E">21.1-1</a>). Basic operations for lists are <code class="func">Length</code> (<a href="chap21_mj.html#X780769238600AFD1">21.17-5</a>), <code class="func">\[\]</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>), and <code class="func">IsBound\[\]</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>) (see <a href="chap21_mj.html#X7B202D147A5C2884">21.2</a>).

Note that the access to the position <var class="Arg">pos</var> in the list <var class="Arg">list</var> via <code class="code"><var class="Arg">list</var>[<var class="Arg">pos</var>]</code> is handled by the call <code class="code">\[\]( <var class="Arg">list</var>, <var class="Arg">pos</var> )</code> to the operation <code class="func">\[\]</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>). To explain the somewhat strange name <code class="code">\[\]</code> of this operation, note that non-alphanumeric characters like <code class="code">[</code> and <code class="code">]</code> may occur in GAP variable names only if they are escaped by a <code class="code">\</code> character.

Analogously, the check <code class="code">IsBound( <var class="Arg">list</var>[<var class="Arg">pos</var>] )</code> whether the position <var class="Arg">pos</var> of the list <var class="Arg">list</var> is bound is handled by the call <code class="code">IsBound\[\]( <var class="Arg">list</var>, <var class="Arg">pos</var> )</code> to the operation <code class="func">IsBound\[\]</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>).

For mutable lists, also assignment to positions and unbinding of positions via the operations <code class="func">\[\]\:\=</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>) and <code class="func">Unbind\[\]</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>) are basic operations. The assignment <code class="code"><var class="Arg">list</var>[<var class="Arg">pos</var>]:= <var class="Arg">val</var></code> is handled by the call <code class="code">\[\]\:\=( <var class="Arg">list</var>, <var class="Arg">pos</var>, <var class="Arg">val</var> )</code>, and <code class="code">Unbind( <var class="Arg">list</var>[<var class="Arg">pos</var>] )</code> is handled by the call <code class="code">Unbind\[\]( <var class="Arg">list</var>, <var class="Arg">pos</var> )</code>.

All other operations for lists, e.g., <code class="func">Add</code> (<a href="chap21_mj.html#X795EC9D67E34DAB0">21.4-2</a>), <code class="func">Append</code> (<a href="chap21_mj.html#X79E31DB27C82D6E1">21.4-5</a>), <code class="func">Sum</code> (<a href="chap21_mj.html#X7A04B71C84CFCC2D">21.20-26</a>), 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 <code class="func">IsList</code> (<a href="chap21_mj.html#X7C4CC4EA8299701E">21.1-1</a>), and installs methods for <code class="func">Length</code> (<a href="chap21_mj.html#X780769238600AFD1">21.17-5</a>), <code class="func">\[\]</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>), and <code class="func">IsBound\[\]</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>). If the new lists shall be mutable, one needs to install also methods for <code class="func">\[\]\:\=</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>) and <code class="func">Unbind\[\]</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>).

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 <q><code class="code">*</code></q> 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. <q><code class="code">=</code></q> because they lie in different families.

<a id="X849D8BC278649EA5" name="X849D8BC278649EA5"></a>

<h4>79.5 Example – Constructing Enumerators</h4>

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 <a href="chap21_mj.html#X7EA3ACE27E43D174">21.23</a>). These are lists containing the elements in a specified ordering, and the operations <code class="func">Position</code> (<a href="chap21_mj.html#X79975EC6783B4293">21.16-1</a>) and list access via <code class="func">\[\]</code> (<a href="chap21_mj.html#X8297BBCD79642BE6">21.2-1</a>) 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 <code class="func">EnumeratorByFunctions</code> (<a href="chap30_mj.html#X85E149177AC547C3">30.3-4</a>).

If the set in question is a domain <var class="Arg">D</var> for which a <code class="func">Size</code> (<a href="chap30_mj.html#X858ADA3B7A684421">30.4-6</a>) 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 <code class="code">ElementNumber</code> and <code class="code">NumberElement</code> of a record, and to call <code class="func">EnumeratorByFunctions</code> (<a href="chap30_mj.html#X85E149177AC547C3">30.3-4</a>) with the domain <var class="Arg">D</var> and this record as arguments. For example, the following lines of code install an <code class="func">Enumerator</code> (<a href="chap30_mj.html#X7EF8910F82B45EC7">30.3-2</a>) method for the case that <var class="Arg">D</var> is the domain of rational integers. (Note that <code class="func">IsIntegers</code> (<a href="chap14_mj.html#X818683B17F8C97F3">14.1-2</a>) is a filter that describes exactly the domain of rational integers.)

