12 Objects and Elements

IsObject returns true if the object obj is an object. Obviously it can never return false.

It can be used as a filter in InstallMethod (see Method Installation in ``Programming in GAP'') when one of the arguments can be anything.

12.2 Elements as equivalence classes

The equality operation ``='' defines an equivalence relation on all GAP objects. The equivalence classes are called elements.

There are basically three reasons to regard different objects as equal. Firstly the same information may be stored in different places. Secondly the same information may be stored in different ways; for example, a polynomial can be stored sparsely or densely. Thirdly different information may be equal modulo a mathematical equivalence relation. For example, in a finitely presented group with the relation a² = 1 the different objects a and a³ describe the same element.

As an example of all three reasons, consider the possibility of storing an integer in several places of the memory, of representing it as a fraction with denominator 1, or of representing it as a fraction with any denominator, and numerator a suitable multiple of the denominator.

12.3 Sets

In GAP there is no category whose definition corresponds to the mathematical property of being a set, however in the manual we will often refer to an object as a set in order to convey the fact that mathematically, we are thinking of it as a set. In particular, two sets A and B are equal if and only if, x Î A Û x Î B.

There are two types of object in GAP which exhibit this kind of behaviour with respect to equality, namely domains (see Section Domains) and lists whose elements are strictly sorted see IsSSortedList (see IsSSortedList). In general, set in this manual will mean an object of one of these types.

More precisely: two domains can be compared with ``='', the answer being true if and only if the sets of elements are equal (regardless of any additional structure) and; a domain and a list can be compared with ``='', the answer being true if and only if the list is equal to the strictly sorted list of elements of the domain.

A discussion about sorted lists and sets can be found in the Reference Manual section ``Sorted Lists and Sets'' Sorted Lists and Sets.

12.4 Domains

An especially important class of objects in GAP are those whose underlying mathematical abstraction is that of a structured set, for example a group, a conjugacy class, or a vector space. Such objects are called domains. The equality relation between domains is always equality as sets, so that two domains are equal if and only if they contain the same elements.

Domains play a central role in GAP. In a sense, the only reason that GAP supports objects such as integers and permutations is the wish to form domains of them and compute the properties of those domains.

Domains are described in Chapter Domains and their Elements.

12.5 Identical Objects

Two objects that are equal as objects (that is they actually refer to the same area of computer memory) and not only w.r.t. the equality relation `=' are called identical. Identical objects do of course describe the same element.

IsIdenticalObj( obj1, obj2 ) tests whether the objects obj1 and obj2 are identical (that is they are either equal immediate objects or are both stored at the same location in memory.

If two copies of a simple constant object (see section Mutability and Copyability) are created, it is not defined whether GAP will actually store two equal but non-identical objects, or just a single object. For mutable objects, however, it is important to know whether two value refer to identical or non-identical objects, and the documentation of operations that return mutable values should make clear whether the values returned are new, or may be identical to values stored elsewhere.

gap> IsIdenticalObj( 10^6, 10^6);
true
gap> IsIdenticalObj( 10^12, 10^12);
false
gap> IsIdenticalObj( true, true);
true

Generally, one may compute with objects but think of the results in terms of the underlying elements because one is not interested in locations in memory, data formats or information beyond underlying equivalence relations. But there are cases where it is important to distinguish the relations identity and equality. This is best illustrated with an example. (The reader who is not familiar with lists in GAP, in particular element access and assignment, is referred to Chapter Lists.)

gap> l1:= [ 1, 2, 3 ];; l2:= [ 1, 2, 3 ];;
gap> l1 = l2;
true
gap> IsIdenticalObj( l1, l2 );
false
gap> l1[3]:= 4;; l1; l2;
[ 1, 2, 4 ]
[ 1, 2, 3 ]
gap> l1 = l2;
false

gap> l1:= [ 1, 2, 3 ];; l2:= l1;;
gap> l1 = l2;
true
gap> IsIdenticalObj( l1, l2 );
true
gap> l1[3]:= 4;; l1; l2;
[ 1, 2, 4 ]
[ 1, 2, 4 ]
gap> l1 = l2;
true

The library also provides:

tests whether the objects obj1 and objs2 are not identical.

12.6 Mutability and Copyability

An object in GAP is said to be immutable if its mathematical value (as defined by =) does not change under any operation. More explicitly, suppose a is immutable and O is some operation on a, then if a = b evaluates to true before executing O(a), a = b also evaluates to true afterwards. (Examples for operations O that change mutable objects are Add and Unbind which are used to change list objects, see Chapter Lists.) An immutable object may change, for example to store new information, or to adopt a more efficient representation, but this does not affect its behaviour under =.

