[ref] 30 Domains and their Elements

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# 30 Domains and their Elements

### Sections

1. Operational Structure of Domains
2. Equality and Comparison of Domains
3. Constructing Domains
4. Changing the Structure
5. Changing the Representation
6. Domain Categories
7. Parents
8. Constructing Subdomains
9. Operations for Domains
10. Attributes and Properties of Elements
11. Comparison Operations for Elements
12. Arithmetic Operations for Elements
13. Relations Between Domains
14. Useful Categories of Elements
15. Useful Categories for all Elements of a Family

Domain is GAP's name for structured sets. The ring of Gaussian integers Z[i] is an example of a domain, the group D12 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` (see Gaussians) and the field of rationals is predefined as `Rationals` (see Rational Numbers). Most domains are constructed by functions, which are called domain constructors (see Constructing Domains). For example the group D12 is constructed by the construction `Group( (1,2,3,4,5,6), (2,6)(3,5) )` (see Group) and the finite field with 16 elements is constructed by `GaloisField( 16 )` (see GaloisField).

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 D12, you had better be able to represent this group in a way that the `Size` 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`, 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 exp( 2 pi/4 ) is written as `E(4)` in GAP.)

The most important facts about domains are stated in Chapter Domains of the GAP Tutorial.

There are only few operations especially for domains (see Operations for Domains), operations such as `Intersection` and `Random` are defined for the more general situation of collections (see Chapter Collections).

## 30.1 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 Properties and Attributes of (General) Mappings).

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 Domain Categories.

## 30.2 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` (see AsSSortedList) 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 Sorted Lists and Sets), i.e., a sorted list without holes or duplicates (see AsSSortedList), 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`.)

## 30.3 Constructing Domains

For several operational structures (see Operational Structure of Domains), GAP provides functions to construct domains with this structure. For example, `Group` returns groups, `VectorSpace` returns vector spaces etc.

- ``Struct`(` arg1`,` arg2`, ... ) F`

  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

- `IsGeneratorsOf`Struct`( [`info`, ]`gens`) O`

  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 ``Struct`ByGenerators` (see below). (So if one wants to omit the check then one should call ``Struct`ByGenerators` directly.)

- `GeneratorsOf`Struct`(` D`) A`

  For a domain D with operational structure corresponding to Struct, the attribute `GeneratorsOf`Struct`` returns a list of corresponding generators of D. If these generators were not yet stored in D then D must know some generators if `GeneratorsOf`Struct`` 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.

- ``Struct`ByGenerators( [`info`, ]`gens`) O`

  Domain construction from generators gens is implemented by operations ``Struct`ByGenerators`, 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 `GeneratorsOf`Struct`` 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 `GeneratorsOf`Struct`` then the list `GeneratorsOf`Struct`(` D`)` is guaranteed to be equal to `[` a`,` b`, ... ]`.

- ``Struct`WithGenerators( [`info`, ]`gens`) O`

  The only difference between ``Struct`ByGenerators` and ``Struct`WithGenerators` is that the latter guarantees that the `GeneratorsOf`Struct`` value of the result is equal to the given generators gens.

- `Closure`Struct`(` D`,` obj`) O`

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

  ## 30.4 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 Domain Categories) then it is reasonable to ask for the group that is equal to M.

- `As`Struct`( [`info`, ]`D`) O`

  If D is a domain that is closed under the operational structure given by Struct then `As`Struct`` returns a domain E that consists of the same elements (that is, ``D`=` E``) and that has this operational structure (that is, `Is`Struct`(` E`)` is `true`); if D is not closed under the structure given by Struct then `As`Struct`` 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 `As`Struct`` is in fact an attribute (see Attributes).

  See the index of the GAP Reference Manual for an overview of the available `As`Struct`` functions.

  ## 30.5 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 Changing the Structure).

  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 Matrix Groups) then one can compute an isomorphism to a polycyclicly presented group G, say (see Chapter Polycyclic Groups); 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.

- `Isomorphism`Rep``Struct`(` D`) A`

  If D is a domain that is closed under the operational structure given by Struct then `Isomorphism`Rep``Struct`` 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` (see IsomorphismPermGroup) takes a group as its argument and returns a group homomorphism (see Group Homomorphisms) onto an isomorphic permutation group (see Chapter Permutation Groups) provided the original group is finite; for infinite groups, `IsomorphismPermGroup` returns `fail`. Similarly, `IsomorphismPcGroup` (see IsomorphismPcGroup) returns a group homomorphism from its argument to a polycyclicly presented group (see Pc Groups) if the argument is polycyclic, and `fail` otherwise.

  See the index of the GAP Reference Manual for an overview of the available `Isomorphism`Rep``Struct`` functions.

