GAP (ref) - Chapter 30: Collections

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30 Collections
30.1 IsCollection (Filter)
30.1-1 IsCollection
30.2 Collection Families
30.2-1 CollectionsFamily
30.2-2 IsCollectionFamily
30.2-3 ElementsFamily
30.2-4 CategoryCollections
30.2-5 DeclareCategoryCollections
30.3 Lists and Collections
30.3-1 IsListOrCollection
30.3-2 Enumerator
30.3-3 EnumeratorSorted
30.3-4 EnumeratorByFunctions
30.3-5 List
30.3-6 SortedList
30.3-7 SSortedList
30.3-8 AsList
30.3-9 AsSortedList
30.3-10 AsSSortedList
30.3-11 Elements
30.4 Attributes and Properties for Collections
30.4-1 IsEmpty
30.4-2 IsFinite
30.4-3 IsTrivial
30.4-4 IsNonTrivial
30.4-5 IsWholeFamily
30.4-6 Size
30.4-7 Representative
30.4-8 RepresentativeSmallest
30.5 Operations for Collections
30.5-1 IsSubset
30.5-2 Intersection
30.5-3 Union
30.5-4 Difference
30.6 Membership Test for Collections
`30.6-1 \in`
30.7 Random Elements
30.7-1 Random
30.7-2 PseudoRandom
30.7-3 RandomList
30.8 Iterators
30.8-1 Iterator
30.8-2 IteratorSorted
30.8-3 IsIterator
30.8-4 IsDoneIterator
30.8-5 NextIterator
30.8-6 IteratorList
30.8-7 TrivialIterator
30.8-8 IteratorByFunctions

### 30 Collections

A collection in GAP consists of elements in the same family (see 13.1). The most important kinds of collections are homogeneous lists (see 21) and domains (see 12.4). Note that a list is never a domain, and a domain is never a list. A list is a collection if and only if it is nonempty and homogeneous.

Basic operations for collections are `Size` (30.4-6) and `Enumerator` (30.3-2); for finite collections, `Enumerator` (30.3-2) admits to delegate the other operations for collections (see 30.4 and 30.5) to functions for lists (see 21). Obviously, special methods depending on the arguments are needed for the computation of e.g. the intersection of two infinite domains.

#### 30.1 IsCollection (Filter)

##### 30.1-1 IsCollection

`‣ IsCollection`( obj ) | ( category )

tests whether an object is a collection.

Some of the functions for lists and collections are described in the chapter about lists, mainly in Section 21.20. In the current chapter, we describe those functions for which the "collection aspect" seems to be more important than the "list aspect".

#### 30.2 Collection Families

##### 30.2-1 CollectionsFamily

`‣ CollectionsFamily`( Fam ) | ( attribute )

For a family Fam, `CollectionsFamily` returns the family of all collections over Fam, that is, of all dense lists and domains that consist of objects in Fam.

The `NewFamily` (13.1-2) call in the standard method of `CollectionsFamily` is executed with second argument `IsCollection` (30.1-1), since every object in the collections family must be a collection, and with third argument the collections categories of the involved categories in the implied filter of Fam.

Note that families (see 13.1) are used to describe relations between objects. Important such relations are that between an element \(e\) and each collection of elements that lie in the same family as \(e\), and that between two collections whose elements lie in the same family. Therefore, all collections of elements in the family Fam form the new family `CollectionsFamily( Fam )`.

##### 30.2-2 IsCollectionFamily

`‣ IsCollectionFamily`( obj ) | ( category )

is `true` if Fam is a family of collections, and `false` otherwise.

##### 30.2-3 ElementsFamily

`‣ ElementsFamily`( Fam ) | ( attribute )

If Fam is a collections family (see `IsCollectionFamily` (30.2-2)) then `ElementsFamily` returns the family from which Fam was created by `CollectionsFamily` (30.2-1). The way a collections family is created, it always has its elements family stored. If Fam is not a collections family then an error is signalled.

    gap> fam:= FamilyObj( (1,2) );;
    gap> collfam:= CollectionsFamily( fam );;
    gap> fam = collfam;  fam = ElementsFamily( collfam );
    false
    true
    gap> collfam = FamilyObj( [ (1,2,3) ] );
    true
    gap> collfam = FamilyObj( Group( () ) );
    true
    gap> collfam = CollectionsFamily( collfam );
    false

##### 30.2-4 CategoryCollections

`‣ CategoryCollections`( filter ) | ( function )

Let filter be a filter that is `true` for all elements of a family Fam, by the construction of Fam. Then `CategoryCollections` returns the collections category of filter. This is a category that is `true` for all elements in `CollectionsFamily( Fam )`.

For example, the construction of `PermutationsFamily` (42.1-3) guarantees that each of its elements lies in the filter `IsPerm` (42.1-1), and each collection of permutations (permutation group or dense list of permutations) lies in the category `CategoryCollections( IsPerm )`. `CategoryCollections( IsPerm )`. Note that this works only if the collections category is created before the collections family. So it is necessary to construct interesting collections categories immediately after the underlying category has been created.

