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# 28 Collections

### Sections

1. Collection Families
2. Lists and Collections
3. Attributes and Properties for Collections
4. Operations for Collections
5. Membership Test for Collections
6. Random Elements
7. Iterators

A collection in GAP consists of elements in the same family (see Families). The most important kinds of collections are homogeneous lists (see Lists) and domains (see Domains). 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` (see Size) and `Enumerator` (see Enumerator); for finite collections, `Enumerator` admits to delegate the other operations for collections (see Attributes and Properties for Collections and Operations for Collections) to functions for lists (see Lists). Obviously, special methods depending on the arguments are needed for the computation of e.g. the intersection of two infinite domains.

- `IsCollection(` obj`) C`

  tests whether an object is a collection.

  Some of the functions for lists and collections have been described in the chapter about lists, mainly in Section Operations for Lists. In this chapter, we describe those functions for which the ``collection aspect'' seems to be more important than the ``list aspect''. As in Chapter Lists, an argument that is a list will be denoted by list, and an argument that is a collection will be denoted by C.

  ## 28.1 Collection Families

- `CollectionsFamily(` Fam`) A`

  For a family Fam, `CollectionsFamily` returns the family of all collections that consist of elements in Fam.

  Note that families (see Families) are used to describe relations between objects. Important such relations are that between an element elm and each collection of elements that lie in the same family as elm, 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`)`.

- `IsCollectionFamily(` Fam`) C`

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

- `ElementsFamily(` Fam`) A`

  returns the family from which the collections family Fam was created by `CollectionsFamily`. The way a collections family is created, it always has its elements family stored. If Fam is not a collections family (see IsCollectionFamily) 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) ] );  collfam = FamilyObj( Group( () ) );
      true
      true
      gap> collfam = CollectionsFamily( collfam );
      false

- `CategoryCollections(` filter`) F`

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

  For example, the construction of `PermutationsFamily` guarantees that each of its elements lies in the filter `IsPerm`, and each collection of permutations lies in the category `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.

  ## 28.2 Lists and Collections

- `IsListOrCollection(` obj`) C`

  Several functions are defined for both lists and collections, for example `Intersection` (see Intersection), `Iterator` (see Iterator), and `Random` (see Random). `IsListOrCollection` is a supercategory of `IsList` and `IsCollection` (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.

  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 Mutability and Copyability), guaranteed to be sorted, or guaranteed to admit list access in constant time (see IsConstantTimeAccessList).

- `Enumerator(` C`) A`
- `Enumerator(` list`) A`

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

  For lists list, the default method is `Immutable`. For collections C that are not lists, there is no default method.

- `EnumeratorSorted(` C`) A`
- `EnumeratorSorted(` list`) A`

  `EnumeratorSorted` returns an immutable list enum. The argument must be a collection C or a list list which may contain holes but whose elements lie in the same family (see Families). `Length(` enum`)` is the number of different elements of C resp. list, and enum contains the different elements in sorted order, w.r.t. `<`. ``enum`[`pos`]` may not execute in constant time (see IsConstantTimeAccessList), 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 ]

- `EnumeratorByFunctions(` D`,` record`) F`
- `EnumeratorByFunctions(` Fam`,` record`) F`

  `EnumeratorByFunctions` returns an immutable, dense, and duplicate-free list enum for which `IsBound`, element access, `Length`, and `Position` 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 Basic Operations for Lists); 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); 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`, 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`, `IsBound`, and `PrintObj` 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).

  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 Basic Operations for Lists); if this component is missing then `Length` 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 Basic Operations for Lists); 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); if this component is missing then `ConstantTimeAccessList` 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 View and Print), 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) 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` (see Enumerator), then it should be installed also as a method for `EnumeratorSorted` (see EnumeratorSorted).

  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 Creating Objects in ``Programming in GAP'').

  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` is supported), but the resulting overhead for the methods seems not to be justified.

  `List(` C`)`
  `List(` list`)`

  This function is described in List, together with the probably more frequently used version which takes a function as second argument and returns the list of function values of the list elements.