  ## 30.6 Domain Categories

  As mentioned in Operational Structure of Domains, the operational structure of a domain is fixed when the domain is constructed. For example, if D was constructed by `Monoid` 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` returns `false` for D. The operational structure determines which operations are applicable for a domain, so for example `SylowSubgroup` is not defined for D and therefore will signal an error.

- `Is`Struct`(` D`)`

  The functions `Is`Struct`` implement the tests whether a domain D has the respective operational structure (upon construction). `Is`Struct`` is a filter (see Types of Objects) that involves certain categories (see Categories) and usually also certain properties (see Properties). For example, `IsGroup` 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` is implied by each domain category,
  - `IsMagma` is implied by each category that describes the closure under multiplication `*`,
  - `IsAdditiveMagma` is implied by each category that describes the closure under addition `+`,
  - `IsMagmaWithOne` implies `IsMagma`; 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` implies `IsMagmaWithOne`; 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 `Is`Struct``, and operations ``Struct`ByGenerators` for constructing a domain from generators, `GeneratorsOf`Struct`` for storing and accessing generators w.r.t. this structure, `Closure`Struct`` for forming the closure, and `As`Struct`` 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 Rings, Fields and Division Rings, Groups, and Vector Spaces, respectively. More general functions for arbitrary collections can be found in Chapter Collections.

  ## 30.7 Parents

- `Parent(` D`) F`
- `SetParent(` D`,` P`) O`
- `HasParent(` D`) F`

  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` (see UseSubsetRelation) 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 In Parent Attributes in ``Extending GAP''.) 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` (see UseSubsetRelation), 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 returns `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`, are described in Section Constructing Subdomains. Whenever a function has this property, the Reference Manual states this explicitly. Note that these functions do not guarantee a certain parent, for example `DerivedSubgroup` (see DerivedSubgroup) 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

  ## 30.8 Constructing Subdomains

  For many domains D, there are functions that construct certain subsets S of D as domains with parent (see Parents) 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``.

- `Sub`struct`(` D`,` gens`) F`

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

- `Sub`struct`NC(` D`,` gens`) F`

  Each function `Sub`struct`` 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 `Sub`struct`NC` instead; the suffix `NC` stands for ``no check''.

- `AsSub`struct`(` D`,` S`) F`

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

- `IsSub`struct`(` D`,` S`) F`

  There is no real need for functions that check whether a domain S is a `Sub`struct`` 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` (see IsSubset) 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 `IsSub`struct`` is available in GAP then it is implemented as first a call to `Is`Struct`` for the second argument and then a call to `IsSubset` for the two arguments.

  ## 30.9 Operations for Domains

  For the meaning of the attributes `Characteristic`, `One`, `Zero` in the case of a domain argument, see Attributes and Properties of Elements.

- `IsGeneralizedDomain(` D`) C`
- `IsDomain(` D`) C`

  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`.

- `GeneratorsOfDomain(` D`) A`

  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 Operational Structure of Domains), the only possible choice of elements is to take all elements of D. See Constructing Domains and Changing the Structure for the concepts of other notions of generation.

- `Domain( [`Fam`, ]`generators`) F`
- `DomainByGenerators(` Fam`,` generators`) O`

  `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 first argument, and the returned (empty) domain lies in the collections family of Fam.

  `DomainByGenerators` is the operation called by `Domain`.

  ## 30.10 Attributes and Properties of Elements

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

- `Characteristic(` obj`) A`

  `Characteristic` returns the characteristic of obj, where obj must either be an additive element, a domain or a family.

  For a domain D, the characteristic is defined if D is closed under addition and has a zero element ``z`= Zero(` D`)` (see Zero); in this case, `Characteristic(` D`)` is the smallest positive integer p such that ``p`*` x`=` z`` for all elements x in D, if such an integer exists, and the integer zero `0` otherwise.

  If a family has a characteristic then this means that all domains of elements in this family have this characteristic. In this case, also each element in the family has this characteristic. (Note that also the zero element z of a finite field in characteristic p has characteristic p, although n * z = z for any integer n.)

- `OneImmutable(` obj`) A`
- `OneAttr(` obj`) AM`
- `One(` obj`) AM`
- `Identity(` obj`) AM`
- `OneMutable(` obj`) O`
- `OneOp(` obj`) O`
- `OneSameMutability(` obj`) O`
- `OneSM(` obj`) O`

  `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). `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`, `Identity`, `OneOp`, and `OneSM` are synonyms as listed above.

  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) 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); use `MultiplicativeNeutralElement` (see MultiplicativeNeutralElement) 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 Rings). 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` (see Representative), but `Representative` is allowed to fetch the identity of a domain D only if `HasOne(` D`)` is `true`.

- `ZeroImmutable(` obj`) A`
- `ZeroAttr(` obj`) AM`
- `Zero(` obj`) AM`
- `ZeroMutable(` obj`) O`
- `ZeroOp(` obj`) O`
- `ZeroSameMutability(` obj`) O`
- `ZeroSM(` obj`) O`

  `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). `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`, `Zero`, `ZeroOp` and `ZeroSM` are synonyms as listed above.

