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31 Domains and their Elements
 31.1 Operational Structure of Domains
 31.2 Equality and Comparison of Domains
 31.3 Constructing Domains
 31.4 Changing the Structure
 31.5 Changing the Representation
 31.6 Domain Categories
 31.7 Parents
 31.8 Constructing Subdomains
 31.9 Operations for Domains
 31.10 Attributes and Properties of Elements
 31.11 Comparison Operations for Elements
 31.12 Arithmetic Operations for Elements
 31.13 Relations Between Domains
 31.14 Useful Categories of Elements
 31.15 Useful Categories for all Elements of a Family

31 Domains and their Elements

Domain is GAP's name for structured sets. The ring of Gaussian integers \(ℤ[\sqrt{{-1}}]\) is an example of a domain, the group \(D_{12}\) of symmetries of a regular hexahedron is another.

The GAP library predefines some domains. For example the ring of Gaussian integers is predefined as GaussianIntegers (60.5-1) (see 60.5) and the field of rationals is predefined as Rationals (17.1-1) (see 17). Most domains are constructed by functions, which are called domain constructors (see 31.3). For example the group \(D_{12}\) is constructed by the construction Group( (1,2,3,4,5,6), (2,6)(3,5) ) (see Group (39.2-1)) and the finite field with 16 elements is constructed by GaloisField( 16 ) (see GaloisField (59.3-2)).

The first place where you need domains in GAP is the obvious one. Sometimes you simply want to deal with a domain. For example if you want to compute the size of the group \(D_{12}\), you had better be able to represent this group in a way that the Size (30.4-6) function can understand.

The second place where you need domains in GAP is when you want to be able to specify that an operation or computation takes place in a certain domain. For example suppose you want to factor 10 in the ring of Gaussian integers. Saying Factors( 10 ) will not do, because this will return the factorization [ 2, 5 ] in the ring of integers. To allow operations and computations to happen in a specific domain, Factors (56.5-9), and many other functions as well, accept this domain as optional first argument. Thus Factors( GaussianIntegers, 10 ) yields the desired result [ 1+E(4), 1-E(4), 2+E(4), 2-E(4) ]. (The imaginary unit \(\sqrt{{-1}}\) is written as E(4) in GAP, see E (18.1-1).)

An introduction to the most important facts about domains is given in Chapter Tutorial: Domains.

There are only few operations especially for domains (see 31.9), operations such as Intersection (30.5-2) and Random (30.7-1) are defined for the more general situation of collections (see Chapter 30).

31.1 Operational Structure of Domains

Domains have an operational structure, that is, a collection of operations under which the domain is closed. For example, a group is closed under multiplication, taking the zeroth power of elements, and taking inverses of elements. The operational structure may be empty, examples of domains without additional structure are the underlying relations of general mappings (see 32.3).

The operations under which a domain is closed are a subset of the operations that the elements of a domain admit. It is possible that the elements admit more operations. For example, matrices can be multiplied and added. But addition plays no role in a group of matrices, and multiplication plays no role in a vector space of matrices. In particular, a matrix group is not closed under addition.

Note that the elements of a domain exist independently of this domain, usually they existed already before the domain was created. So it makes sense to say that a domain is generated by some elements with respect to certain operations.

Of course, different sets of operations yield different notions of generation. For example, the group generated by some matrices is different from the ring generated by these matrices, and these two will in general be different from the vector space generated by the same matrices, over a suitable field.

The other way round, the same set of elements may be obtained by generation w.r.t. different notions of generation. For example, one can get the group generated by two elements \(g\) and \(h\) also as the monoid generated by the elements \(g\), \(g^{{-1}}\), \(h\), \(h^{{-1}}\); if both \(g\) and \(h\) have finite order then of course the group generated by \(g\) and \(h\) coincides with the monoid generated by \(g\) and \(h\).

Additionally to the operational structure, a domain can have properties. For example, the multiplication of a group is associative, and the multiplication in a field is commutative.

Note that associativity and commutativity depend on the set of elements for which one considers the multiplication, i.e., it depends on the domain. For example, the multiplication in a full matrix ring over a field is not commutative, whereas its restriction to the set of diagonal matrices is commutative.

One important difference between the operational structure and the properties of a domain is that the operational structure is fixed when the domain is constructed, whereas properties can be discovered later. For example, take a domain whose operational structure is given by closure under multiplication. If it is discovered that the inverses of all its elements also do (by chance) lie in this domain, being closed under taking inverses is not added to the operational structure. But a domain with operational structure of multiplication, taking the identity, and taking inverses will be treated as a group as soon as the multiplication is found out to be associative for this domain.

The operational structures available in GAP form a hierarchy, which is explicitly formulated in terms of domain categories, see 31.6.

31.2 Equality and Comparison of Domains

Equality and comparison of domains are defined as follows.

Two domains are considered equal if and only if the sets of their elements as computed by AsSSortedList (30.3-10)) are equal. Thus, in general = behaves as if each domain operand were replaced by its set of elements. Except that = will also sometimes, but not always, work for infinite domains, for which of course GAP cannot compute the set of elements. Note that this implies that domains with different algebraic structure may well be equal. As a special case of this, either operand of = may also be a proper set (see 21.19), i.e., a sorted list without holes or duplicates (see AsSSortedList (30.3-10)), and = will return true if and only if this proper set is equal to the set of elements of the argument that is a domain.

No general ordering of arbitrary domains via < is defined in GAP 4. This is because a well-defined < for domains or, more general, for collections, would have to be compatible with = and would need to be transitive and antisymmetric in order to be used to form ordered sets. In particular, < would have to be independent of the algebraic structure of its arguments because this holds for =, and thus there would be hardly a situation where one could implement an efficient comparison method. (Note that in the case that two domains are comparable with <, the result is in general not compatible with the set theoretical subset relation, which can be decided with IsSubset (30.5-1).)

31.3 Constructing Domains

For several operational structures (see 31.1), GAP provides functions to construct domains with this structure (note that such functions do not exist for all operational structures). For example, Group (39.2-1) returns groups, VectorSpace (61.2-1) returns vector spaces etc.:

Struct( arg1, arg2, ... )

The syntax of these functions may vary, dependent on the structure in question. Usually a domain is constructed as the closure of some elements under the given operations, that is, the domain is given by its generators. For example, a group can be constructed from a list of generating permutations or matrices or whatever is admissible as group elements, and a vector space over a given field \(F\) can be constructed from \(F\) and a list of appropriate vectors.

