Every GAP object has a type. The type of an object is the information which is used to decide whether an operation is admissible or possible with that object as an argument, and if so, how it is to be performed (see Chapter Method Selection in ``Programming in GAP'').
For example, the types determine whether two objects can be multiplied and what function is called to compute the product. Analogously, the type of an object determines whether and how the size of the object can be computed. It is sometimes useful in discussing the type system, to identify types with the set of objects that have this type. Partial types can then also be regarded as sets, such that any type is the intersection of its parts.
The type of an object consists of two main parts, which describe different aspects of the object.
The family determines the relation of the object to other objects. For example, all permutations form a family. Another family consists of all collections of permutations, this family contains the set of permutation groups as a subset. A third family consists of all rational functions with coefficients in a certain family.
The other part of a type is a collection of filters (actually stored as a bit-list indicating, from the complete set of possible filters, which are included in this particular type). These filters are all treated equally by the method selection, but, from the viewpoint of their creation and use, they can be divided (with a small number of unimportant exceptions) into categories, representations, attribute testers and properties. Each of these is described in more detail below.
This chapter does not describe how types and their constituent parts can be created. Information about this topic can be found in ``Programming in GAP'' in Section Creating New Objects.)
Note: Detailed understanding of the type system is not required to use GAP. It can be helpful, however, to understand how things work and why GAP behaves the way it does.
A discussion of the type system can be found in BreuerLinton98.
The family of an object determines its relationship to other objects.
More precisely, the families form a partition of all GAP objects
such that the following two conditions hold: objects that are equal
w.r.t. `=' lie in the same family; and the family of the result of
an operation depends only on the families of its operands.
The first condition means that a family can be regarded as a set of
elements instead of a set of objects. Note that this does not hold
for categories and representations (see below), two objects that are
equal w.r.t. `=' need not lie in the same categories and
representations. For example, a sparsely represented matrix can be
equal to a densely represented matrix. Similarly, each domain is
equal w.r.t. `=' to the sorted list of its elements, but a domain
is not a list, and a list is not a domain.
FamilyObj( obj ) F
returns the family of the object obj.
The family of the object obj is itself an object, its family is the
FamilyOfFamilies.
It should be emphasized that families may be created when they are needed. For example, the family of elements of a finitely presented group is created only after the presentation has been constructed. Thus families are the dynamic part of the type system, that is, the part that is not fixed after the initialisation of GAP.
Families can be parametrized. For example, the elements of each finitely presented group form a family of their own. Here the family of elements and the finitely presented group coincide when viewed as sets. Note that elements in different finitely presented groups lie in different families. This distinction allows GAP to forbid multiplications of elements in different finitely presented groups.
As a special case, families can be parametrized by other families. An important example is the family of collections that can be formed for each family. A collection consists of objects that lie in the same family, it is either a nonempty dense list of objects from the same family or a domain.
Note that every domain is a collection, that is, it is not possible to
construct domains whose elements lie in different families. For
example, a polynomial ring over the rationals cannot contain the
integer 0 because the family that contains the integers does not
contain polynomials. So one has to distinguish the integer zero from
each zero polynomial.
Let us look at this example from a different viewpoint. A polynomial ring and its coefficients ring lie in different families, hence the coefficients ring cannot be embedded ``naturally'' into the polynomial ring in the sense that it is a subset. But it is possible to allow, e.g., the multiplication of an integer and a polynomial over the integers. The relation between the arguments, namely that one is a coefficient and the other a polynomial, can be detected from the relation of their families. Moreover, this analysis is easier than in a situation where the rationals would lie in one family together with all polynomials over the rationals, because then the relation of families would not distinguish the multiplication of two polynomials, the multiplication of two coefficients, and the multiplication of a coefficient with a polynomial. So the wish to describe relations between elements can be taken as a motivation for the introduction of families.
A filter is a special unary GAP function that returns either true
or false, depending on whether or not the argument lies in the set defined
by the filter.
Filters are used to express different aspects of information about a GAP
object, which are described below
(see Categories, Representation, Attributes,
Setter and Tester for Attributes, Properties, Other Filters).
Presently any filter in GAP is implemented as a function which
corresponds to a set of positions in the bitlist which forms part of the
type of each GAP object, and returns true if and only if the bitlist
of the type of the argument has the value true at all of these positions.