  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) 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); use `AdditiveNeutralElement` (see AdditiveNeutralElement) 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` (see Representative), but `Representative` is allowed to fetch the zero of a domain D only if `HasZero(` D`)` is `true`.

- `MultiplicativeZeroOp(` elt`) O`

  returns the element z in the family F of elt with the property that z * m = z = m * z holds for all m Î F, if such an element is known.

  Families of elements in the category IsMultiplicativeElementWithZero often arise from adjoining a new zero to an existing magma. See InjectionZeroMagma for details.

- `IsOne(` elm`) P`

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

- `IsZero(` elm`) P`

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

- `IsIdempotent(` elt`) P`

  true iff elt is its own square. (Even if IsZero(elt) is also true.)

- `InverseImmutable(` elm`) A`
- `InverseAttr(` elm`) AM`
- `Inverse(` elm`) AM`
- `InverseMutable(` elm`) O`
- `InverseOp(` elm`) O`
- `InverseSameMutability(` elm`) O`
- `InverseSM(` elm`) O`

  `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 Fail) 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 Technical Matters Concerning General Mappings). For a bijective mapping f whose source and range differ, `InverseGeneralMapping` (see InverseGeneralMapping) 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`, `Inverse`, `InverseOp` and `InverseSM` are synonyms as listed above.

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

- `AdditiveInverseImmutable(` elm`) A`
- `AdditiveInverseAttr(` elm`) AM`
- `AdditiveInverse(` elm`) AM`
- `AdditiveInverseMutable(` elm`) O`
- `AdditiveInverseOp(` elm`) O`
- `AdditiveInverseSameMutability(` elm`) O`
- `AdditiveInverseSM(` elm`) O`

  `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). `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`, `AdditiveInverse`, `AdditiveInverseOp` and `AdditiveInverseSM` are synonyms as listed above.

  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`.

- `Order(` elm`) A`

  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`, but it also might run into an infinite loop trying to test the order.

  ## 30.11 Comparison Operations for Elements

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

- `\=(` left-expr`,` right-expr`) O`
- `\<(` left-expr`,` right-expr`) O`

  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 Comparisons.

  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(` obj`) P`
- `CanEasilyCompareElementsFamily(` fam`) F`
- `CanEasilySortElements(` obj`) P`
- `CanEasilySortElementsFamily(` fam`) F`

  `CanEasilyCompareElements` indicates whether the elements in the family fam of obj can be easily compared with `=`. (In some cases element comparisons are very hard, for example in cases where no normal forms for the elements exist.)

  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` 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 `<`. This property implies `CanEasilyCompareElements`, as the ordering must be total.

  ## 30.12 Arithmetic Operations for Elements

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

- `\+(` left-expr`,` right-expr`) O`
- `\*(` left-expr`,` right-expr`) O`
- `\/(` left-expr`,` right-expr`) O`
- `\^(` left-expr`,` right-expr`) O`
- `\mod(` left-expr`,` right-expr`) O`

  For details about special methods for `\mod`, consult the index entries for ``mod''.

- `LeftQuotient(` elm1`,` elm2`) O`

  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.

- `Comm(` elm1`,` elm2`) O`

  returns the commutator of elm1 and elm2. The commutator is defined as the product elm1 -1 * elm2 -1 * elm1 * 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)

- `LieBracket(` elm1`,` elm2`) O`

  returns the element ``elm1`*` elm2`-` elm2`*` elm1``.

  The addition `\+` is assumed to be associative but not assumed to be commutative (see IsAdditivelyCommutative). The multiplication `\*` is not assumed to be commutative or associative (see IsCommutative, IsAssociative).

- `Sqrt(` obj`) O`

  `Sqrt` returns a square root of obj, that is, an object x with the property that x ·x = obj holds. If such an x is not unique then the choice of x depends on the type of obj. For example, `ER` (see ER) is the `Sqrt` method for rationals (see IsRat).

  ## 30.13 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) of two domains, or a Sylow subgroup of a given group (see SylowSubgroup).

  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 Constructing Subdomains for the exception), this is the right moment to take advantage of the subset relation.

  Analogous relations occur when a factor structure is created from a domain and a subset, and when a domain isomorphic to a given one is created.

- `UseSubsetRelation(` super`,` sub`) O`

  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` 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

- `UseIsomorphismRelation(` old`,` new`) O`

  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 `TryNextMethod()`. This default method deals with the information that is available by the calls of `InstallIsomorphismMaintenance` 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

- `UseFactorRelation(` numer`,` denom`,` factor`) O`

  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 `TryNextMethod()`. This default method deals with the information that is available by the calls of `InstallFactorMaintenance` 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

  The following functions 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 Creating Attributes and Properties in ``Programming in GAP''. For the attributes already available, such as `IsFinite` and `IsCommutative`, the maintenances are already notified.