The idea of generation and generators in GAP is that the domain returned by a function such as Group, Algebra, or FreeLeftModule contains the given generators. This implies that the generators of a group must know how they are multiplied and inverted, the generators of a module must know how they are added and how scalar multiplication works, and so on. Thus one cannot use for example permutations as generators of a vector space.

The function Struct first checks whether the arguments admit the construction of a domain with the desired structure. This is done by calling the operation

IsGeneratorsOfStruct( [info, ]gens )

where arglist is the list of given generators and info an argument of Struct, for example the field of scalars in the case that a vector space shall be constructed. If the check failed then Struct returns fail, otherwise it returns the result of StructByGenerators (see below). (So if one wants to omit the check then one should call StructByGenerators directly.)

GeneratorsOfStruct( D)

For a domain D with operational structure corresponding to Struct, the attribute GeneratorsOfStruct returns a list of corresponding generators of D. If these generators were not yet stored in D then D must know some generators if GeneratorsOfStruct shall have a chance to compute the desired result; for example, monoid generators of a group can be computed from known group generators (and vice versa). Note that several notions of generation may be meaningful for a given domain, so it makes no sense to ask for "the generators of a domain". Further note that the generators may depend on other information about D. For example the generators of a vector space depend on the underlying field of scalars; the vector space generators of a vector space over the field with four elements need not generate the same vector space when this is viewed as a space over the field with two elements.

StructByGenerators( [info, ]gens )

Domain construction from generators gens is implemented by operations StructByGenerators, which are called by the simple functions Struct; methods can be installed only for the operations. Note that additional information info may be necessary to construct the domain; for example, a vector space needs the underlying field of scalars in addition to the list of vector space generators. The GeneratorsOfStruct value of the returned domain need not be equal to gens. But if a domain D is printed as Struct([a, b, ...]) and if there is an attribute GeneratorsOfStruct then the list GeneratorsOfStruct( D ) is guaranteed to be equal to [ a, b, ... ].

StructWithGenerators( [info, ]gens )

The only difference between StructByGenerators and StructWithGenerators is that the latter guarantees that the GeneratorsOfStruct value of the result is equal to the given generators gens.

ClosureStruct( D, obj )

For constructing a domain as the closure of a given domain with an element or another domain, one can use the operation ClosureStruct. It returns the smallest domain with operational structure corresponding to Struct that contains D as a subset and obj as an element.

31.4 Changing the Structure

The same set of elements can have different operational structures. For example, it may happen that a monoid \(M\) does in fact contain the inverses of all of its elements; if \(M\) has not been constructed as a group (see 31.6) then it is reasonable to ask for the group that is equal to \(M\).

AsStruct( [info, ]D )

If D is a domain that is closed under the operational structure given by Struct then AsStruct returns a domain E that consists of the same elements (that is, D = E) and that has this operational structure (that is, IsStruct( E ) is true); if D is not closed under the structure given by Struct then AsStruct returns fail.

If additional information besides generators are necessary to define D then the argument info describes the value of this information for the desired domain. For example, if we want to view D as a vector space over the field with two elements then we may call AsVectorSpace( GF(2), D ); this allows us to change the underlying field of scalars, for example if D is a vector space over the field with four elements. Again, if D is not equal to a domain with the desired structure and additional information then fail is returned.

In the case that no additional information info is related to the structure given by Struct, the operation AsStruct is in fact an attribute (see 13.5).

See the index of the GAP Reference Manual for an overview of the available AsStruct functions.

31.5 Changing the Representation

Often it is useful to answer questions about a domain via computations in a different but isomorphic domain. In the sense that this approach keeps the structure and changes the underlying set of elements, it can be viewed as a counterpart of keeping the set of elements and changing its structure (see 31.4).

One reason for doing so can be that computations with the elements in the given domain are not very efficient. For example, if one is given a solvable matrix group (see Chapter 44) then one can compute an isomorphism to a polycyclicly presented group \(G\), say (see Chapter 45); the multiplication of two matrices –which is essentially determined by the dimension of the matrices– is much more expensive than the multiplication of two elements in \(G\) –which is essentially determined by the composition length of \(G\).

IsomorphismRepStruct( D )

If D is a domain that is closed under the operational structure given by Struct then IsomorphismRepStruct returns a mapping hom from D to a domain \(E\) having structure given by Struct, such that hom respects the structure Struct and Rep describes the representation of the elements in \(E\). If no domain \(E\) with the required properties exists then fail is returned.

For example, IsomorphismPermGroup (43.3-1) takes a group as its argument and returns a group homomorphism (see 40) onto an isomorphic permutation group (see Chapter 43) provided the original group is finite; for infinite groups, IsomorphismPermGroup (43.3-1) returns fail. Similarly, IsomorphismPcGroup (46.5-2) returns a group homomorphism from its argument to a polycyclicly presented group (see 46) if the argument is polycyclic, and fail otherwise.

See the index of the GAP Reference Manual for an overview of the available IsomorphismRepStruct functions.

31.6 Domain Categories

As mentioned in 31.1, the operational structure of a domain is fixed when the domain is constructed. For example, if D was constructed by Monoid (51.2-2) then D is in general not regarded as a group in GAP, even if D is in fact closed under taking inverses. In this case, IsGroup (39.2-7) returns false for D. The operational structure determines which operations are applicable for a domain, so for example SylowSubgroup (39.13-1) is not defined for D and therefore will signal an error.

IsStruct( D )

The functions IsStruct implement the tests whether a domain D has the respective operational structure (upon construction). IsStruct is a filter (see 13) that involves certain categories (see 13.3) and usually also certain properties (see 13.7). For example, IsGroup (39.2-7) is equivalent to IsMagmaWithInverses and IsAssociative, the first being a category and the second being a property.

Implications between domain categories describe the hierarchy of operational structures available in GAP. Here are some typical examples.

Each operational structure Struct has associated with it a domain category IsStruct, and operations StructByGenerators for constructing a domain from generators, GeneratorsOfStruct for storing and accessing generators w.r.t. this structure, ClosureStruct for forming the closure, and AsStruct for getting a domain with the desired structure from one with weaker operational structure and for testing whether a given domain can be regarded as a domain with Struct.