The intersection (or meet) of two filters filt1, filt2 is again a filter, it can be formed as
See and!for filters for more details.
For example, IsList and IsEmpty is a filter that returns true if
its argument is an empty list, and false otherwise. The filter
IsGroup is defined as the intersection of the category
IsMagmaWithInverses and the property IsAssociative.
A filter that is not the meet of other filters is called a simple filter. For example, each attribute tester (see Setter and Tester for Attributes) is a simple filter. Each simple filter corresponds to a position in the bitlist currently used as part of the data structure representing a type.
Every filter filt has a rank, which is used to define a ranking of the methods installed for an operation, see Section Method Installation in ``Programming in GAP''. The rank of a filter can be accessed as
RankFilter( filt ) F
For simple filters, an incremental rank is defined when the filter is created, see the sections about the creation of filters Creating Categories, Creating Representations, Creating Attributes and Properties, Creating Other Filters; all in ``Programming in GAP''. For an arbitrary filter, its rank is given by the sum of the incremental ranks of the involved simple filters; in addition to the implied filters, these are also the required filters of attributes (again see the sections about the creation of filters). In other words, for the purpose of computing the rank and only for this purpose, attribute testers are treated as if they would imply the requirements of their attributes.
NamesFilter( filt ) F
NamesFilter returns a list of names of the implied simple filters
of the filter filt, these are all those simple filters imp such that
every object in filt also lies in imp.
For implications between filters, see ShowImpliedFilters as well as
sections Logical Implications, Creating Categories,
Creating Representations, Creating Attributes and Properties in
``Programming in GAP''
ShowImpliedFilters( filter ) F
Displays information about the filters that may be implied by
filter. They are given by their names. ShowImpliedFilters first
displays the names of all filters that are unconditionally implied by
filter. It then displays implications that require further filters to
be present (indicating by + the required further filters).
The function displays only first-level implications, implications that
follow in turn are not displayed (though GAP will do these).
gap> ShowImpliedFilters(IsMatrix); Implies: IsGeneralizedRowVector IsNearAdditiveElementWithInverse IsAdditiveElement IsMultiplicativeElement May imply with: +IsGF2MatrixRep IsOrdinaryMatrix +CategoryCollections(CategoryCollections(IsAdditivelyCommutativeElement)) IsAdditivelyCommutativeElement +IsInternalRep IsOrdinaryMatrix
The categories of an object are filters (see Filters) determine what
operations an object admits.
For example, all integers form a category, all rationals form a category,
and all rational functions form a category.
An object which claims to lie in a certain category is accepting the
requirement that it should have methods for certain operations (and
perhaps that their behaviour should satisfy certain axioms).
For example, an object lying in the category IsList must have methods
for Length, IsBound\[\] and the list element access operation
\[\].
An object can lie in several categories. For example, a row vector
lies in the categories IsList and IsVector; each list lies in the
category IsCopyable, and depending on whether or not it is mutable,
it may lie in the category IsMutable. Every domain lies in the
category IsDomain.
Of course some categories of a mutable object may change when the
object is changed. For example, after assigning values to positions
of a mutable non-dense list, this list may become part of the category
IsDenseList.
However, if an object is immutable then the set of categories it lies in is fixed.
All categories in the library are created during initialization, in particular they are not created dynamically at runtime.
The following list gives an overview of important categories of arithmetic objects. Indented categories are to be understood as subcategories of the non indented category listed above it.
IsObject
IsExtLElement
IsExtRElement
IsMultiplicativeElement
IsMultiplicativeElementWithOne
IsMultiplicativeElementWithInverse
IsExtAElement
IsAdditiveElement
IsAdditiveElementWithZero
IsAdditiveElementWithInverse
Every object lies in the category IsObject.
The categories IsExtLElement and IsExtRElement contain objects
that can be multiplied with other objects via * from the left and
from the right, respectively. These categories are required for the
operands of the operation *.
The category IsMultiplicativeElement contains objects that can be
multiplied from the left and from the right with objects from the same
family. IsMultiplicativeElementWithOne contains objects obj for
which a multiplicatively neutral element can be obtained by taking the
zeroth power obj^0. IsMultiplicativeElementWithInverse contains
objects obj for which a multiplicative inverse can be obtained by
forming obj^-1.