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

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 <a href="chap79_mj.html#X866E223484649E5A">79.2</a> and <a href="chap79_mj.html#X834893D07FAA6FD2">79.3</a>.)

<div class="example"><pre>
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
</pre></div>

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 <a href="chap79_mj.html#X82E86CF37B123FD4">79.1</a>). 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 <code class="code">ListsFamily</code>.

An example in the GAP library is an enumerator for the set of \(k\)-tuples over a finite set; the function is called <code class="func">EnumeratorOfTuples</code> (<a href="chap16_mj.html#X7BA135297E8DA819">16.2-9</a>).

<div class="example"><pre>
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
</pre></div>

We see that the enumerator is a homogeneous list that stores individual functions <code class="code">ElementNumber</code>, <code class="code">NumberElement</code>, <code class="code">Length</code>, and <code class="code">PrintObj</code>; besides that, the data components \(S\) and \(k\) are contained.

<a id="X7F6BF6CE7AD04EFC" name="X7F6BF6CE7AD04EFC"></a>

<h4>79.6 Example – Constructing Iterators</h4>

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 <a href="chap30_mj.html#X85A3F00985453F95">30.8</a>. A general setup for implementing new iterators is provided by <code class="func">IteratorByFunctions</code> (<a href="chap30_mj.html#X82677D8F817D6701">30.8-8</a>).

All one has to do is to write down the functions for <code class="func">NextIterator</code> (<a href="chap30_mj.html#X879F62F77D1D1179">30.8-5</a>), <code class="func">IsDoneIterator</code> (<a href="chap30_mj.html#X8055FC557B5D899E">30.8-4</a>), and <code class="func">ShallowCopy</code> (<a href="chap12_mj.html#X846BC7107C352031">12.7-1</a>), and to call <code class="func">IteratorByFunctions</code> (<a href="chap30_mj.html#X82677D8F817D6701">30.8-8</a>) with this record as argument. For example, the following lines of code install an <code class="func">Iterator</code> (<a href="chap30_mj.html#X83ADF8287ED0668E">30.8-1</a>) method for the case that the argument is the domain of rational integers.

(Note that <code class="func">IsIntegers</code> (<a href="chap14_mj.html#X818683B17F8C97F3">14.1-2</a>) is a filter that describes exactly the domain of rational integers.)

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

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 <a href="chap79_mj.html#X866E223484649E5A">79.2</a> and <a href="chap79_mj.html#X834893D07FAA6FD2">79.3</a>.)

<div class="example"><pre>
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
</pre></div>

Note that the <code class="code">ShallowCopy</code> 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 <code class="func">IteratorByFunctions</code> (<a href="chap30_mj.html#X82677D8F817D6701">30.8-8</a>) in order to create the desired shallow copy.

<a id="X829629E87E30090C" name="X829629E87E30090C"></a>

<h4>79.7 Arithmetic Issues in the Implementation of New Kinds of Lists</h4>

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 <code class="func">IsGeneralizedRowVector</code> (<a href="chap21_mj.html#X87ABCEE9809585A0">21.12-1</a>) and <code class="func">IsMultiplicativeGeneralizedRowVector</code> (<a href="chap21_mj.html#X7FBCA5B58308C158">21.12-2</a>) in the types of the lists, the latter can be achieved by setting the filter <code class="func">IsListDefault</code> (<a href="chap21_mj.html#X7BAD12E67BFC90DE">21.12-3</a>). (for details, see <a href="chap21_mj.html#X84D642967B8546B7">21.12</a>). An example for <q>wrapped lists</q> 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 <code class="func">IsListDefault</code> (<a href="chap21_mj.html#X7BAD12E67BFC90DE">21.12-3</a>) 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 <code class="func">IsGeneralizedRowVector</code> (<a href="chap21_mj.html#X87ABCEE9809585A0">21.12-1</a>) and <code class="func">IsMultiplicativeGeneralizedRowVector</code> (<a href="chap21_mj.html#X7FBCA5B58308C158">21.12-2</a>), 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 <a href="chap21_mj.html#X7E6A1F66781BE923">21.13</a> and <a href="chap21_mj.html#X782ED7F27D8C7FC1">21.14</a>).

