GAP (ref) - Chapter 31: Domains and their Elements

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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
31.7-1 Parent
31.8 Constructing Subdomains
31.9 Operations for Domains
31.9-1 IsGeneralizedDomain
31.9-2 GeneratorsOfDomain
31.9-3 Domain
31.10 Attributes and Properties of Elements
31.10-1 Characteristic
31.10-2 OneImmutable
31.10-3 ZeroImmutable
31.10-4 MultiplicativeZeroOp
31.10-5 IsOne
31.10-6 IsZero
31.10-7 IsIdempotent
31.10-8 InverseImmutable
31.10-9 AdditiveInverseImmutable
31.10-10 Order
31.11 Comparison Operations for Elements
31.11-1 \= and \<
31.11-2 CanEasilyCompareElements
31.12 Arithmetic Operations for Elements
31.12-1 \+, \*, \/, \^, \mod
31.12-2 LeftQuotient
31.12-3 Comm
31.12-4 LieBracket
31.12-5 Sqrt
31.13 Relations Between Domains
31.13-1 UseSubsetRelation
31.13-2 UseFactorRelation
31.13-3 UseIsomorphismRelation
31.13-4 InstallSubsetMaintenance
31.13-5 InstallFactorMaintenance
31.13-6 InstallIsomorphismMaintenance
31.14 Useful Categories of Elements
31.14-1 IsExtAElement
31.14-2 IsNearAdditiveElement
31.14-3 IsAdditiveElement
31.14-4 IsNearAdditiveElementWithZero
31.14-5 IsAdditiveElementWithZero
31.14-6 IsNearAdditiveElementWithInverse
31.14-7 IsAdditiveElementWithInverse
31.14-8 IsExtLElement
31.14-9 IsExtRElement
31.14-10 IsMultiplicativeElement
31.14-11 IsMultiplicativeElementWithOne
31.14-12 IsMultiplicativeElementWithZero
31.14-13 IsMultiplicativeElementWithInverse
31.14-14 IsVector
31.14-15 IsNearRingElement
31.14-16 IsRingElement
31.14-17 IsNearRingElementWithOne
31.14-18 IsRingElementWithOne
31.14-19 IsNearRingElementWithInverse
31.14-20 IsRingElementWithInverse
31.15 Useful Categories for all Elements of a Family
31.15-1 IsAssociativeElement
31.15-2 IsAdditivelyCommutativeElement
31.15-3 IsCommutativeElement
31.15-4 IsFiniteOrderElement
31.15-5 IsJacobianElement
31.15-6 IsZeroSquaredElement

### 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.

- `IsDomain` (31.9-1) is implied by each domain category,

- `IsMagma` (35.1-1) is implied by each category that describes the closure under multiplication `*`,

- `IsAdditiveMagma` (55.1-4) is implied by each category that describes the closure under addition `+`,

- `IsMagmaWithOne` (35.1-2) implies `IsMagma` (35.1-1); a magma-with-one is a magma such that each element (and thus also the magma itself) can be asked for its zeroth power,

- `IsMagmaWithInverses` (35.1-4) implies `IsMagmaWithOne` (35.1-2); a magma-with-inverses is a magma such that each element can be asked for its inverse; important special cases are groups, which in addition are associative,

- a ring is a magma that is also an additive group,

- a ring-with-one is a ring that is also a magma-with-one,

- a division ring is a ring-with-one that is also closed under taking inverses of nonzero elements,

- a field is a commutative division ring.

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 generators of D with respect to the trivial operational structure, that is interpreting D as a set. The returned list may contain repetitions.

See 31.3 and for `GeneratorsOfStruct` methods with respect to other available operational structures.

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.

    gap> G := DihedralGroup(IsPermGroup, 4);;
    gap> GeneratorsOfGroup(G);
    [ (1,2), (3,4) ]
    gap> GeneratorsOfDomain(G);
    [ (), (3,4), (1,2), (1,2)(3,4) ]
    gap> F := FreeGroup("x");
    <free group on the generators [ x ]>
    gap> GeneratorsOfGroup(F);
    [ x ]
    gap> GeneratorsOfDomain(F);
    Error, resulting list would be too large (length infinity)

##### 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` (13.1-2) 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 or property is created, see `NewAttribute` (13.5-3) and `NewProperty` (13.7-4). 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` (78.5-1). 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` (78.5-1). 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 `MultiplicativeZeroOp` (31.10-4).

    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` (13.1-2). 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 78.8).

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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