Likewise, the categories IsExtAElement, IsAdditiveElement,
IsAdditiveElementWithZero, and IsAdditiveElementWithInverse
contain objects that can be added via + to other objects, objects
that can be added to objects of the same family, objects for which an
additively neutral element can be obtained by multiplication with
zero, and objects for which an additive inverse can be obtained by
multiplication with -1.
So a vector lies in IsExtLElement, IsExtRElement, and
IsAdditiveElementWithInverse. A ring element must additionally lie
in IsMultiplicativeElement.
As stated above it is not guaranteed by the categories of objects
whether the result of an operation with these objects as arguments is
defined. For example, the category IsMatrix is a subcategory of
IsMultiplicativeElementWithInverse. Clearly not every matrix has a
multiplicative inverse. But the category IsMatrix makes each matrix
an admissible argument of the operation Inverse, which may sometimes
return 'fail'. Likewise, two matrices can be multiplied only if they
are of appropriate shapes.
Analogous to the categories of arithmetic elements, there are categories of domains of these elements.
IsObject
IsDomain
IsMagma
IsMagmaWithOne
IsMagmaWithInversesIfNonzero
IsMagmaWithInverses
IsAdditiveMagma
IsAdditiveMagmaWithZero
IsAdditiveMagmaWithInverses
IsExtLSet
IsExtRSet
Of course IsDomain is a subcategory of IsObject. A domain that is
closed under multiplication * is called a magma and it lies in the
category IsMagma. If a magma is closed under taking the identity,
it lies in IsMagmaWithOne, and if it is also closed under taking
inverses, it lies in IsMagmaWithInverses. The category
IsMagmaWithInversesIfNonzero denotes closure under taking inverses
only for nonzero elements, every division ring lies in this category.
Note that every set of categories constitutes its own notion of generation, for example a group may be generated as a magma with inverses by some elements, but to generate it as a magma with one it may be necessary to take the union of these generators and their inverses.
CategoriesOfObject( object ) O
returns a list of the names of the categories in which object lies.
gap> g:=Group((1,2),(1,2,3));; gap> CategoriesOfObject(g); [ "IsListOrCollection", "IsCollection", "IsExtLElement", "CategoryCollections(IsExtLElement)", "IsExtRElement", "CategoryCollections(IsExtRElement)", "CategoryCollections(IsMultiplicativeElement)", "CategoryCollections(IsMultiplicativeElementWithOne)", "CategoryCollections(IsMultiplicativeElementWithInverse)", "CategoryCollections(IsAssociativeElement)", "CategoryCollections(IsFiniteOrderElement)", "IsGeneralizedDomain", "CategoryCollections(IS_PERM)", "IsMagma", "IsMagmaWithOne", "IsMagmaWithInversesIfNonzero", "IsMagmaWithInverses" ]
The representation of an object is a set of filters (see Filters) that determines how an object is actually represented. For example, a matrix or a polynomial can be stored sparsely or densely; all dense polynomials form a representation. An object which claims to lie in a certain representation is accepting the requirement that certain fields in the data structure be present and have specified meanings.
GAP distinguishes four essentially different ways to represent
objects. First there are the representations IsInternalRep for
internal objects such as integers and permutations, and
IsDataObjectRep for other objects that are created and whose data
are accessible only by kernel functions. The data structures
underlying such objects cannot be manipulated at the GAP level.
All other objects are either in the representation
IsComponentObjectRep or in the representation
IsPositionalObjectRep, see Component Objects
and Positional Objects in ``Programming in GAP''.
An object can belong to several representations in the sense that it
lies in several subrepresentations of IsComponentObjectRep or of
IsPositionalObjectRep. The representations to which an object
belongs should form a chain and either two representations are disjoint
or one is contained in the other. So the subrepresentations of
IsComponentObjectRep and IsPositionalObjectRep each form
trees. In the language of Object Oriented Programming, we support only
single inheritance for representations.
These trees are typically rather shallow, since for one representation to be contained in another implies that all the components of the data structure implied by the containing representation, are present in, and have the same meaning in, the smaller representation (whose data structure presumably contains some additional components).
Objects may change their representation, for example a mutable list of characters can be converted into a string.
All representations in the library are created during initialization, in particular they are not created dynamically at runtime.