The functions applicable to domains with the various structures are described in the corresponding chapters of the Reference Manual. For example, functions for rings, fields, groups, and vector spaces are described in Chapters 56, 58, 39, and 61, respectively. More general functions for arbitrary collections can be found in Chapter 30.

31.7 Parents

31.7-1 Parent
‣ Parent( D )( function )
‣ SetParent( D, P )( operation )
‣ HasParent( D )( filter )

It is possible to assign to a domain D one other domain P containing D as a subset, in order to exploit this subset relation between D and P. Note that P need not have the same operational structure as D, for example P may be a magma and D a field.

The assignment is done by calling SetParent, and P is called the parent of D. If D has already a parent, calls to SetParent will be ignored.

If D has a parent P –this can be checked with HasParent– then P can be used to gain information about D. First, the call of SetParent causes UseSubsetRelation (31.13-1) to be called. Second, for a domain D with parent, information relative to the parent can be stored in D; for example, there is an attribute NormalizerInParent for storing Normalizer( P, D ) in the case that D is a group. (More about such parent dependent attributes can be found in 85.2.) Note that because of this relative information, one cannot change the parent; that is, one can set the parent only once, subsequent calls to SetParent for the same domain D are ignored. Further note that contrary to UseSubsetRelation (31.13-1), also knowledge about the parent P might be used that is discovered after the SetParent call.

A stored parent can be accessed using Parent. If D has no parent then Parent returns D itself, and HasParent will return false also after a call to Parent. So Parent is not an attribute, the underlying attribute to store the parent is ParentAttr.

Certain functions that return domains with parent already set, for example Subgroup (39.3-1), are described in Section 31.8. Whenever a function has this property, the GAP Reference Manual states this explicitly. Note that these functions do not guarantee a certain parent, for example DerivedSubgroup (39.12-3) for a perfect group \(G\) may return \(G\) itself, and if \(G\) had already a parent then this is not replaced by \(G\). As a rule of thumb, GAP avoids to set a domain as its own parent, which is consistent with the behaviour of Parent, at least until a parent is set explicitly with SetParent.

gap> g:= Group( (1,2,3), (1,2) );; h:= Group( (1,2) );;
gap> HasParent( g );  HasParent( h );
false
false
gap> SetParent( h, g );
gap> Parent( g );  Parent( h );
Group([ (1,2,3), (1,2) ])
Group([ (1,2,3), (1,2) ])
gap> HasParent( g );  HasParent( h );
false
true

31.8 Constructing Subdomains

For many domains D, there are functions that construct certain subsets S of D as domains with parent (see 31.7) already set to D. For example, if G is a group that contains the elements in the list gens then Subgroup( G, gens ) returns a group S that is generated by the elements in gens and with Parent( S ) = G.

Substruct( D, gens )

More general, if D is a domain whose algebraic structure is given by the function Struct (for example Group, Algebra, Field) then the function Substruct (for example Subgroup, Subalgebra, Subfield) returns domains with structure Struct and parent set to the first argument.

SubstructNC( D, gens )

Each function Substruct checks that the Struct generated by gens is in fact a subset of D. If one wants to omit this check then one can call SubstructNC instead; the suffix NC stands for "no check".

AsSubstruct( D, S )

first constructs AsStruct( [info, ]S ), where info depends on D and S, and then sets the parent (see 31.7) of this new domain to D.

IsSubstruct( D, S )

There is no real need for functions that check whether a domain S is a Substruct of a domain D, since this is equivalent to the checks whether S is a Struct and S is a subset of D. Note that in many cases, only the subset relation is what one really wants to check, and that appropriate methods for the operation IsSubset (30.5-1) are available for many special situations, such as the test whether a group is contained in another group, where only generators need to be checked.

If a function IsSubstruct is available in GAP then it is implemented as first a call to IsStruct for the second argument and then a call to IsSubset (30.5-1) for the two arguments.

31.9 Operations for Domains

For the meaning of the attributes Characteristic (31.10-1), One (31.10-2), Zero (31.10-3) in the case of a domain argument, see 31.10.

31.9-1 IsGeneralizedDomain
‣ IsGeneralizedDomain( obj )( category )
‣ IsDomain( obj )( category )

For some purposes, it is useful to deal with objects that are similar to domains but that are not collections in the sense of GAP because their elements may lie in different families; such objects are called generalized domains. An instance of generalized domains are "operation domains", for example any \(G\)-set for a permutation group \(G\) consisting of some union of points, sets of points, sets of sets of points etc., under a suitable action.

IsDomain is a synonym for IsGeneralizedDomain and IsCollection.

31.9-2 GeneratorsOfDomain
‣ GeneratorsOfDomain( D )( attribute )

For a domain D, GeneratorsOfDomain returns a list containing all elements of D, perhaps with repetitions. Note that if the domain D shall be generated by a list of some elements w.r.t. the empty operational structure (see 31.1), the only possible choice of elements is to take all elements of D. See 31.3 and 31.4 for concepts of other notions of generation.

For many domains that have natural generators by construction (for example, the natural generators of a free group of rank two are the two generators stored as value of the attribute GeneratorsOfGroup (39.2-4), and the natural generators of a free associative algebra are those generators stored as value of the attribute GeneratorsOfAlgebra (62.9-1)), each natural generator can be accessed using the . operator. For a domain D, D.i returns the \(i\)-th generator if \(i\) is a positive integer, and if name is the name of a generator of D then D.name returns this generator.

31.9-3 Domain
‣ Domain( [Fam, ]generators )( function )
‣ DomainByGenerators( Fam, generators )( operation )

Domain returns the domain consisting of the elements in the homogeneous list generators. If generators is empty then a family Fam must be entered as the first argument, and the returned (empty) domain lies in the collections family of Fam.

DomainByGenerators is the operation called by Domain.

31.10 Attributes and Properties of Elements

The following attributes and properties for elements and domains correspond to the operational structure.

31.10-1 Characteristic
‣ Characteristic( obj )( attribute )

Characteristic returns the characteristic of obj.

If obj is a family, all of whose elements lie in IsAdditiveElementWithZero (31.14-5) then its characteristic is the least positive integer \(n\), if any, such that IsZero(n*x) is true for all x in the family obj, otherwise it is \(0\).