##### 30.2-5 DeclareCategoryCollections

`‣ DeclareCategoryCollections`( name ) | ( function )

Calls `CategoryCollections` (30.2-4) on the category that is bound to the global variable with name name to obtain its collections category, and binds the latter to the global variable with name `nname`. This name is defined as follows: If name is of the form `SomethingCollection` then `nname` is set to `SomethingCollColl`, if name is of the form `SomethingColl` then `nname` is set to `SomethingCollColl`, otherwise we set `nname` to `nameCollection`.

#### 30.3 Lists and Collections

The following functions take a list or collection as argument, and return a corresponding list. They differ in whether or not the result is mutable or immutable (see 12.6), guaranteed to be sorted, or guaranteed to admit list access in constant time (see `IsConstantTimeAccessList` (21.1-6)).

##### 30.3-1 IsListOrCollection

`‣ IsListOrCollection`( obj ) | ( category )

Several functions are defined for both lists and collections, for example `Intersection` (30.5-2), `Iterator` (30.8-1), and `Random` (30.7-1). `IsListOrCollection` is a supercategory of `IsList` (21.1-1) and `IsCollection` (30.1-1) (that is, all lists and collections lie in this category), which is used to describe the arguments of functions such as the ones listed above.

##### 30.3-2 Enumerator

`‣ Enumerator`( listorcoll ) | ( attribute )

`Enumerator` returns an immutable list \(enum\). If the argument is a list (which may contain holes), then `Length(` \(enum\)`)` is the length of this list, and \(enum\) contains the elements (and holes) of this list in the same order. If the argument is a collection that is not a list, then `Length(` \(enum\)`)` is the number of different elements of C, and \(enum\) contains the different elements of the collection in an unspecified order, which may change for repeated calls of `Enumerator`. \(enum[pos]\) may not execute in constant time (see `IsConstantTimeAccessList` (21.1-6)), and the size of \(enum\) in memory is as small as is feasible.

For lists, the default method is `Immutable` (12.6-3). For collections that are not lists, there is no default method.

##### 30.3-3 EnumeratorSorted

`‣ EnumeratorSorted`( listorcoll ) | ( attribute )

`EnumeratorSorted` returns an immutable list \(enum\). The argument must be a collection or a list listorcoll which may contain holes but whose elements lie in the same family (see 13.1). `Length(` \(enum\)`)` is the number of different elements of the argument, and \(enum\) contains the different elements in sorted order, w.r.t. `<`. \(enum[pos]\) may not execute in constant time (see `IsConstantTimeAccessList` (21.1-6)), and the size of \(enum\) in memory is as small as is feasible.

    gap> Enumerator( [ 1, 3,, 2 ] );
    [ 1, 3,, 2 ]
    gap> enum:= Enumerator( Rationals );;  elm:= enum[ 10^6 ];
    -69/907
    gap> Position( enum, elm );
    1000000
    gap> IsMutable( enum );  IsSortedList( enum );
    false
    false
    gap> IsConstantTimeAccessList( enum );
    false
    gap> EnumeratorSorted( [ 1, 3,, 2 ] );
    [ 1, 2, 3 ]

##### 30.3-4 EnumeratorByFunctions

`‣ EnumeratorByFunctions`( D, record ) | ( function )

`‣ EnumeratorByFunctions`( Fam, record ) | ( function )

`EnumeratorByFunctions` returns an immutable, dense, and duplicate-free list \(enum\) for which `IsBound` (21.5-1), element access via `\[\]` (21.2-1), `Length` (21.17-5), and `Position` (21.16-1) are computed via prescribed functions.

Let record be a record with at least the following components.

`ElementNumber`

  a function taking two arguments enum and pos, which returns `enum[ pos ]` (see 21.2); it can be assumed that the argument pos is a positive integer, but pos may be larger than the length of enum (in which case an error must be signalled); note that the result must be immutable since enum itself is immutable,

`NumberElement`

  a function taking two arguments enum and elm, which returns `Position( enum, elm )` (see `Position` (21.16-1)); it cannot be assumed that elm is really contained in enum (and `fail` must be returned if not); note that for the three argument version of `Position` (21.16-1), the method that is available for duplicate-free lists suffices.

Further (data) components may be contained in record which can be used by these function.

If the first argument is a domain D then enum lists the elements of D (in general enum is not sorted), and methods for `Length` (21.17-5), `IsBound` (21.5-1), and `PrintObj` (6.3-5) may use D.

If one wants to describe the result without creating a domain then the elements are given implicitly by the functions in record, and the first argument must be a family Fam which will become the family of enum; if enum is not homogeneous then Fam must be `ListsFamily`, otherwise it must be the collections family of any element in enum. In this case, additionally the following component in record is needed.

`Length`

  a function taking the argument enum, which returns the length of enum (see `Length` (21.17-5)).

The following components are optional; they are used if they are present but default methods are installed for the case that they are missing.