Examples of subrepresentations of IsPositionalObjectRep are
IsModulusRep, which is used for residue classes in the ring of
integers, and IsDenseCoeffVectorRep, which is used for elements of
algebras that are defined by structure constants.
An important subrepresentation of IsComponentObjectRep is
IsAttributeStoringRep, which is used for many domains and some other
objects. It provides automatic storing of all attribute values (see
below).
RepresentationsOfObject( object ) O
returns a list of the names of the representations object has.
gap> g:=Group((1,2),(1,2,3));; gap> RepresentationsOfObject(g); [ "IsComponentObjectRep", "IsAttributeStoringRep" ]
The attributes of an object are filters (see Filters) that describe knowledge about it.
An attribute is a unary operation without side-effects.
An object may store values of its attributes once they have been computed, and claim that it knows these values. Note that ``store'' and ``know'' have to be understood in the sense that it is very cheap to get such a value when the attribute is called again.
The stored value of an attribute is in general immutable (see Mutability and Copyability), except if the attribute had been specally constructed as ``mutable attribute''.
It depends on the representation of an object (see Representation)
which attribute values it stores. An object in the representation
IsAttributeStoringRep stores all attribute values once they are
computed. Moreover, for an object in this representation, subsequent
calls to an attribute will return the same object; this is achieved
via a special method for each attribute setter that stores the
attribute value in an object in IsAttributeStoringRep, and a special
method for the attribute itself that fetches the stored attribute
value. (These methods are called the ``system setter'' and the
``system getter'' of the attribute, respectively.)indexsystem
getterindexsystem setter
Note also that it is impossible to get rid of a stored attribute value because the system may have drawn conclusions from the old attribute value, and just removing the value might leave the data structures in an inconsistent state. If necessary, a new object can be constructed.
Properties are a special form of attributes that have the value true
or false, see section Properties.
All attributes in the library are created during initialization, in particular they are not created dynamically at runtime.
Examples of attributes for multiplicative elements are Inverse, One,
and Order.
Size is an attribute for domains, Centre is an attribute for magmas,
and DerivedSubgroup is an attribute for groups.
KnownAttributesOfObject( object ) O
returns a list of the names of the attributes whose values are known for object.
gap> g:=Group((1,2),(1,2,3));;Size(g);; gap> KnownAttributesOfObject(g); [ "Size", "OneImmutable", "NrMovedPoints", "MovedPoints", "GeneratorsOfMagmaWithInverses", "MultiplicativeNeutralElement", "Pcgs", "GeneralizedPcgs", "StabChainMutable", "StabChainOptions" ]
Several attributes have methods for more than one argument. For example
IsTransitive (see IsTransitive!for group actions) is an attribute for
a G-set that
can also be called for the two arguments, being a group G and its operation
domain. If attributes are called with more than one argument then the
return value is not stored in any of the arguments.
For every attribute two further operations, the attribute setter and the attribute tester are defined.
To check whether an object belongs to an attribute attr, the tester
Tester( attr ) O
of the attribute is used;
this is a filter (see Filters) that returns true or false,
depending on whether or not the value of attr for the object is known.
For example, Tester( Size )( obj ) is true if the size of the object
obj is known.
To store a value for the attribute attr in an object, the setter
Setter( attr ) O
of the attribute is used.
The setter is called automatically when the attribute value has been
computed for the first time.
One can also call the setter explicitly,
for example, Setter( Size )( obj, val ) sets val as size of the
object obj if the size was not yet known.
For each attribute attr that is declared with DeclareAttribute
resp. DeclareProperty (see Global Variables in the Library in
``Programming in GAP''),
tester and setter are automatically made accessible by the names
Hasattr and Setattr, respectively.
For example, the tester for Size is called HasSize,
and the setter is called SetSize.
gap> g:=Group((1,2,3,4),(1,2));;Size(g); 24 gap> HasSize(g); true gap> SetSize(g,99); gap> Size(g); 24
For two properties prop1 and prop2,
the intersection prop1 and prop2 (see Filters) is again a property
for which a setter and a tester exist.