If obj is a collections family of a family \(g\) which has a characteristic, then the characteristic of obj is the same as the characteristic of \(g\).

For other families obj the characteristic is not defined and fail will be returned.

For any object obj which is in the filter IsAdditiveElementWithZero (31.14-5) or in the filter IsAdditiveMagmaWithZero (55.1-5) the characteristic of obj is the same as the characteristic of its family if that is defined and undefined otherwise.

For all other objects obj the characteristic is undefined and may return fail or a "no method found" error.

31.10-2 OneImmutable
‣ OneImmutable( obj )( attribute )
‣ OneAttr( obj )( attribute )
‣ One( obj )( attribute )
‣ Identity( obj )( attribute )
‣ OneMutable( obj )( operation )
‣ OneOp( obj )( operation )
‣ OneSameMutability( obj )( operation )
‣ OneSM( obj )( operation )

OneImmutable, OneMutable, and OneSameMutability return the multiplicative neutral element of the multiplicative element obj.

They differ only w.r.t. the mutability of the result. OneImmutable is an attribute and hence returns an immutable result. OneMutable is guaranteed to return a new mutable object whenever a mutable version of the required element exists in GAP (see IsCopyable (12.6-1)). OneSameMutability returns a result that is mutable if obj is mutable and if a mutable version of the required element exists in GAP; for lists, it returns a result of the same immutability level as the argument. For instance, if the argument is a mutable matrix with immutable rows, it returns a similar object.

If obj is a multiplicative element then OneSameMutability( obj ) is equivalent to obj^0.

OneAttr, One and Identity are synonyms of OneImmutable. OneSM is a synonym of OneSameMutability. OneOp is a synonym of OneMutable.

If obj is a domain or a family then One is defined as the identity element of all elements in obj, provided that all these elements have the same identity. For example, the family of all cyclotomics has the identity element 1, but a collections family (see CollectionsFamily (30.2-1)) may contain matrices of all dimensions and then it cannot have a unique identity element. Note that One is applicable to a domain only if it is a magma-with-one (see IsMagmaWithOne (35.1-2)); use MultiplicativeNeutralElement (35.4-10) otherwise.

The identity of an object need not be distinct from its zero, so for example a ring consisting of a single element can be regarded as a ring-with-one (see 56). This is particularly useful in the case of finitely presented algebras, where any factor of a free algebra-with-one is again an algebra-with-one, no matter whether or not it is a zero algebra.

The default method of One for multiplicative elements calls OneMutable (note that methods for OneMutable must not delegate to One); so other methods to compute identity elements need to be installed only for OneOp and (in the case of copyable objects) OneSameMutability.

For domains, One may call Representative (30.4-7), but Representative (30.4-7) is allowed to fetch the identity of a domain D only if HasOne( D ) is true.

31.10-3 ZeroImmutable
‣ ZeroImmutable( obj )( attribute )
‣ ZeroAttr( obj )( attribute )
‣ Zero( obj )( attribute )
‣ ZeroMutable( obj )( operation )
‣ ZeroOp( obj )( operation )
‣ ZeroSameMutability( obj )( operation )
‣ ZeroSM( obj )( operation )

ZeroImmutable, ZeroMutable, and ZeroSameMutability all return the additive neutral element of the additive element obj.

They differ only w.r.t. the mutability of the result. ZeroImmutable is an attribute and hence returns an immutable result. ZeroMutable is guaranteed to return a new mutable object whenever a mutable version of the required element exists in GAP (see IsCopyable (12.6-1)). ZeroSameMutability returns a result that is mutable if obj is mutable and if a mutable version of the required element exists in GAP; for lists, it returns a result of the same immutability level as the argument. For instance, if the argument is a mutable matrix with immutable rows, it returns a similar object.

ZeroSameMutability( obj ) is equivalent to 0 * obj.

ZeroAttr and Zero are synonyms of ZeroImmutable. ZeroSM is a synonym of ZeroSameMutability. ZeroOp is a synonym of ZeroMutable.

If obj is a domain or a family then Zero is defined as the zero element of all elements in obj, provided that all these elements have the same zero. For example, the family of all cyclotomics has the zero element 0, but a collections family (see CollectionsFamily (30.2-1)) may contain matrices of all dimensions and then it cannot have a unique zero element. Note that Zero is applicable to a domain only if it is an additive magma-with-zero (see IsAdditiveMagmaWithZero (55.1-5)); use AdditiveNeutralElement (55.3-5) otherwise.

The default method of Zero for additive elements calls ZeroMutable (note that methods for ZeroMutable must not delegate to Zero); so other methods to compute zero elements need to be installed only for ZeroMutable and (in the case of copyable objects) ZeroSameMutability.

For domains, Zero may call Representative (30.4-7), but Representative (30.4-7) is allowed to fetch the zero of a domain D only if HasZero( D ) is true.

31.10-4 MultiplicativeZeroOp
‣ MultiplicativeZeroOp( elt )( operation )

Returns: A multiplicative zero element.

for an element elt in the category IsMultiplicativeElementWithZero (31.14-12), MultiplicativeZeroOp returns the element \(z\) in the family \(F\) of elt with the property that \(z * m = z = m * z\) holds for all \(m \in F\), if such an element can be determined.

Families of elements in the category IsMultiplicativeElementWithZero (31.14-12) often arise from adjoining a new zero to an existing magma. See InjectionZeroMagma (35.2-13) or MagmaWithZeroAdjoined (35.2-13) for details.

gap> G:=AlternatingGroup(5);;
gap> x:=Representative(MagmaWithZeroAdjoined(G));
<group with 0 adjoined elt: ()>
gap> MultiplicativeZeroOp(x);
<group with 0 adjoined elt: 0>

31.10-5 IsOne
‣ IsOne( elm )( property )

is true if elm = One( elm ), and false otherwise.

31.10-6 IsZero
‣ IsZero( elm )( property )

is true if elm = Zero( elm ), and false otherwise.

31.10-7 IsIdempotent
‣ IsIdempotent( elt )( property )

returns true iff elt is its own square. (Even if IsZero (31.10-6) returns true for elt.)