There are two points here to note. Firstly, ``operation'' above refers to the functions and methods which can legitimately be applied to the object, and not the !. operation whereby virtually any aspect of any GAP level object may be changed. The second point which follows from this, is that when implementing new types of objects, it is the programmer's responsibility to ensure that the functions and methods they write never change immutable objects mathematically.

In fact, most objects with which one deals in GAP are immutable. For instance, the permutation (1,2) will never become a different permutation or a non-permutation (although a variable which previously had (1,2) stored in it may subsequently have some other value).

For many purposes, however, mutable objects are useful. These objects may be changed to represent different mathematical objects during their life. For example, mutable lists can be changed by assigning values to positions or by unbinding values at certain positions. Similarly, one can assign values to components of a mutable record, or unbind them.

If a mutable form of an object obj can be made in GAP, the object is called copyable. Examples of copyable objects are of course lists and records. A new mutable version of the object can always be obtained by the operation ShallowCopy (see Duplication of Objects).

Objects for which only an immutable form exists in GAP are called constants. Examples of constants are integers, permutations, and domains. Called with a constant as argument, Immutable and ShallowCopy return this argument.

tests whether obj is mutable.

If an object is mutable then it is also copyable (see IsCopyable), and a ShallowCopy (see ShallowCopy) method should be supplied for it. Note that IsMutable must not be implied by another filter, since otherwise Immutable would be able to create paradoxical objects in the sense that IsMutable for such an object is false but the filter that implies IsMutable is true.

In many situations, however, one wants to ensure that objects are immutable. For example, take the identity of a matrix group. Since this matrix may be referred to as the identity of the group in several places, it would be fatal to modify its entries, or add or unbind rows. We can obtain an immutable copy of an object with:

returns an immutable structural copy (see StructuralCopy) of obj in which the subobjects are immutable copies of the subobjects of obj. If obj is immutable then Immutable returns obj itself.

GAP will complain with an error if one tries to change an immutable object.

One can turn the (mutable or immutable) object obj into an immutable one with MakeImmutable; note that this also makes all subobjects of obj immutable, so one should call MakeImmutable only if obj and its mutable subobjects are newly created. If one is not sure about this, Immutable should be used.

Note that it is not possible to turn an immutable object into a mutable one; only mutable copies can be made (see Duplication of Objects).

Using Immutable, it is possible to store an immutable identity matrix or an immutable list of generators, and to pass around references to this immutable object safely. Only when a mutable copy is really needed does the actual object have to be duplicated. Compared to the situation without immutable objects, much unnecessary copying is avoided this way. Another advantage of immutability is that lists of immutable objects may remember whether they are sorted (see Sorted Lists and Sets), which is not possible for lists of mutable objects.

Since the operation Immutable must work for any object in GAP, it follows that an immutable form of every object must be possible, even if it is not sensible, and user-defined objects must allow for the possibility of becoming immutable without notice.

Another interesting example of mutable (and thus copyable) objects is provided by iterators, see Iterators. (Of course an immutable form of an iterator is not very useful, but clearly Immutable will yield such an object.) Every call of NextIterator changes a mutable iterator until it is exhausted, and this is the only way to change an iterator. ShallowCopy for an iterator iter is defined so as to return a mutable iterator that has no mutable data in common with iter, and that behaves equally to iter w.r.t. IsDoneIterator and (if iter is mutable) NextIterator. Note that this meaning of the ``shallow copy'' of an iterator that is returned by ShallowCopy is not as obvious as for lists and records, and must be explicitly defined.

Many operations return immutable results, among those in particular attributes (see Attributes). Examples of attributes are Size, Zero, AdditiveInverse, One, and Inverse. Arithmetic operations, such as the binary infix operations +, -, *, /, ^, mod, the unary -, and operations such as Comm and LeftQuotient, return mutable results, except if all arguments are immutable. So the product of two matrices or of a vector and a matrix is immutable if and only if the two matrices or both the vector and the matrix are immutable (see also Arithmetic for Lists). There is one exception to this rule, which arises where the result is less deeply nested that at least one of the argument, where mutable arguments may sometimes lead to an immutable result. For instance, a mutable matrix with immutable rows, multiplied by an immutable vector gives an immutable vector result. The exact rules are given in Arithmetic for Lists.