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

- `SortedList(` C`) O`
- `SortedList(` list`) O`

  `SortedList` returns a new mutable and dense list new. The argument must be a collection C or a list list which may contain holes but whose elements lie in the same family (see Families). `Length(` new`)` is the number of elements of C resp. list, and new contains the elements in sorted order, w.r.t. `<=`. ``new`[`pos`]` executes in constant time (see IsConstantTimeAccessList), 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 ]

- `SSortedList(` C`) O`
- `SSortedList(` list`) O`
- `Set(` C`) O`

  `SSortedList` (``strictly sorted list'') returns a new dense, mutable, and duplicate free list new. The argument must be a collection C or a list list which may contain holes but whose elements lie in the same family (see Families). `Length(` new`)` is the number of different elements of C resp. list, and new contains the different elements in strictly sorted order, w.r.t. `<`. ``new`[`pos`]` executes in constant time (see IsConstantTimeAccessList), 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( [ 1, 2, 1,, 3, 2 ] );
      [ 1, 2, 3 ]

- `AsList(` C`) A`
- `AsList(` list`) A`

  `AsList` returns a immutable list imm. If the argument is a list list (which may contain holes), then `Length(` imm`)` is `Length(` list`)`, and imm contains the elements (and holes) of list in the same order. If the argument is a collection C that is not a list, then `Length(` imm`)` is the number of different elements of C, and imm contains the different elements of C in an unspecified order, which may change for repeated calls of `AsList`. ``imm`[`pos`]` executes in constant time (see IsConstantTimeAccessList), 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`.

      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) ]

- `AsSortedList(` C`) A`
- `AsSortedList(` list`) A`

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

  The only difference to the operation `SortedList` (see SortedList) 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

- `AsSSortedList(` C`) A`
- `AsSSortedList(` list`) A`
- `AsSet(` C`) A`

  `AsSSortedList` (``as strictly sorted list'') returns a dense, immutable, and duplicate free list imm. The argument must be a collection C or a list list which may contain holes but whose elements lie in the same family (see Families). `Length(` imm`)` is the number of different elements of C resp. list, and imm contains the different elements in strictly sorted order, w.r.t. `<`. ``imm`[`pos`]` executes in constant time (see IsConstantTimeAccessList), 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` instead if you do not require the result to be sorted.

  The only difference to the operation `SSortedList` (see SSortedList) 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) ]

- `Elements(` C`) F`

  `Elements` does the same as `AsSSortedList` (see AsSSortedList), 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` (see AsList). If one is really interested in the strictly sorted list of elements in C then one should use `AsSet` or `AsSSortedList` instead.

  ## 28.3 Attributes and Properties for Collections

- `IsEmpty(` C`) P`
- `IsEmpty(` list`) P`

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

- `IsFinite(` C`) P`

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

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

  Methods for `IsFinite` may call `Size`, but methods for `Size` must not call `IsFinite`.

- `IsTrivial(` C`) P`

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

- `IsNonTrivial(` C`) P`

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

      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

- `IsWholeFamily(` C`) P`

  `IsWholeFamily` returns `true` if the collection C contains the whole family (see Families) 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

- `Size(` C`) A`
- `Size(` list`) A`

  `Size` returns the size of the collection C, which is either an integer or `infinity`. The argument may also be a list list, in which case the result is the length of list (see Length).

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

  Methods for `IsFinite` may call `Size`, but methods for `Size` must not call `IsFinite`.

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

- `Representative(` C`) A`

  `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` (see Random) is that `Representative` is free to choose a value that is cheap to compute, while `Random` 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 Domains and their Elements. 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.

- `RepresentativeSmallest(` C`) A`

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

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

  ## 28.4 Operations for Collections

- `IsSubset(` C1`,` C2`) O`

  `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 Sorted Lists and Sets).

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

- `Intersection(` C1`,` C2`... ) F`
- `Intersection(` list`) F`
- `Intersection2(` C1`,` C2`) O`

  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` (see Set) 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> Intersection( CyclotomicField(9), CyclotomicField(12) )
      >       # this is one of the rare cases where the intersection of two infinite
      >    ;  # domains works (`CF' is a shorthand for `CyclotomicField')
      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> Intersection( D12, [ (), (1,2)(3,4), (1,3)(4,6), (1,4)(5,6) ] )
      >       # although the result is mathematically a group it is returned as a
      >    ;  # proper set because the second argument is not regarded as a group
      [ (), (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 ]

- `Union(` C1`,` C2`... ) F`
- `Union(` list`) F`
- `Union2(` C1`,` C2`) O`

  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` (see Set) 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, 2, 3, 4 ]
      gap> Union( [ ] );
      [  ]

- `Difference(` C1`,` C2`) O`

  `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` (see Set) 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) ]

  ## 28.5 Membership Test for Collections

- ``obj `in` C``
- `\in(` obj`,` C`) O`

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

  The infix version of the command 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

  ## 28.6 Random Elements

- `Random(` C`) O`
- `Random(` list`) O`

  `Random` returns a (pseudo-)random element of the collection C respectively the list list.