`IsBound\[\]`

  a function taking two arguments enum and k, which returns `IsBound( enum[ k ] )` (see 21.2); if this component is missing then `Length` (21.17-5) is used for computing the result,

`Membership`

  a function taking two arguments elm and enum, which returns `true` is elm is an element of enum, and `false` otherwise (see 21.2); if this component is missing then `NumberElement` is used for computing the result,

`AsList`

  a function taking one argument enum, which returns a list with the property that the access to each of its elements will take roughly the same time (see `IsConstantTimeAccessList` (21.1-6)); if this component is missing then `ConstantTimeAccessList` (21.17-6) is used for computing the result,

`ViewObj` and `PrintObj`

  two functions that print what one wants to be printed when `View( enum )` or `Print( enum )` is called (see 6.3), if the `ViewObj` component is missing then the `PrintObj` method is used as a default.

If the result is known to have additional properties such as being strictly sorted (see `IsSSortedList` (21.17-4)) then it can be useful to set these properties after the construction of the enumerator, before it is used for the first time. And in the case that a new sorted enumerator of a domain is implemented via `EnumeratorByFunctions`, and this construction is installed as a method for the operation `Enumerator` (30.3-2), then it should be installed also as a method for `EnumeratorSorted` (30.3-3).

Note that it is not checked that `EnumeratorByFunctions` really returns a dense and duplicate-free list. `EnumeratorByFunctions` does not make a shallow copy of record, this record is changed in place, see 79.1.

It would be easy to implement a slightly generalized setup for enumerators that need not be duplicate-free (where the three argument version of `Position` (21.16-1) is supported), but the resulting overhead for the methods seems not to be justified.

##### 30.3-5 List

`‣ List`( C ) | ( function )

For a collection C (see 30) that is not a list, `List` returns a new mutable list new such that `Length( new )` is the number of different elements of C, and new contains the different elements of C in an unspecified order which may change for repeated calls. `new[pos]` executes in constant time (see `IsConstantTimeAccessList` (21.1-6)), and the size of new is proportional to its length. The generic method for this case is `ShallowCopy( Enumerator( C ) )`.

Developers who wish to adapt this for custom list or collection types need to install suitable methods for the operation `ListOp`.

    gap> l:= List( Group( (1,2,3) ) );
    [ (), (1,3,2), (1,2,3) ]
    gap> IsMutable( l );  IsSortedList( l );  IsConstantTimeAccessList( l );
    true
    false
    true

(See also `List` (21.20-18).)

##### 30.3-6 SortedList

`‣ SortedList`( listorcoll[, func] ) | ( operation )

`SortedList` returns a new mutable and dense list new. The argument must be a collection or list listorcoll which may contain holes but whose elements lie in the same family (see 13.1). `Length( new )` is the number of elements of listorcoll, and new contains the elements in sorted order, w.r.t. `<` or func if it is specified. For details, please refer to `Sort` (21.18-1). `new[pos]` executes in constant time (see `IsConstantTimeAccessList` (21.1-6)), and the size of new in memory is proportional to its length.

    gap> l:= SortedList( Group( (1,2,3) ) );
    [ (), (1,2,3), (1,3,2) ]
    gap> IsMutable( l );  IsSortedList( l );  IsConstantTimeAccessList( l );
    true
    true
    true
    gap> SortedList( [ 1, 2, 1,, 3, 2 ] );
    [ 1, 1, 2, 2, 3 ]

##### 30.3-7 SSortedList

`‣ SSortedList`( listorcoll[, fun] ) | ( operation )

`‣ Set`( C[, fun] ) | ( operation )

`SSortedList` ("strictly sorted list") returns a new dense, mutable, and duplicate free list new. The argument must be a collection or list listorcoll which may contain holes.

If the optional argument fun is not given then `Length( new )` is the number of different elements of listorcoll, and new contains the different elements in strictly sorted order, w.r.t. `\<` (31.11-1). For that, any two entries of listorcoll must be comparable via `\<` (31.11-1). (Typically, the entries lie in the same family, see 13.1.)

If fun is given then it must be a unary function. In this case, fun is applied to all elements of listorcoll, new contains the different return values in strictly sorted order, and `Length( new )` is the number of different such values. For that, any two return values must be comparable via `\<` (31.11-1).

`new[pos]` executes in constant time (see `IsConstantTimeAccessList` (21.1-6)), and the size of new in memory is proportional to its length.