For example, if one implements block matrices in <code class="func">IsMultiplicativeGeneralizedRowVector</code> (<a href="chap21_mj.html#X7FBCA5B58308C158">21.12-2</a>) 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 <code class="func">IsMultiplicativeGeneralizedRowVector</code> (<a href="chap21_mj.html#X7FBCA5B58308C158">21.12-2</a>) 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 <code class="func">IsGeneralizedRowVector</code> (<a href="chap21_mj.html#X87ABCEE9809585A0">21.12-1</a>) and <code class="func">IsMultiplicativeGeneralizedRowVector</code> (<a href="chap21_mj.html#X7FBCA5B58308C158">21.12-2</a>) 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 <code class="func">IsGeneralizedRowVector</code> (<a href="chap21_mj.html#X87ABCEE9809585A0">21.12-1</a>) (<code class="func">IsMultiplicativeGeneralizedRowVector</code> (<a href="chap21_mj.html#X7FBCA5B58308C158">21.12-2</a>)) lies in <code class="func">IsAttributeStoringRep</code> (<a href="chap13_mj.html#X7A951C33839AF2C1">13.5-5</a>), 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 <q>wrapped matrices</q> 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.

<a id="X7EBB961E7FE1B0EB" name="X7EBB961E7FE1B0EB"></a>

<h4>79.8 External Representation</h4>

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 <q><code class="code">=</code></q> then the result must be equal to the old result w.r.t. <q><code class="code">=</code></q>.

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 <q>standard representation</q>, and then access the data in the way defined for this representation, simply because it may be impossible to choose such a <q>standard representation</q> uniformly for all objects in the given family.

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

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 <var class="Arg">obj</var> encodes <var class="Arg">obj</var> in such a way that from this data and the family of <var class="Arg">obj</var>, one can create an object that is equal to <var class="Arg">obj</var>. Usually the external representation of an object is a list or a record.

Although the external representation of <var class="Arg">obj</var> is by definition independent of the actually available representations for <var class="Arg">obj</var>, it is usual that a representation of <var class="Arg">obj</var> exists for which the computation of the external representation is obtained by just <q>unpacking</q> <var class="Arg">obj</var>, in the sense that the desired data is stored in a component or a position of <var class="Arg">obj</var>, if <var class="Arg">obj</var> is a component object (see <a href="chap79_mj.html#X866E223484649E5A">79.2</a>) or a positional object (see <a href="chap79_mj.html#X834893D07FAA6FD2">79.3</a>).

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

<a id="X8542B32A8206118C" name="X8542B32A8206118C"></a>

<h5>79.8-1 ExtRepOfObj</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExtRepOfObj</code>( <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ObjByExtRep</code>( <var class="Arg">fam</var>, <var class="Arg">data</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<code class="func">ExtRepOfObj</code> returns the external representation of its argument, and <code class="func">ObjByExtRep</code> returns an object in the family <var class="Arg">fam</var> that has external representation <var class="Arg">data</var>.

Of course, <code class="code">ObjByExtRep( FamilyObj( <var class="Arg">obj</var> ), ExtRepOfObj( <var class="Arg">obj</var> ) )</code> must be equal to <var class="Arg">obj</var> w.r.t. the operation <code class="func">\=</code> (<a href="chap31_mj.html#X7EF67D047F03CA6F">31.11-1</a>). 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.

<a id="X8090219A7C76AF55" name="X8090219A7C76AF55"></a>

<h4>79.9 Mutability and Copying</h4>

Any GAP object is either mutable or immutable. This can be tested with the function <code class="func">IsMutable</code> (<a href="chap12_mj.html#X7999AD1D7A4F1F46">12.6-2</a>). 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 <code class="func">Objectify</code> (<a href="chap79_mj.html#X7CB5C12E813F512B">79.1-1</a>)).