- `InstallSubsetMaintenance(` opr`,` super_req`,` sub_req`) F`

  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) 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`.

- `InstallIsomorphismMaintenance(` opr`,` old_req`,` new_req`) F`

  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) 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`.

- `InstallFactorMaintenance(` opr`,` numer_req`,` denom_req`,` factor_req`) F`

  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) 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` (see UseIsomorphismRelation) will transfer a known `IsFinite` value because of the installed factor maintained method.

  ## 30.14 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 Magmas) makes sense only if its elements can be multiplied, and the latter is indicated by the category `IsMultiplicativeElement` 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 Categories.

- `IsExtAElement(` obj`) C`

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

- `IsNearAdditiveElement(` obj`) C`

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

- `IsAdditiveElement(` obj`) C`

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

- `IsNearAdditiveElementWithZero(` obj`) C`

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

- `IsAdditiveElementWithZero(` obj`) C`

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

- `IsNearAdditiveElementWithInverse(` obj`) C`

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

- `IsAdditiveElementWithInverse(` obj`) C`

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

- `IsExtLElement(` obj`) C`

  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 Families).

- `IsExtRElement(` obj`) C`

  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 Families).

- `IsMultiplicativeElement(` obj`) C`

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

- `IsMultiplicativeElementWithOne(` obj`) C`

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

- `IsMultiplicativeElementWithZero(` elt`) C`

  Elements in a family which can be the operands of the `*` and the operation MultiplicativeZero.

- `IsMultiplicativeElementWithInverse(` obj`) C`

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

- `IsVector(` obj`) C`

  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` is implied by the join of its collections category and `IsList`, the family of a list entry may not imply `IsAdditiveElementWithInverse` for all its elements.

- `IsNearRingElement(` obj`) C`

  `IsNearRingElement` is just a synonym for the join of `IsNearAdditiveElementWithInverse` and `IsMultiplicativeElement`.

- `IsRingElement(` obj`) C`

  `IsRingElement` is just a synonym for the join of `IsAdditiveElementWithInverse` and `IsMultiplicativeElement`.

- `IsNearRingElementWithOne(` obj`) C`

  `IsNearRingElementWithOne` is just a synonym for the join of `IsNearAdditiveElementWithInverse` and `IsMultiplicativeElementWithOne`.

- `IsRingElementWithOne(` obj`) C`

  `IsRingElementWithOne` is just a synonym for the join of `IsAdditiveElementWithInverse` and `IsMultiplicativeElementWithOne`.

- `IsNearRingElementWithInverse(` obj`) C`

- `IsRingElementWithInverse(` obj`) C`
- `IsScalar(` obj`) C`

  `IsRingElementWithInverse` and `IsScalar` are just synonyms for the join of `IsAdditiveElementWithInverse` and `IsMultiplicativeElementWithInverse`.

  More special categories of this kind are described in the contexts where they arise, they are `IsRowVector` (see IsRowVector), `IsMatrix` (see IsMatrix), `IsOrdinaryMatrix` (see IsOrdinaryMatrix), and `IsLieMatrix` (see IsLieMatrix).

  ## 30.15 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 Creating Families in ``Programming in GAP''). 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).

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

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

- `IsAssociativeElement(` obj`) C`
- `IsAssociativeElementCollection(` obj`) C`
- `IsAssociativeElementCollColl(` obj`) C`

  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) whose entries are also in `IsAssociativeElement`.

- `IsAdditivelyCommutativeElement(` obj`) C`
- `IsAdditivelyCommutativeElementCollection(` obj`) C`
- `IsAdditivelyCommutativeElementCollColl(` obj`) C`
- `IsAdditivelyCommutativeElementFamily(` obj`) C`

  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.

- `IsCommutativeElement(` obj`) C`
- `IsCommutativeElementCollection(` obj`) C`
- `IsCommutativeElementCollColl(` obj`) C`

  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.

- `IsFiniteOrderElement(` obj`) C`
- `IsFiniteOrderElementCollection(` obj`) C`
- `IsFiniteOrderElementCollColl(` obj`) C`

  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.

- `IsJacobianElement(` obj`) C`
- `IsJacobianElementCollection(` obj`) C`
- `IsJacobianElementCollColl(` obj`) C`

  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) lies in this category.

- `IsZeroSquaredElement(` obj`) C`
- `IsZeroSquaredElementCollection(` obj`) C`
- `IsZeroSquaredElementCollColl(` obj`) C`

  An element obj in the category `IsZeroSquaredElement` knows that ``obj`^2 = Zero(` obj`)`. For example, each Lie matrix (see IsLieMatrix) 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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  GAP 4 manual
  May 2002