31.10-8 InverseImmutable
‣ InverseImmutable( elm )( attribute )
‣ InverseAttr( elm )( attribute )
‣ Inverse( elm )( attribute )
‣ InverseMutable( elm )( operation )
‣ InverseOp( elm )( operation )
‣ InverseSameMutability( elm )( operation )
‣ InverseSM( elm )( operation )

InverseImmutable, InverseMutable, and InverseSameMutability all return the multiplicative inverse of an element elm, that is, an element inv such that elm * inv = inv * elm = One( elm ) holds; if elm is not invertible then fail (see 20.2) is returned.

Note that the above definition implies that a (general) mapping is invertible in the sense of Inverse only if its source equals its range (see 32.14). For a bijective mapping \(f\) whose source and range differ, InverseGeneralMapping (32.2-3) can be used to construct a mapping \(g\) with the property that \(f\) * \(g\) is the identity mapping on the source of \(f\) and \(g\) * \(f\) is the identity mapping on the range of \(f\).

The operations differ only w.r.t. the mutability of the result. InverseImmutable is an attribute and hence returns an immutable result. InverseMutable is guaranteed to return a new mutable object whenever a mutable version of the required element exists in GAP. InverseSameMutability returns a result that is mutable if elm is mutable and if a mutable version of the required element exists in GAP; for lists, it returns a result of the same immutability level as the argument. For instance, if the argument is a mutable matrix with immutable rows, it returns a similar object.

InverseSameMutability( elm ) is equivalent to elm^-1.

InverseAttr and Inverse are synonyms of InverseImmutable. InverseSM is a synonym of InverseSameMutability. InverseOp is a synonym of InverseMutable.

The default method of InverseImmutable calls InverseMutable (note that methods for InverseMutable must not delegate to InverseImmutable); other methods to compute inverses need to be installed only for InverseMutable and (in the case of copyable objects) InverseSameMutability.

31.10-9 AdditiveInverseImmutable
‣ AdditiveInverseImmutable( elm )( attribute )
‣ AdditiveInverseAttr( elm )( attribute )
‣ AdditiveInverse( elm )( attribute )
‣ AdditiveInverseMutable( elm )( operation )
‣ AdditiveInverseOp( elm )( operation )
‣ AdditiveInverseSameMutability( elm )( operation )
‣ AdditiveInverseSM( elm )( operation )

AdditiveInverseImmutable, AdditiveInverseMutable, and AdditiveInverseSameMutability all return the additive inverse of elm.

They differ only w.r.t. the mutability of the result. AdditiveInverseImmutable is an attribute and hence returns an immutable result. AdditiveInverseMutable is guaranteed to return a new mutable object whenever a mutable version of the required element exists in GAP (see IsCopyable (12.6-1)). AdditiveInverseSameMutability returns a result that is mutable if elm is mutable and if a mutable version of the required element exists in GAP; for lists, it returns a result of the same immutability level as the argument. For instance, if the argument is a mutable matrix with immutable rows, it returns a similar object.

AdditiveInverseSameMutability( elm ) is equivalent to -elm.

AdditiveInverseAttr and AdditiveInverse are synonyms of AdditiveInverseImmutable. AdditiveInverseSM is a synonym of AdditiveInverseSameMutability. AdditiveInverseOp is a synonym of AdditiveInverseMutable.

The default method of AdditiveInverse calls AdditiveInverseMutable (note that methods for AdditiveInverseMutable must not delegate to AdditiveInverse); so other methods to compute additive inverses need to be installed only for AdditiveInverseMutable and (in the case of copyable objects) AdditiveInverseSameMutability.

31.10-10 Order
‣ Order( elm )( attribute )

is the multiplicative order of elm. This is the smallest positive integer \(n\) such that elm ^ \(n\) = One( elm ) if such an integer exists. If the order is infinite, Order may return the value infinity (18.2-1), but it also might run into an infinite loop trying to test the order.

31.11 Comparison Operations for Elements

Binary comparison operations have been introduced already in 4.12. The underlying operations for which methods can be installed are the following.

31.11-1 \= and \<
‣ \=( left-expr, right-expr )( operation )
‣ \<( left-expr, right-expr )( operation )

Note that the comparisons via <>, <=, >, and >= are delegated to the operations \= and \<.

In general, objects in different families cannot be compared with \<. For the reason and for exceptions from this rule, see 4.12.

31.11-2 CanEasilyCompareElements
‣ CanEasilyCompareElements( obj )( property )
‣ CanEasilyCompareElementsFamily( fam )( function )
‣ CanEasilySortElements( obj )( property )
‣ CanEasilySortElementsFamily( fam )( function )

For some objects a "normal form" is hard to compute and thus equality of elements of a domain might be expensive to test. Therefore GAP provides a (slightly technical) property with which an algorithm can test whether an efficient equality test is available for elements of a certain kind.

CanEasilyCompareElements indicates whether the elements in the family fam of obj can be easily compared with \= (31.11-1).

The default method for this property is to ask the family of obj, the default method for the family is to return false.

The ability to compare elements may depend on the successful computation of certain information. (For example for finitely presented groups it might depend on the knowledge of a faithful permutation representation.) This information might change over time and thus it might not be a good idea to store a value false too early in a family. Instead the function CanEasilyCompareElementsFamily should be called for the family of obj which returns false if the value of CanEasilyCompareElements is not known for the family without computing it. (This is in fact what the above mentioned family dispatch does.)

If a family knows ab initio that it can compare elements this property should be set as implied filter and filter for the family (the 3rd and 4th argument of NewFamily (79.7-1) respectively). This guarantees that code which directly asks the family gets a right answer.

The property CanEasilySortElements and the function CanEasilySortElementsFamily behave exactly in the same way, except that they indicate that objects can be compared via \< (31.11-1). This property implies CanEasilyCompareElements, as the ordering must be total.

31.12 Arithmetic Operations for Elements

Binary arithmetic operations have been introduced already in 4.13. The underlying operations for which methods can be installed are the following.

31.12-1 \+, \*, \/, \^, \mod
‣ \+( left-expr, right-expr )( operation )
‣ \*( left-expr, right-expr )( operation )
‣ \/( left-expr, right-expr )( operation )
‣ \^( left-expr, right-expr )( operation )
‣ \mod( left-expr, right-expr )( operation )

For details about special methods for \*, \/, \^ and \mod, consult the appropriate index entries for them.

31.12-2 LeftQuotient
‣ LeftQuotient( elm1, elm2 )( operation )

returns the product elm1^(-1) * elm2. For some types of objects (for example permutations) this product can be evaluated more efficiently than by first inverting elm1 and then forming the product with elm2.