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

<div class="example"><pre>
gap> l := [1,2,4];
[ 1, 2, 4 ]
gap> MakeImmutable(l);
[ 1, 2, 4 ]
gap> l[3] := 5;
Error, List Assignment: <list> must be a mutable list
</pre></div>

For external objects, the situation is different. An external object which claims to be immutable (i.e. its type does not contain <code class="func">IsMutable</code> (<a href="chap12_mj.html#X7999AD1D7A4F1F46">12.6-2</a>)) should not admit any methods which change the element it represents. The kernel does not prevent the use of <code class="code">!.</code> and <code class="code">![</code> to change the underlying data structure. This is used for instance by the code that stores attribute values for reuse. In general, these <code class="code">!</code> 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.

<div class="example"><pre>
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"
</pre></div>

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

<div class="example"><pre>
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
</pre></div>

When we consider copying objects, another filter <code class="func">IsCopyable</code> (<a href="chap12_mj.html#X811EFD727EBD1ADC">12.6-1</a>), enters the game and we find that <code class="func">ShallowCopy</code> (<a href="chap12_mj.html#X846BC7107C352031">12.7-1</a>) and <code class="func">StructuralCopy</code> (<a href="chap12_mj.html#X7C1E70587EBDD2CB">12.7-2</a>) 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 <code class="func">ShallowCopy</code> (<a href="chap12_mj.html#X846BC7107C352031">12.7-1</a>), 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.

<code class="func">ShallowCopy</code> (<a href="chap12_mj.html#X846BC7107C352031">12.7-1</a>) applied to a constant simply returns the constant.

<code class="func">StructuralCopy</code> (<a href="chap12_mj.html#X7C1E70587EBDD2CB">12.7-2</a>) is expected to be much less used than <code class="func">ShallowCopy</code> (<a href="chap12_mj.html#X846BC7107C352031">12.7-1</a>). 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).

<div class="example"><pre>
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) ]
</pre></div>

There are two other related functions: <code class="func">Immutable</code> (<a href="chap12_mj.html#X7F0ABF2C870B0CBB">12.6-3</a>), which makes a new immutable object which shares no mutable subobjects with its input and <code class="func">MakeImmutable</code> (<a href="chap12_mj.html#X80CE136D804097C7">12.6-4</a>) which changes an object and its mutable subobjects in place to be immutable. It should only be used on <q>new</q> objects that you have just created, and which cannot share mutable subobjects with anything else.

Both <code class="func">Immutable</code> (<a href="chap12_mj.html#X7F0ABF2C870B0CBB">12.6-3</a>) and <code class="func">MakeImmutable</code> (<a href="chap12_mj.html#X80CE136D804097C7">12.6-4</a>) work on external objects by just resetting the <code class="func">IsMutable</code> (<a href="chap12_mj.html#X7999AD1D7A4F1F46">12.6-2</a>) 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:

<ol>
<li>You decide if your objects should be mutable or copyable or constants, by fixing whether their type includes <code class="func">IsMutable</code> (<a href="chap12_mj.html#X7999AD1D7A4F1F46">12.6-2</a>) or <code class="func">IsCopyable</code> (<a href="chap12_mj.html#X811EFD727EBD1ADC">12.6-1</a>).

</li>
<li>You install methods for your objects respecting that decision:

<dl>
<dt>for constants:</dt>
<dd>no methods change the underlying elements;

</dd>
<dt>for copyables:</dt>
<dd>you provide a method for <code class="func">ShallowCopy</code> (<a href="chap12_mj.html#X846BC7107C352031">12.7-1</a>);

</dd>
<dt>for mutables:</dt>
<dd>you may have methods that change the underlying elements and these should explicitly require <code class="func">IsMutable</code> (<a href="chap12_mj.html#X7999AD1D7A4F1F46">12.6-2</a>).

</dd>
</dl>
</li>
</ol>
<a id="X87E29BA57C8208A4" name="X87E29BA57C8208A4"></a>

<h4>79.10 Global Variables in the Library</h4>

Global variables in the GAP library are usually read-only in order to prevent them from being overwritten accidentally. See also Section <a href="chap4_mj.html#X816FBEEA85782EC2">4.9</a>.

<a id="X828E14ED7EE39522" name="X828E14ED7EE39522"></a>

<h5>79.10-1 DeclareGlobalName</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DeclareGlobalName</code>( <var class="Arg">name</var> )</td><td class="tdright">( function )</td></tr></table></div>
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