  The distribution of elements returned by `Random` depends on the argument. For a list list, all elements are equally likely. The same holds usually for finite collections C that are not lists. For infinite collections C 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. If speed at the cost of equal distribution is desired, the operation `PseudoRandom` should be used instead.

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

      gap> Random(Rationals);
      -4
      gap> g:= Group( (1,2,3) );;  Random( g );  Random( g );
      (1,3,2)
      (1,3,2)

- `StateRandom( ) F`
- `RestoreStateRandom(` obj`) F`

  For debugging purposes, it can be desirable to reset the random number generator to a state it had before. `StateRandom` returns a GAP object that represents the current state of the random number generator used by `RandomList`.

  By calling `RestoreStateRandom` with this object as argument, the random number is reset to this same state.

  (The same result can be obtained by accessing the two global variables `R_N` and `R_X`.)

  (The format of the object used to represent the random generator seed is not guaranteed to be stable betweed different machines or versions of GAP.

      gap> seed:=StateRandom();;
      gap> List([1..10],i->Random(Integers));
      [ -3, 2, 5, 1, 0, -2, 4, 3, 5, 3 ]
      gap> List([1..10],i->Random(Integers));
      [ 1, -2, 1, -1, -2, 4, -1, -3, -1, 1 ]
      gap> RestoreStateRandom(seed);
      gap> List([1..10],i->Random(Integers));
      [ -3, 2, 5, 1, 0, -2, 4, 3, 5, 3 ]

- `PseudoRandom(` C`) O`
- `PseudoRandom(` list`) O`

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

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

  Many random methods in the library are eventually based on the function `RandomList`. As `RandomList` is restricted to lists of up to 228 elements, this may create problems for very large collections. Also note that the method used by `RandomList` is intended to provide a fast algorithm rather than to produce high quality randomness for statistical purposes.

  If you implement your own `Random` methods we recommend that they initialize their seed to a defined value when they are loaded to permit to reproduce calculations even if they involved random elements.

- `RandomList(` list`) F`

  For a dense list list of up to 228 elements, `RandomList` returns a (pseudo-)random element with equal distribution.

  The algorithm used is an additive number generator (Algorithm A in section 3.2.2 of TACP2 with lag 30)

  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 `StateRandom` for a description on how to reset the random number generator to a previous state.

  ## 28.7 Iterators

- `Iterator(` C`) O`
- `Iterator(` list`) O`

  Iterators provide a possibility to loop over the elements of a (countable) collection C or a list list, 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 is a list list (which may contain holes), then iter iterates over the elements (but not the holes) of list in the same order (see IteratorList for details). If this is a collection C but not a list then iter iterates over the elements of C in an unspecified order, which may change for repeated calls of `Iterator`. Because iterators returned by `Iterator` are mutable (see Mutability and Copyability), each call of `Iterator` for the same argument returns a new iterator. Therefore `Iterator` is not an attribute (see Attributes).

  The only operations for iterators are `IsDoneIterator`, `NextIterator`, and `ShallowCopy`. In particular, it is only possible to access the next element of the iterator with `NextIterator` if there is one, and this can be checked with `IsDoneIterator` (see NextIterator). For an iterator iter, `ShallowCopy(` iter`)` is a mutable iterator new that iterates over the remaining elements independent of iter; the results of `IsDoneIterator` for iter and new are equal, and if iter is mutable then also the results of `NextIterator` 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 list (see IteratorList 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 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(` list`)`. For collections 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 Enumerators for the facility to compute a list from C, which provides a (partial) mapping from C to the positive integers.

      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 ]

- `IteratorSorted(` C`) O`
- `IteratorSorted(` list`) O`

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

  For collections C that are not lists, the generic method is `IteratorList( EnumeratorSorted(` C`) )`.

- `IsIterator(` obj`) C`

  Every iterator lies in the category `IsIterator`.

- `IsDoneIterator(` iter`) O`

  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.

- `NextIterator(` iter`) O`

  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 if `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.

- `IteratorList(` list`) F`

  `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.

- `TrivialIterator(` elm`) F`

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

      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

- `IteratorByFunctions(` record`) F`

  `IteratorByFunctions` returns a (mutable) iterator iter for which `NextIterator`, `IsDoneIterator`, and `ShallowCopy` 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); for that, the components of iter are changed,

  `IsDoneIterator`
    a function taking one argument iter, which returns `IsDoneIterator(` iter`)` (see IsDoneIterator);

  `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` and `IsDoneIterator`.
  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 Creating Objects in ``Programming in GAP'').

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  GAP 4 manual
  May 2005