`Set` is simply a synonym for `SSortedList`.

    gap> l:= SSortedList( Group( (1,2,3) ) );
    [ (), (1,2,3), (1,3,2) ]
    gap> IsMutable( l );  IsSSortedList( l );  IsConstantTimeAccessList( l );
    true
    true
    true
    gap> SSortedList( Group( (1,2,3) ), Order );
    [ 1, 3 ]
    gap> SSortedList( [ 1, 2, 1,, 3, 2 ] );
    [ 1, 2, 3 ]
    gap> SSortedList( [ 1, 2, 1,, 3, 2 ], x -> x^2 );
    [ 1, 4, 9 ]

##### 30.3-8 AsList

`‣ AsList`( listorcoll ) | ( attribute )

`AsList` returns a immutable list imm. If the argument is a list (which may contain holes), then `Length( imm )` is the `Length` (21.17-5) value of this list, and imm contains the elements (and holes) of the list in the same order. If the argument is a collection that is not a list, then `Length( imm )` is the number of different elements of this collection, and imm contains the different elements of the collection in an unspecified order, which may change for repeated calls of `AsList`. `imm[pos]` executes in constant time (see `IsConstantTimeAccessList` (21.1-6)), and the size of imm in memory is proportional to its length.

If you expect to do many element tests in the resulting list, it might be worth to use a sorted list instead, using `AsSSortedList` (30.3-10).

    gap> l:= AsList( [ 1, 3, 3,, 2 ] );
    [ 1, 3, 3,, 2 ]
    gap> IsMutable( l );  IsSortedList( l );  IsConstantTimeAccessList( l );
    false
    false
    true
    gap> AsList( Group( (1,2,3), (1,2) ) );
    [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]

##### 30.3-9 AsSortedList

`‣ AsSortedList`( listorcoll ) | ( attribute )

`AsSortedList` returns a dense and immutable list imm. The argument must be a collection or list listorcoll which may contain holes but whose elements lie in the same family (see 13.1). `Length( imm )` is the number of elements of the argument, and imm contains the elements in sorted order, w.r.t. `<=`. `new[pos]` executes in constant time (see `IsConstantTimeAccessList` (21.1-6)), and the size of imm in memory is proportional to its length.

The only difference to the operation `SortedList` (30.3-6) is that `AsSortedList` returns an immutable list.

    gap> l:= AsSortedList( [ 1, 3, 3,, 2 ] );
    [ 1, 2, 3, 3 ]
    gap> IsMutable( l );  IsSortedList( l );  IsConstantTimeAccessList( l );
    false
    true
    true
    gap> IsSSortedList( l );
    false

##### 30.3-10 AsSSortedList

`‣ AsSSortedList`( listorcoll ) | ( attribute )

`‣ AsSet`( listorcoll ) | ( attribute )

`AsSSortedList` ("as strictly sorted list") returns a dense, immutable, and duplicate free list imm. The argument must be a collection or list listorcoll which may contain holes but whose elements lie in the same family (see 13.1). `Length( imm )` is the number of different elements of listorcoll, and imm contains the different elements in strictly sorted order, w.r.t. `\<` (31.11-1). `imm[pos]` executes in constant time (see `IsConstantTimeAccessList` (21.1-6)), and the size of imm in memory is proportional to its length.

Because the comparisons required for sorting can be very expensive for some kinds of objects, you should use `AsList` (30.3-8) instead if you do not require the result to be sorted.

The only difference to the operation `SSortedList` (30.3-7) is that `AsSSortedList` returns an immutable list.

`AsSet` is simply a synonym for `AsSSortedList`.

In general a function that returns a set of elements is free, in fact encouraged, to return a domain instead of the proper set of its elements. This allows one to keep a given structure, and moreover the representation by a domain object is usually more space efficient. `AsSSortedList` must of course not do this, its only purpose is to create the proper set of elements.

    gap> l:= AsSSortedList( l );
    [ 1, 2, 3 ]
    gap> IsMutable( l );  IsSSortedList( l );  IsConstantTimeAccessList( l );
    false
    true
    true
    gap> AsSSortedList( Group( (1,2,3), (1,2) ) );
    [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]

##### 30.3-11 Elements

`‣ Elements`( C ) | ( function )

`Elements` does the same as `AsSSortedList` (30.3-10), that is, the return value is a strictly sorted list of the elements in the list or collection C.

`Elements` is only supported for backwards compatibility. In many situations, the sortedness of the "element list" for a collection is in fact not needed, and one can save a lot of time by asking for a list that is not necessarily sorted, using `AsList` (30.3-8). If one is really interested in the strictly sorted list of elements in C then one should use `AsSet` (30.3-10) or `AsSSortedList` (30.3-10) instead.

#### 30.4 Attributes and Properties for Collections

##### 30.4-1 IsEmpty

`‣ IsEmpty`( listorcoll ) | ( property )

`IsEmpty` returns `true` if the collection or list listorcoll is empty (that is it contains no elements), and `false` otherwise.

##### 30.4-2 IsFinite

`‣ IsFinite`( C ) | ( property )

`IsFinite` returns `true` if the collection C is finite, and `false` otherwise.

The default method for `IsFinite` checks the size (see `Size` (30.4-6)) of C.

Methods for `IsFinite` may call `Size` (30.4-6), but methods for `Size` (30.4-6) must not call `IsFinite`.

##### 30.4-3 IsTrivial

`‣ IsTrivial`( C ) | ( property )

`IsTrivial` returns `true` if the collection C consists of exactly one element.