Setting the value of this intersection to true for a GAP object
means to set the values of prop1 and prop2 to true for this object.
gap> prop:= IsFinite and IsCommutative; <Operation "<<and-filter>>"> gap> g:= Group( (1,2,3), (4,5) );; gap> Tester( prop )( g ); false gap> Setter( prop )( g, true ); gap> Tester( prop )( g ); prop( g ); true true
It is not allowed to set the value of such an intersection to false
for an object.
gap> Setter( prop )( Rationals, false ); You cannot set an "and-filter" except to true not in any function Entering break read-eval-print loop ... you can 'quit;' to quit to outer loop, or you can type 'return true;' to set all components true (but you might really want to reset just one component) to continue brk>
AttributeValueNotSet( attr, obj ) F
If the value of the attribute attr is already stored for obj,
AttributeValueNotSet simply returns this value.
Otherwise the value of attr( obj ) is computed and returned
without storing it in obj.
This can be useful when ``large'' attribute values (such as element lists)
are needed only once and shall not be stored in the object.
gap> HasAsSSortedList(g); false gap> AttributeValueNotSet(AsSSortedList,g); [ (), (4,5), (1,2,3), (1,2,3)(4,5), (1,3,2), (1,3,2)(4,5) ] gap> HasAsSSortedList(g); false
The normal behaviour of attributes (when called with just one argument)
is that once a method has been selected and executed, and has returned a
value the setter of the attribute is called, to (possibly) store the
computed value. In special circumstances, this behaviour can be altered
dynamically on an attribute-by-attribute basis, using the functions
DisableAttributeValueStoring and EnableAttributeValueStoring.
In general, the code in the library assumes, for efficiency, but not for correctness, that attribute values will be stored (in suitable objects), so disabling storing may cause substantial computations to be repeated.
InfoAttributes V
This info class (together with InfoWarning; see InfoWarning) is used
for messages about attribute storing being disabled (at level 2) or
enabled (level 3). It may be used in the future for other messages
concerning changes to attribute behaviour.
DisableAttributeValueStoring( attr ) F
disables the usual call of Setter( attr ) when a method for attr
returns a value. In consequence the values will never be stored. Note
that attr must be an attribute and not a property.
EnableAttributeValueStoring( attr ) F
enables the usual call of Setter( attr ) when a method for attr
returns a value. In consequence the values may be stored. This will
usually have no effect unless DisableAttributeValueStoring has
previously been used for attr. Note that attr must be an attribute
and not a property.
gap> g := Group((1,2,3,4,5),(1,2,3)); Group([ (1,2,3,4,5), (1,2,3) ]) gap> KnownAttributesOfObject(g); [ "LargestMovedPoint", "GeneratorsOfMagmaWithInverses", "MultiplicativeNeutralElement" ] gap> SetInfoLevel(InfoAttributes,3); gap> DisableAttributeValueStoring(Size); #I Disabling value storing for Size gap> Size(g); 60 gap> KnownAttributesOfObject(g); [ "OneImmutable", "LargestMovedPoint", "NrMovedPoints", "MovedPoints", "GeneratorsOfMagmaWithInverses", "MultiplicativeNeutralElement", "StabChainMutable", "StabChainOptions" ] gap> Size(g); 60 gap> EnableAttributeValueStoring(Size); #I Enabling value storing for Size gap> Size(g); 60 gap> KnownAttributesOfObject(g); [ "Size", "OneImmutable", "LargestMovedPoint", "NrMovedPoints", "MovedPoints", "GeneratorsOfMagmaWithInverses", "MultiplicativeNeutralElement", "StabChainMutable", "StabChainOptions" ]
The properties of an object are those of its attributes (see Attributes)
whose values can only be true or false.
The main difference between attributes and properties is that a
property defines two sets of objects, namely the usual set of all
objects for which the value is known, and the set of all objects for
which the value is known to be true.
(Note that it makes no sense to consider a third set, namely the set
of objects for which the value of a property is true whether or not
it is known, since there may be objects for which the containment in
this set cannot be decided.)
For a property prop, the containment of an object obj in the first
set is checked again by applying Tester( prop ) to obj,
and obj lies in the second set if and only if
Tester( prop )( obj ) and prop( obj ) is true.
If a property value is known for an immutable object then this value is also stored, as part of the type of the object. To some extent, property values of mutable objects also can be stored, for example a mutable list all of whose entries are immutable can store whether it is strictly sorted. When the object is mutated (for example by list assignment) the type may need to be adjusted.