31.12-3 Comm
‣ Comm( elm1, elm2 )( operation )

returns the commutator of elm1 and elm2. The commutator is defined as the product \(\textit{elm1}^{{-1}} * \textit{elm2}^{{-1}} * \textit{elm1} * \textit{elm2}\).

gap> a:= (1,3)(4,6);; b:= (1,6,5,4,3,2);;
gap> Comm( a, b );
(1,5,3)(2,6,4)
gap> LeftQuotient( a, b );
(1,2)(3,6)(4,5)

31.12-4 LieBracket
‣ LieBracket( elm1, elm2 )( operation )

returns the element elm1 * elm2 - elm2 * elm1.

The addition \+ (31.12-1) is assumed to be associative but not assumed to be commutative (see IsAdditivelyCommutative (55.3-1)). The multiplication \* (31.12-1) is not assumed to be commutative or associative (see IsCommutative (35.4-9), IsAssociative (35.4-7)).

31.12-5 Sqrt
‣ Sqrt( obj )( operation )

Sqrt returns a square root of obj, that is, an object \(x\) with the property that \(x \cdot x = \textit{obj}\) holds. If such an \(x\) is not unique then the choice of \(x\) depends on the type of obj. For example, ER (18.4-2) is the Sqrt method for rationals (see IsRat (17.2-1)).

31.13 Relations Between Domains

Domains are often constructed relative to other domains. The probably most usual case is to form a subset of a domain, for example the intersection (see Intersection (30.5-2)) of two domains, or a Sylow subgroup of a given group (see SylowSubgroup (39.13-1)).

In such a situation, the new domain can gain knowledge by exploiting that several attributes are maintained under taking subsets. For example, the intersection of an arbitrary domain with a finite domain is clearly finite, a Sylow subgroup of an abelian group is abelian, too, and so on.

Since usually the new domain has access to the knowledge of the old domain(s) only when it is created (see 31.8 for the exception), this is the right moment to take advantage of the subset relation, using UseSubsetRelation (31.13-1).

Analogous relations occur when a factor structure is created from a domain and a subset (see UseFactorRelation (31.13-2)), and when a domain isomorphic to a given one is created (see UseIsomorphismRelation (31.13-3)).

The functions InstallSubsetMaintenance (31.13-4), InstallIsomorphismMaintenance (31.13-6), and InstallFactorMaintenance (31.13-5) are used to tell GAP under what conditions an attribute is maintained under taking subsets, or forming factor structures or isomorphic domains. This is used only when a new attribute is created, see 79.3. For the attributes already available, such as IsFinite (30.4-2) and IsCommutative (35.4-9), the maintenances are already notified.

31.13-1 UseSubsetRelation
‣ UseSubsetRelation( super, sub )( operation )

Methods for this operation transfer possibly useful information from the domain super to its subset sub, and vice versa.

UseSubsetRelation is designed to be called automatically whenever substructures of domains are constructed. So the methods must be cheap, and the requirements should be as sharp as possible!

To achieve that all applicable methods are executed, all methods for this operation except the default method must end with TryNextMethod(). This default method deals with the information that is available by the calls of InstallSubsetMaintenance (31.13-4) in the GAP library.

gap> g:= Group( (1,2), (3,4), (5,6) );; h:= Group( (1,2), (3,4) );;
gap> IsAbelian( g );  HasIsAbelian( h );
true
false
gap> UseSubsetRelation( g, h );;  HasIsAbelian( h );  IsAbelian( h );
true
true

31.13-2 UseFactorRelation
‣ UseFactorRelation( numer, denom, factor )( operation )

Methods for this operation transfer possibly useful information from the domain numer or its subset denom to the domain factor that is isomorphic to the factor of numer by denom, and vice versa. denom may be fail, for example if factor is just known to be a factor of numer but denom is not available as a GAP object; in this case those factor relations are used that are installed without special requirements for denom.

UseFactorRelation is designed to be called automatically whenever factor structures of domains are constructed. So the methods must be cheap, and the requirements should be as sharp as possible!

To achieve that all applicable methods are executed, all methods for this operation except the default method must end with a call to TryNextMethod (78.4-1). This default method deals with the information that is available by the calls of InstallFactorMaintenance (31.13-5) in the GAP library.

gap> g:= Group( (1,2,3,4), (1,2) );; h:= Group( (1,2,3), (1,2) );;
gap> IsSolvableGroup( g );  HasIsSolvableGroup( h );
true
false
gap> UseFactorRelation(g, Subgroup( g, [ (1,2)(3,4), (1,3)(2,4) ] ), h);;
gap> HasIsSolvableGroup( h );  IsSolvableGroup( h );
true
true

31.13-3 UseIsomorphismRelation
‣ UseIsomorphismRelation( old, new )( operation )

Methods for this operation transfer possibly useful information from the domain old to the isomorphic domain new.

UseIsomorphismRelation is designed to be called automatically whenever isomorphic structures of domains are constructed. So the methods must be cheap, and the requirements should be as sharp as possible!

To achieve that all applicable methods are executed, all methods for this operation except the default method must end with a call to TryNextMethod (78.4-1). This default method deals with the information that is available by the calls of InstallIsomorphismMaintenance (31.13-6) in the GAP library.

gap> g:= Group( (1,2) );;  h:= Group( [ [ -1 ] ] );;
gap> Size( g );  HasSize( h );
2
false
gap> UseIsomorphismRelation( g, h );;  HasSize( h );  Size( h );
true
2

31.13-4 InstallSubsetMaintenance
‣ InstallSubsetMaintenance( opr, super_req, sub_req )( function )

opr must be a property or an attribute. The call of InstallSubsetMaintenance has the effect that for a domain \(D\) in the filter super_req, and a domain \(S\) in the filter sub_req, the call UseSubsetRelation\(( D, S )\) (see UseSubsetRelation (31.13-1)) sets a known value of opr for \(D\) as value of opr also for \(S\). A typical example for which InstallSubsetMaintenance is applied is given by opr = IsFinite, super_req = IsCollection and IsFinite, and sub_req = IsCollection.

If opr is a property and the filter super_req lies in the filter opr then we can use also the following inverse implication. If \(D\) is in the filter whose intersection with opr is super_req and if \(S\) is in the filter sub_req, \(S\) is a subset of \(D\), and the value of opr for \(S\) is false then the value of opr for \(D\) is also false.