##### 30.4-4 IsNonTrivial

`‣ IsNonTrivial`( C ) | ( property )

`IsNonTrivial` returns `true` if the collection C is empty or consists of at least two elements (see `IsTrivial` (30.4-3)).

    gap> IsEmpty( [] ); IsEmpty( [ 1 .. 100 ] ); IsEmpty( Group( (1,2,3) ) );
    true
    false
    false
    gap> IsFinite( [ 1 .. 100 ] );  IsFinite( Integers );
    true
    false
    gap> IsTrivial( Integers );  IsTrivial( Group( () ) );
    false
    true
    gap> IsNonTrivial( Integers );  IsNonTrivial( Group( () ) );
    true
    false

##### 30.4-5 IsWholeFamily

`‣ IsWholeFamily`( C ) | ( property )

`IsWholeFamily` returns `true` if the collection C contains the whole family (see 13.1) of its elements.

    gap> IsWholeFamily( Integers )
    >    ;  # all rationals and cyclotomics lie in the family
    false
    gap> IsWholeFamily( Integers mod 3 )
    >    ;  # all finite field elements in char. 3 lie in this family
    false
    gap> IsWholeFamily( Integers mod 4 );
    true
    gap> IsWholeFamily( FreeGroup( 2 ) );
    true

##### 30.4-6 Size

`‣ Size`( listorcoll ) | ( attribute )

`Size` returns the size of the list or collection listorcoll, which is either an integer or `infinity` (18.2-1). If the argument is a list then the result is its length (see `Length` (21.17-5)).

The default method for `Size` checks the length of an enumerator of listorcoll.

Methods for `IsFinite` (30.4-2) may call `Size`, but methods for `Size` must not call `IsFinite` (30.4-2).

    gap> Size( [1,2,3] );  Size( Group( () ) );  Size( Integers );
    3
    1
    infinity

##### 30.4-7 Representative

`‣ Representative`( C ) | ( attribute )

`Representative` returns a representative of the collection C.

Note that `Representative` is free in choosing a representative if there are several elements in C. It is not even guaranteed that `Representative` returns the same representative if it is called several times for one collection. The main difference between `Representative` and `Random` (30.7-1) is that `Representative` is free to choose a value that is cheap to compute, while `Random` (30.7-1) must make an effort to randomly distribute its answers.

If C is a domain then there are methods for `Representative` that try to fetch an element from any known generator list of C, see 31. Note that `Representative` does not try to compute generators of C, thus `Representative` may give up and signal an error if C has no generators stored at all.

##### 30.4-8 RepresentativeSmallest

`‣ RepresentativeSmallest`( C ) | ( attribute )

returns the smallest element in the collection C, w.r.t. the ordering `\<` (31.11-1). While the operation defaults to comparing all elements, better methods are installed for some collections.

    gap> Representative( Rationals );
    0
    gap> Representative( [ -1, -2 .. -100 ] );
    -1
    gap> RepresentativeSmallest( [ -1, -2 .. -100 ] );
    -100

#### 30.5 Operations for Collections

##### 30.5-1 IsSubset

`‣ IsSubset`( C1, C2 ) | ( operation )

`IsSubset` returns `true` if C2, which must be a collection, is a subset of C1, which also must be a collection, and `false` otherwise.

C2 is considered a subset of C1 if and only if each element of C2 is also an element of C1. That is `IsSubset` behaves as if implemented as `IsSubsetSet( AsSSortedList( C1 ), AsSSortedList( C2 ) )`, except that it will also sometimes, but not always, work for infinite collections, and that it will usually work much faster than the above definition. Either argument may also be a proper set (see 21.19).

    gap> IsSubset( Rationals, Integers );
    true
    gap> IsSubset( Integers, [ 1, 2, 3 ] );
    true
    gap> IsSubset( Group( (1,2,3,4) ), [ (1,2,3) ] );
    false

##### 30.5-2 Intersection

`‣ Intersection`( C1, C2, ... ) | ( function )

`‣ Intersection`( list ) | ( function )

`‣ Intersection2`( C1, C2 ) | ( operation )

In the first form `Intersection` returns the intersection of the collections C1, C2, etc. In the second form list must be a nonempty list of collections and `Intersection` returns the intersection of those collections. Each argument or element of list respectively may also be a homogeneous list that is not a proper set, in which case `Intersection` silently applies `Set` (30.3-7) to it first.

The result of `Intersection` is the set of elements that lie in every of the collections C1, C2, etc. If the result is a list then it is mutable and new, i.e., not identical to any of C1, C2, etc.

Methods can be installed for the operation `Intersection2` that takes only two arguments. `Intersection` calls `Intersection2`.

