28 Collections

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

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 ).

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

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

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

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 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 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 ]

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

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

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

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

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 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.

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 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

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.

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

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.

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

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.

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

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 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 );
()
(1,2,3)

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

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.

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

28.7 Iterators

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 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 ) ).

Every iterator lies in the category IsIterator.

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.

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 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.

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

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GAP 4 manual
May 2002