31.13-5 InstallFactorMaintenance
‣ InstallFactorMaintenance( opr, numer_req, denom_req, factor_req )( function )

opr must be a property or an attribute. The call of InstallFactorMaintenance has the effect that for collections \(N\), \(D\), \(F\) in the filters numer_req, denom_req, and factor_req, respectively, the call UseFactorRelation\(( N, D, F )\) (see UseFactorRelation (31.13-2)) sets a known value of opr for \(N\) as value of opr also for \(F\). A typical example for which InstallFactorMaintenance is applied is given by opr = IsFinite, numer_req = IsCollection and IsFinite, denom_req = IsCollection, and factor_req = IsCollection.

For the other direction, if numer_req involves the filter opr then a known false value of opr for \(F\) implies a false value for \(D\) provided that \(D\) lies in the filter obtained from numer_req by removing opr.

Note that an implication of a factor relation holds in particular for the case of isomorphisms. So one need not install an isomorphism maintained method when a factor maintained method is already installed. For example, UseIsomorphismRelation (31.13-3) will transfer a known IsFinite (30.4-2) value because of the installed factor maintained method.

31.13-6 InstallIsomorphismMaintenance
‣ InstallIsomorphismMaintenance( opr, old_req, new_req )( function )

opr must be a property or an attribute. The call of InstallIsomorphismMaintenance has the effect that for a domain \(D\) in the filter old_req, and a domain \(E\) in the filter new_req, the call UseIsomorphismRelation\(( D, E )\) (see UseIsomorphismRelation (31.13-3)) sets a known value of opr for \(D\) as value of opr also for \(E\). A typical example for which InstallIsomorphismMaintenance is applied is given by opr = Size, old_req = IsCollection, and new_req = IsCollection.

31.14 Useful Categories of Elements

This section and the following one are rather technical, and may be interesting only for those GAP users who want to implement new kinds of elements.

It deals with certain categories of elements that are useful mainly for the design of elements, from the viewpoint that one wants to form certain domains of these elements. For example, a domain closed under multiplication * (a so-called magma, see Chapter 35) makes sense only if its elements can be multiplied, and the latter is indicated by the category IsMultiplicativeElement (31.14-10) for each element. Again note that the underlying idea is that a domain is regarded as generated by given elements, and that these elements carry information about the desired domain. For general information on categories and their hierarchies, see 13.3.

More special categories of this kind are described in the contexts where they arise, they are IsRowVector (23.1-1), IsMatrix (24.2-1), IsOrdinaryMatrix (24.2-2), and IsLieMatrix (24.2-3).

31.14-1 IsExtAElement
‣ IsExtAElement( obj )( category )

An external additive element is an object that can be added via + with other elements (not necessarily in the same family, see 13.1).

31.14-2 IsNearAdditiveElement
‣ IsNearAdditiveElement( obj )( category )

A near-additive element is an object that can be added via + with elements in its family (see 13.1); this addition is not necessarily commutative.

31.14-3 IsAdditiveElement
‣ IsAdditiveElement( obj )( category )

An additive element is an object that can be added via + with elements in its family (see 13.1); this addition is commutative.

31.14-4 IsNearAdditiveElementWithZero
‣ IsNearAdditiveElementWithZero( obj )( category )

A near-additive element-with-zero is an object that can be added via + with elements in its family (see 13.1), and that is an admissible argument for the operation Zero (31.10-3); this addition is not necessarily commutative.

31.14-5 IsAdditiveElementWithZero
‣ IsAdditiveElementWithZero( obj )( category )

An additive element-with-zero is an object that can be added via + with elements in its family (see 13.1), and that is an admissible argument for the operation Zero (31.10-3); this addition is commutative.

31.14-6 IsNearAdditiveElementWithInverse
‣ IsNearAdditiveElementWithInverse( obj )( category )

A near-additive element-with-inverse is an object that can be added via + with elements in its family (see 13.1), and that is an admissible argument for the operations Zero (31.10-3) and AdditiveInverse (31.10-9); this addition is not necessarily commutative.

31.14-7 IsAdditiveElementWithInverse
‣ IsAdditiveElementWithInverse( obj )( category )

An additive element-with-inverse is an object that can be added via + with elements in its family (see 13.1), and that is an admissible argument for the operations Zero (31.10-3) and AdditiveInverse (31.10-9); this addition is commutative.

31.14-8 IsExtLElement
‣ IsExtLElement( obj )( category )

An external left element is an object that can be multiplied from the left, via *, with other elements (not necessarily in the same family, see 13.1).

31.14-9 IsExtRElement
‣ IsExtRElement( obj )( category )

An external right element is an object that can be multiplied from the right, via *, with other elements (not necessarily in the same family, see 13.1).

31.14-10 IsMultiplicativeElement
‣ IsMultiplicativeElement( obj )( category )

A multiplicative element is an object that can be multiplied via * with elements in its family (see 13.1).

31.14-11 IsMultiplicativeElementWithOne
‣ IsMultiplicativeElementWithOne( obj )( category )

A multiplicative element-with-one is an object that can be multiplied via * with elements in its family (see 13.1), and that is an admissible argument for the operation One (31.10-2).

31.14-12 IsMultiplicativeElementWithZero
‣ IsMultiplicativeElementWithZero( elt )( category )

Returns: true or false.

This is the category of elements in a family which can be the operands of * (multiplication) and the operation MultiplicativeZero (35.4-11).

gap> S:=Semigroup(Transformation( [ 1, 1, 1 ] ));;
gap> M:=MagmaWithZeroAdjoined(S);
<<commutative transformation semigroup of degree 3 with 1 generator>
  with 0 adjoined>
gap> x:=Representative(M);
<semigroup with 0 adjoined elt: Transformation( [ 1, 1, 1 ] )>
gap> IsMultiplicativeElementWithZero(x);
true
gap> MultiplicativeZeroOp(x);
<semigroup with 0 adjoined elt: 0>

31.14-13 IsMultiplicativeElementWithInverse
‣ IsMultiplicativeElementWithInverse( obj )( category )

A multiplicative element-with-inverse is an object that can be multiplied via * with elements in its family (see 13.1), and that is an admissible argument for the operations One (31.10-2) and Inverse (31.10-8). (Note the word "admissible": an object in this category does not necessarily have an inverse, Inverse (31.10-8) may return fail.)