Methods for `Intersection2` should try to maintain as much structure as possible, for example the intersection of two permutation groups is again a permutation group.

    gap> # this is one of the rare cases where the intersection of two
    gap> # infinite domains works ('CF' is a shorthand for 'CyclotomicField'):
    gap> Intersection( CyclotomicField(9), CyclotomicField(12) );
    CF(3)
    gap> D12 := Group( (2,6)(3,5), (1,2)(3,6)(4,5) );;
    gap> Intersection( D12, Group( (1,2), (1,2,3,4,5) ) );
    Group([ (1,5)(2,4) ])
    gap> Intersection( D12, [ (1,3)(4,6), (1,2)(3,4) ] )
    >    ;  # note that the second argument is not a proper set
    [ (1,3)(4,6) ]
    gap> # although the result is mathematically a group it is returned as a
    gap> # proper set because the second argument is not regarded as a group:
    gap> Intersection( D12, [ (), (1,2)(3,4), (1,3)(4,6), (1,4)(5,6) ] );
    [ (), (1,3)(4,6) ]
    gap> Intersection( Group( () ), [1,2,3] );
    [  ]
    gap> Intersection( [2,4,6,8,10], [3,6,9,12,15], [5,10,15,20,25] )
    >    ;  # two or more lists or collections as arguments are legal
    [  ]
    gap> Intersection( [ [1,2,4], [2,3,4], [1,3,4] ] )
    >    ;  # or one list of lists or collections
    [ 4 ]

##### 30.5-3 Union

`‣ Union`( C1, C2, ... ) | ( function )

`‣ Union`( list ) | ( function )

`‣ Union2`( C1, C2 ) | ( operation )

In the first form `Union` returns the union of the collections C1, C2, etc. In the second form list must be a list of collections and `Union` returns the union of those collections. Each argument or element of list respectively may also be a homogeneous list that is not a proper set, in which case `Union` silently applies `Set` (30.3-7) to it first.

The result of `Union` is the set of elements that lie in any of the collections C1, C2, etc. If the result is a list then it is mutable and new, i.e., not identical to any of C1, C2, etc.

Methods can be installed for the operation `Union2` that takes only two arguments. `Union` calls `Union2`.

    gap> Union( [ (1,2,3), (1,2,3,4) ], Group( (1,2,3), (1,2) ) );
    [ (), (2,3), (1,2), (1,2,3), (1,2,3,4), (1,3,2), (1,3) ]
    gap> Union( [2,4,6,8,10], [3,6,9,12,15], [5,10,15,20,25] )
    >    ;  # two or more lists or collections as arguments are legal
    [ 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 20, 25 ]
    gap> Union( [ [1,2,4], [2,3,4], [1,3,4] ] )
    >    ;  # or one list of lists or collections
    [ 1 .. 4 ]
    gap> Union( [ ] );
    [  ]

When computing the Union of lists or sets of small integers and ranges, every attempt is made to return the result as a range and to avoid expanding ranges provided as input.

##### 30.5-4 Difference

`‣ Difference`( C1, C2 ) | ( operation )

`Difference` returns the set difference of the collections C1 and C2. Either argument may also be a homogeneous list that is not a proper set, in which case `Difference` silently applies `Set` (30.3-7) to it first.

The result of `Difference` is the set of elements that lie in C1 but not in C2. Note that C2 need not be a subset of C1. The elements of C2, however, that are not elements of C1 play no role for the result. If the result is a list then it is mutable and new, i.e., not identical to C1 or C2.

    gap> Difference( [ (1,2,3), (1,2,3,4) ], Group( (1,2,3), (1,2) ) );
    [ (1,2,3,4) ]

#### 30.6 Membership Test for Collections

##### `30.6-1 \in`

`‣ \in`( obj, C ) | ( operation )

returns `true` if the object obj lies in the collection C, and `false` otherwise.

The infix version of the command

obj `in` C

calls the operation `\in`, for which methods can be installed.

    gap> 13 in Integers;  [ 1, 2 ] in Integers;
    true
    false
    gap> g:= Group( (1,2) );;  (1,2) in g;  (1,2,3) in g;
    true
    false

#### 30.7 Random Elements

The method used by GAP to obtain random elements may depend on the type object.

Most methods which produce random elements in GAP use a global random number generator (see `GlobalMersenneTwister` (14.7-4)). This random number generator is (deliberately) initialized to the same values when GAP is started, so different runs of GAP with the same input will always produce the same result, even if random calculations are involved.

See `Reset` (14.7-3) for a description of how to reset the random number generator to a previous state.

##### 30.7-1 Random

`‣ Random`( listorcoll ) | ( operation )

`‣ Random`( from, to ) | ( operation )

`Random` returns a (pseudo-)random element of the dense, nonempty list or nonempty collection listorcoll. The behaviour for non-dense or empty lists, and for empty collections (see `IsDenseList` (21.1-2), `IsEmpty` (30.4-1)) is undefined.

As lists or ranges are restricted in length (\(2^{28}-1\) or \(2^{60}-1\) depending on your system), the second form returns a random integer in the range from to to (inclusive) for arbitrary integers from and to. The behaviour in the case that from is larger than to is undefined.

See Section 14.7 for more about computing random elements, in particular for `Random` (14.7-2) methods that take a random source as the first argument.