It should be noted that 0 * obj is equivalent to ZeroSM( obj ), -obj is equivalent to AdditiveInverseSM( obj ), obj^0 is equivalent to OneSM( obj), and obj^-1 is equivalent to InverseSM( obj ). The ``SM'' stands for ``same mutability'', and indicates that the result is mutable if and only if the argument is mutable.

The operations ZeroOp, AdditiveInverseOp, OneOp, and InverseOp return mutable results whenever a mutable version of the result exists, contrary to the attributes Zero, AdditiveInverse, One, and Inverse.

If one introduces new arithmetic objects then one need not install methods for the attributes One, Zero, etc. The methods for the associated operations OneOp and ZeroOp will be called, and then the results made immutable.

All methods installed for the arithmetic operations must obey the rule about the mutability of the result. This means that one may try to avoid the perhaps expensive creation of a new object if both operands are immutable, and of course no problems of this kind arise at all in the (usual) case that the objects in question do not admit a mutable form, i.e., that these objects are not copyable.

In a few, relatively low-level algorithms, one wishes to treat a matrix partly as a data structure, and manipulate and change its entries. For this, the matrix needs to be mutable, and the rule that attribute values are immutable is an obstacle. For these situations, a number of additional operations are provided, for example TransposedMatMutable constructs a mutable matrix (contrary to the attribute TransposedMat), while TriangulizeMat modifies a mutable matrix (in place) into upper triangular form.

Note that being immutable does not forbid an object to store knowledge. For example, if it is found out that an immutable list is strictly sorted then the list may store this information. More precisely, an immutable object may change in any way, provided that it continues to represent the same mathematical object.

12.7 Duplication of Objects

If GAP supports a mutable form of the object obj (see Mutability and Copyability) then this is obtained by ShallowCopy. Otherwise ShallowCopy returns obj itself.

The subobjects of ShallowCopy( obj ) are identical to the subobjects of obj. Note that if the object returned by ShallowCopy is mutable then it is always a new object. In particular, if the return value is mutable, then it is not identical with the argument obj, no matter whether obj is mutable or immutable. But of course the object returned by ShallowCopy is equal to obj w.r.t. the equality operator =.

Since ShallowCopy is an operation, the concrete meaning of ``subobject'' depends on the type of obj. But for any copyable object obj, the definition should reflect the idea of ``first level copying''.

The definition of ShallowCopy for lists (in particular for matrices) can be found in Duplication of Lists.

In a few situations, one wants to make a structural copy scp of an object obj. This is defined as follows. scp and obj are identical if obj is immutable. Otherwise, scp is a mutable copy of obj such that each subobject of scp is a structural copy of the corresponding subobject of obj. Furthermore, if two subobjects of obj are identical then also the corresponding subobjects of scp are identical.

gap> obj:= [ [ 0, 1 ] ];;
gap> obj[2]:= obj[1];;
gap> obj[3]:= Immutable( obj[1] );;
gap> scp:= StructuralCopy( obj );;
gap> scp = obj; IsIdenticalObj( scp, obj );
true
false
gap> IsIdenticalObj( scp[1], obj[1] );
false
gap> IsIdenticalObj( scp[3], obj[3] );
true
gap> IsIdenticalObj( scp[1], scp[2] );
true

That both ShallowCopy and StructuralCopy return the argument obj itself if it is not copyable is consistent with this definition, since there is no way to change obj by modifying the result of any of the two functions, because in fact there is no way to change this result at all.

12.8 Other Operations Applicable to any Object

There are a number of general operations which can be applied, in principle, to any object in GAP. Some of these are documented elsewhere -- see String, PrintObj and Display. Others are mainly somewhat technical.

for a suitable object obj sets that object to have name name (a string).

returns the name, a string, previously assigned to obj via a call to SetName (see SetName). The name of an object is used only for viewing the object via this name.

There are no methods installed for computing names of objects, but the name may be set for suitable objects, using SetName.

gap> g := Group((1,2,3),(1,2));
Group([ (1,2,3), (1,2) ])
gap> SetName(g, "S3");
gap> g;
S3
gap> Name(g);
"S3"

For debugging purposes, it may be useful to check the consistency of an object obj that is composed from other (composed) objects.

There is a default method of IsInternallyConsistent, with rank zero, that returns true. So it is possible (and recommended) to check the consistency of subobjects of obj recursively by IsInternallyConsistent.

(Note that IsInternallyConsistent is not an attribute.)

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GAP 4 manual
May 2002