31.14-14 IsVector
‣ IsVector( obj )( category )

A vector is an additive-element-with-inverse that can be multiplied from the left and right with other objects (not necessarily of the same type). Examples are cyclotomics, finite field elements, and of course row vectors (see below).

Note that not all lists of ring elements are regarded as vectors, for example lists of matrices are not vectors. This is because although the category IsAdditiveElementWithInverse (31.14-7) is implied by the meet of its collections category and IsList (21.1-1), the family of a list entry may not imply IsAdditiveElementWithInverse (31.14-7) for all its elements.

31.14-15 IsNearRingElement
‣ IsNearRingElement( obj )( category )

IsNearRingElement is just a synonym for the meet of IsNearAdditiveElementWithInverse (31.14-6) and IsMultiplicativeElement (31.14-10).

31.14-16 IsRingElement
‣ IsRingElement( obj )( category )

IsRingElement is just a synonym for the meet of IsAdditiveElementWithInverse (31.14-7) and IsMultiplicativeElement (31.14-10).

31.14-17 IsNearRingElementWithOne
‣ IsNearRingElementWithOne( obj )( category )

IsNearRingElementWithOne is just a synonym for the meet of IsNearAdditiveElementWithInverse (31.14-6) and IsMultiplicativeElementWithOne (31.14-11).

31.14-18 IsRingElementWithOne
‣ IsRingElementWithOne( obj )( category )

IsRingElementWithOne is just a synonym for the meet of IsAdditiveElementWithInverse (31.14-7) and IsMultiplicativeElementWithOne (31.14-11).

31.14-19 IsNearRingElementWithInverse
‣ IsNearRingElementWithInverse( obj )( category )

IsNearRingElementWithInverse is just a synonym for the meet of IsNearAdditiveElementWithInverse (31.14-6) and IsMultiplicativeElementWithInverse (31.14-13).

31.14-20 IsRingElementWithInverse
‣ IsRingElementWithInverse( obj )( category )
‣ IsScalar( obj )( category )

IsRingElementWithInverse and IsScalar are just synonyms for the meet of IsAdditiveElementWithInverse (31.14-7) and IsMultiplicativeElementWithInverse (31.14-13).

31.15 Useful Categories for all Elements of a Family

The following categories of elements are to be understood mainly as categories for all objects in a family, they are usually used as third argument of NewFamily (see 79.7). The purpose of each of the following categories is then to guarantee that each collection of its elements automatically lies in its collections category (see CategoryCollections (30.2-4)).

For example, the multiplication of permutations is associative, and it is stored in the family of permutations that each permutation lies in IsAssociativeElement (31.15-1). As a consequence, each magma consisting of permutations (more precisely: each collection that lies in the family CollectionsFamily( PermutationsFamily ), see CollectionsFamily (30.2-1)) automatically lies in CategoryCollections( IsAssociativeElement ). A magma in this category is always known to be associative, via a logical implication (see 78.7).

Similarly, if a family knows that all its elements are in the categories IsJacobianElement (31.15-5) and IsZeroSquaredElement (31.15-6), then each algebra of these elements is automatically known to be a Lie algebra (see Chapter 62).

31.15-1 IsAssociativeElement
‣ IsAssociativeElement( obj )( category )
‣ IsAssociativeElementCollection( obj )( category )
‣ IsAssociativeElementCollColl( obj )( category )

An element obj in the category IsAssociativeElement knows that the multiplication of any elements in the family of obj is associative. For example, all permutations lie in this category, as well as those ordinary matrices (see IsOrdinaryMatrix (24.2-2)) whose entries are also in IsAssociativeElement.

31.15-2 IsAdditivelyCommutativeElement
‣ IsAdditivelyCommutativeElement( obj )( category )
‣ IsAdditivelyCommutativeElementCollection( obj )( category )
‣ IsAdditivelyCommutativeElementCollColl( obj )( category )
‣ IsAdditivelyCommutativeElementFamily( obj )( category )

An element obj in the category IsAdditivelyCommutativeElement knows that the addition of any elements in the family of obj is commutative. For example, each finite field element and each rational number lies in this category.

31.15-3 IsCommutativeElement
‣ IsCommutativeElement( obj )( category )
‣ IsCommutativeElementCollection( obj )( category )
‣ IsCommutativeElementCollColl( obj )( category )

An element obj in the category IsCommutativeElement knows that the multiplication of any elements in the family of obj is commutative. For example, each finite field element and each rational number lies in this category.

31.15-4 IsFiniteOrderElement
‣ IsFiniteOrderElement( obj )( category )
‣ IsFiniteOrderElementCollection( obj )( category )
‣ IsFiniteOrderElementCollColl( obj )( category )

An element obj in the category IsFiniteOrderElement knows that it has finite multiplicative order. For example, each finite field element and each permutation lies in this category. However the value may be false even if obj has finite order, but if this was not known when obj was constructed.

Although it is legal to set this filter for any object with finite order, this is really useful only in the case that all elements of a family are known to have finite order.

31.15-5 IsJacobianElement
‣ IsJacobianElement( obj )( category )
‣ IsJacobianElementCollection( obj )( category )
‣ IsJacobianElementCollColl( obj )( category )
‣ IsRestrictedJacobianElement( obj )( category )
‣ IsRestrictedJacobianElementCollection( obj )( category )
‣ IsRestrictedJacobianElementCollColl( obj )( category )

An element obj in the category IsJacobianElement knows that the multiplication of any elements in the family \(F\) of obj satisfies the Jacobi identity, that is, \(x * y * z + z * x * y + y * z * x\) is zero for all \(x\), \(y\), \(z\) in \(F\).

For example, each Lie matrix (see IsLieMatrix (24.2-3)) lies in this category.

31.15-6 IsZeroSquaredElement
‣ IsZeroSquaredElement( obj )( category )
‣ IsZeroSquaredElementCollection( obj )( category )
‣ IsZeroSquaredElementCollColl( obj )( category )

An element obj in the category IsZeroSquaredElement knows that obj^2 = Zero( obj ). For example, each Lie matrix (see IsLieMatrix (24.2-3)) lies in this category.

Although it is legal to set this filter for any zero squared object, this is really useful only in the case that all elements of a family are known to have square zero.

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