Important properties for domains are IsAssociative, IsCommutative,
IsAnticommutative, IsLDistributive, and IsRDistributive,
which mean that the multiplication of elements in the domain satisfies
(a * b ) * c = a * ( b * c ), a * b = b * a,
a * b = − ( b * a ), a * ( b + c ) = a * b + a * c,
and ( a + b ) * c = a * c + b * c, respectively,
for all a, b, c in the domain.
KnownPropertiesOfObject( object ) O
returns a list of the names of the properties whose values are known for object.
KnownTruePropertiesOfObject( object ) O
returns a list of the names of the properties known to be true for
object.
gap> g:=Group((1,2),(1,2,3));; gap> KnownPropertiesOfObject(g); [ "IsFinite", "CanEasilyCompareElements", "CanEasilySortElements", "IsDuplicateFree", "IsGeneratorsOfMagmaWithInverses", "IsAssociative", "IsSimpleSemigroup", "IsFinitelyGeneratedGroup", "IsSubsetLocallyFiniteGroup", "KnowsHowToDecompose" ] gap> Size(g); 6 gap> KnownPropertiesOfObject(g); [ "IsEmpty", "IsTrivial", "IsNonTrivial", "IsFinite", "CanEasilyCompareElements", "CanEasilySortElements", "IsDuplicateFree", "IsGeneratorsOfMagmaWithInverses", "IsAssociative", "IsSimpleSemigroup", "IsFinitelyGeneratedGroup", "IsSubsetLocallyFiniteGroup", "KnowsHowToDecompose", "IsPerfectGroup", "IsSolvableGroup", "IsPolycyclicGroup" ] gap> KnownTruePropertiesOfObject(g); [ "IsNonTrivial", "IsFinite", "CanEasilyCompareElements", "CanEasilySortElements", "IsDuplicateFree", "IsGeneratorsOfMagmaWithInverses", "IsAssociative", "IsSimpleSemigroup", "IsFinitelyGeneratedGroup", "IsSubsetLocallyFiniteGroup", "KnowsHowToDecompose", "IsSolvableGroup", "IsPolycyclicGroup" ]
There are situations where one wants to express a kind of knowledge that is based on some heuristic.
For example, the filters (see Filters) CanEasilyTestMembership and
CanEasilyComputePcgs are defined in the GAP library.
Note that such filters do not correspond to a mathematical concept,
contrary to properties (see Properties).
Also it need not be defined what ``easily'' means for an arbitrary GAP
object, and in this case one cannot compute the value for an arbitrary
GAP object.
In order to access this kind of knowledge as a part of the type of an object,
GAP provides filters for which the value is false by default,
and it is changed to true in certain situations,
either explicitly (for the given object) or via a logical implication
(see Logical Implications in ``Programming in GAP'')
from other filters.
For example, a true value of CanEasilyComputePcgs for a group
means that certain methods are applicable that use a pcgs
(see Polycyclic Generating Systems) for the group.
There are logical implications to set the filter value to true
for permutation groups that are known to be solvable,
and for groups that have already a (sufficiently nice) pcgs stored.
In the case one has a solvable matrix group and wants to enable methods
that use a pcgs, one can set the CanEasilyComputePcgs value to
true for this particular group.
A filter filt of the kind described here is different from
the other filters introduced in the previous sections.
In particular, filt is not a category (see Categories) or a property
(see Properties) because its value may change for a given object,
and filt is not a representation (see Representation) because it has
nothing to do with the way an object is made up from some data.
filt is similar to an attribute tester
(see Setter and Tester for Attributes),
the only difference is that filt does not refer to an attribute value;
note that filt is also used in the same way as an attribute tester;
namely, the true value may be required for certain methods to be
applicable.
We stated above (see Types of Objects) that, for an object obj, its type is formed from its family and its filters. There is a also a third component, used in a few situations, namely defining data of the type.
TypeObj( obj ) F
returns the type of the object obj.
The type of an object is itself an object.
Two types are equal if and only if the two families are identical, the filters are equal, and, if present, also the defining data of the types are equal.
The last part of the type, defining data, has not been mentioned before and seems to be of minor importance. It can be used, e.g., for cosets U g of a group U, where the type of each coset may contain the group U as defining data. As a consequence, two such cosets mod U and V can have the same type only if U = V. The defining data of the type type can be accessed as
DataType( type ) F
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GAP 4 manual
March 2006