The distribution of elements returned by `Random` depends on the argument. For a dense, nonempty list the distribution is uniform (all elements are equally likely). The same holds usually for finite collections that are not lists. For infinite collections some reasonable distribution is used.

See the chapters of the various collections to find out which distribution is being used.

For some collections ensuring a reasonable distribution can be difficult and require substantial runtime (for example for large finite groups). If speed is more important than a guaranteed distribution, the operation `PseudoRandom` (30.7-2) should be used instead.

Note that `Random` is of course not an attribute.

    gap> Random([1..6]);
    6
    gap> Random(1, 2^100);
    866227015645295902682304086250
    gap> g:= Group( (1,2,3) );;  Random( g );  Random( g );
    (1,3,2)
    ()
    gap> Random(Rationals);
    -4

##### 30.7-2 PseudoRandom

`‣ PseudoRandom`( listorcoll ) | ( operation )

`PseudoRandom` returns a pseudo random element of the list or collection listorcoll, which can be roughly described as follows. For a list, `PseudoRandom` returns the same as `Random` (30.7-1). For collections that are not lists, the elements returned by `PseudoRandom` are not necessarily equally distributed, even for finite collections; the idea is that `Random` (30.7-1) returns elements according to a reasonable distribution, `PseudoRandom` returns elements that are cheap to compute but need not satisfy this strong condition, and `Representative` (30.4-7) returns arbitrary elements, probably the same element for each call.

##### 30.7-3 RandomList

`‣ RandomList`( [rs, ]list ) | ( function )

For a dense list list, `RandomList` returns a (pseudo-)random element with equal distribution.

The random source rs (see 14.7) is used to choose a random number. If rs is absent, this function uses the `GlobalMersenneTwister` (14.7-4) to produce the random elements (a source of high quality random numbers).

    gap> RandomList( [ 1 .. 6 ] );
    3
    gap> elms:= AsList( Group( (1,2,3) ) );;
    gap> RandomList( elms );  RandomList( elms );
    (1,2,3)
    (1,3,2)
    gap> rs:= RandomSource( IsMersenneTwister, 1 );
    <RandomSource in IsMersenneTwister>
    gap> RandomList( rs, elms );
    (1,2,3)

#### 30.8 Iterators

##### 30.8-1 Iterator

`‣ Iterator`( listorcoll ) | ( operation )

`‣ IsStandardIterator`( listorcoll ) | ( filter )

Iterators provide a possibility to loop over the elements of a (countable) collection or list listorcoll, without repetition. For many collections \(C\), an iterator of \(C\) need not store all elements of \(C\), for example it is possible to construct an iterator of some infinite domains, such as the field of rational numbers.

`Iterator` returns a mutable iterator \(iter\) for its argument. If this argument is a list (which may contain holes), then \(iter\) iterates over the elements (but not the holes) of this list in the same order (see `IteratorList` (30.8-6) for details). If this argument is a collection but not a list then \(iter\) iterates over the elements of this collection in an unspecified order, which may change for repeated calls of `Iterator`. Because iterators returned by `Iterator` are mutable (see 12.6), each call of `Iterator` for the same argument returns a new iterator. Therefore `Iterator` is not an attribute (see 13.5).

The only operations for iterators are `IsDoneIterator` (30.8-4), `NextIterator` (30.8-5), and `ShallowCopy` (12.7-1). In particular, it is only possible to access the next element of the iterator with `NextIterator` (30.8-5) if there is one, and this can be checked with `IsDoneIterator` (30.8-4) For an iterator \(iter\), `ShallowCopy` (12.7-1) returns a mutable iterator \(new\) that iterates over the remaining elements independent of \(iter\); the results of `IsDoneIterator` (30.8-4) for \(iter\) and \(new\) are equal, and if \(iter\) is mutable then also the results of `NextIterator` (30.8-5) for \(iter\) and \(new\) are equal; note that `=` is not defined for iterators, so the equality of two iterators cannot be checked with `=`.

When `Iterator` is called for a mutable collection \(C\) then it is not defined whether \(iter\) respects changes to \(C\) occurring after the construction of \(iter\), except if the documentation explicitly promises a certain behaviour. The latter is the case if the argument is a mutable list (see `IteratorList` (30.8-6) for subtleties in this case).

It is possible to have `for`-loops run over mutable iterators instead of lists.

In some situations, one can construct iterators with a special succession of elements, see `IteratorByBasis` (61.6-6) for the possibility to loop over the elements of a vector space w.r.t. a given basis.

For lists, `Iterator` is implemented by `IteratorList` (30.8-6). For collections \(C\) that are not lists, the default method is `IteratorList( Enumerator(` \(C\)`) )`. Better methods depending on \(C\) should be provided if possible.

For random access to the elements of a (possibly infinite) collection, enumerators are used. See 21.23 for the facility to compute a list from \(C\), which provides a (partial) mapping from \(C\) to the positive integers.

The filter `IsStandardIterator` means that the iterator is implemented as a component object and has components `IsDoneIterator` and `NextIterator` which are bound to the methods of the operations of the same name for this iterator.

    gap> iter:= Iterator( GF(5) );
    <iterator>
    gap> l:= [];;
    gap> for i in iter do Add( l, i ); od; l;
    [ 0*Z(5), Z(5)^0, Z(5), Z(5)^2, Z(5)^3 ]
    gap> iter:= Iterator( [ 1, 2, 3, 4 ] );;  l:= [];;
    gap> for i in iter do
    >      new:= ShallowCopy( iter );
    >      for j in new do Add( l, j ); od;
    >    od; l;
    [ 2, 3, 4, 3, 4, 4 ]

##### 30.8-2 IteratorSorted

`‣ IteratorSorted`( listorcoll ) | ( operation )

`IteratorSorted` returns a mutable iterator. The argument must be a collection or a list that is not necessarily dense but whose elements lie in the same family (see 13.1). It loops over the different elements in sorted order.

For a collection \(C\) that is not a list, the generic method is `IteratorList( EnumeratorSorted(` C`) )`.

##### 30.8-3 IsIterator

`‣ IsIterator`( obj ) | ( category )

Every iterator lies in the category `IsIterator`.

##### 30.8-4 IsDoneIterator

`‣ IsDoneIterator`( iter ) | ( operation )

If iter is an iterator for the list or collection \(C\) then `IsDoneIterator( iter )` is `true` if all elements of \(C\) have been returned already by `NextIterator( iter )`, and `false` otherwise.

##### 30.8-5 NextIterator

`‣ NextIterator`( iter ) | ( operation )

Let iter be a mutable iterator for the list or collection \(C\). If `IsDoneIterator( iter )` is `false` then `NextIterator` is applicable to iter, and the result is the next element of \(C\), according to the succession defined by iter.

If `IsDoneIterator( iter )` is `true` then it is not defined what happens when `NextIterator` is called for iter; that is, it may happen that an error is signalled or that something meaningless is returned, or even that GAP crashes.

    gap> iter:= Iterator( [ 1, 2, 3, 4 ] );
    <iterator>
    gap> sum:= 0;;
    gap> while not IsDoneIterator( iter ) do
    >      sum:= sum + NextIterator( iter );
    >    od;
    gap> IsDoneIterator( iter ); sum;
    true
    10
    gap> ir:= Iterator( Rationals );;
    gap> l:= [];; for i in [1..20] do Add( l, NextIterator( ir ) ); od; l;
    [ 0, 1, -1, 1/2, 2, -1/2, -2, 1/3, 2/3, 3/2, 3, -1/3, -2/3, -3/2, -3,
      1/4, 3/4, 4/3, 4, -1/4 ]
    gap> for i in ir do
    >      if DenominatorRat( i ) > 10 then break; fi;
    >    od;
    gap> i;
    1/11

##### 30.8-6 IteratorList

`‣ IteratorList`( list ) | ( function )

`IteratorList` returns a new iterator that allows iteration over the elements of the list list (which may have holes) in the same order.

If list is mutable then it is in principle possible to change list after the call of `IteratorList`. In this case all changes concerning positions that have not yet been reached in the iteration will also affect the iterator. For example, if list is enlarged then the iterator will iterate also over the new elements at the end of the changed list.

Note that changes of list will also affect all shallow copies of list.

##### 30.8-7 TrivialIterator

`‣ TrivialIterator`( elm ) | ( function )

is a mutable iterator for the collection `[ elm ]` that consists of exactly one element elm (see `IsTrivial` (30.4-3)).

##### 30.8-8 IteratorByFunctions

`‣ IteratorByFunctions`( record ) | ( function )

`IteratorByFunctions` returns a (mutable) iterator iter for which `NextIterator` (30.8-5), `IsDoneIterator` (30.8-4), and `ShallowCopy` (12.7-1) are computed via prescribed functions.

Let record be a record with at least the following components.

`NextIterator`

  a function taking one argument iter, which returns the next element of iter (see `NextIterator` (30.8-5)); for that, the components of iter are changed,

`IsDoneIterator`

  a function taking one argument iter, which returns the `IsDoneIterator` (30.8-4) value of iter,

`ShallowCopy`

  a function taking one argument iter, which returns a record for which `IteratorByFunctions` can be called in order to create a new iterator that is independent of iter but behaves like iter w.r.t. the operations `NextIterator` (30.8-5) and `IsDoneIterator` (30.8-4).

`ViewObj` and `PrintObj`

  two functions that print what one wants to be printed when `View( iter )` or `Print( item )` is called (see 6.3), if the `ViewObj` component is missing then the `PrintObj` method is used as a default.

Further (data) components may be contained in record which can be used by these function.

`IteratorByFunctions` does not make a shallow copy of record, this record is changed in place (see Section 79.1).

Iterators constructed with `IteratorByFunctions` are in the filter `IsStandardIterator` (30.8-1).

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