Regina Calculation Engine
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Represents a finitely generated abelian group. More...
#include <algebra/abeliangroup.h>
Public Member Functions | |
AbelianGroup () | |
Creates a new trivial group. More... | |
AbelianGroup (const AbelianGroup &cloneMe)=default | |
Creates a clone of the given group. More... | |
AbelianGroup (const MatrixInt &M, const MatrixInt &N) | |
Creates an abelian group as the homology of a chain complex. More... | |
AbelianGroup (const MatrixInt &M, const MatrixInt &N, const Integer &p) | |
Creates an abelian group as the homology of a chain complex, using mod-p coefficients. More... | |
void | addRank (int extraRank=1) |
Increments the rank of the group by the given integer. More... | |
void | addTorsionElement (const Integer °ree, unsigned mult=1) |
Adds the given torsion element to the group. More... | |
void | addTorsionElement (unsigned long degree, unsigned mult=1) |
Adds the given torsion element to the group. More... | |
void | addTorsionElements (const std::multiset< Integer > &torsion) |
Adds the given set of torsion elements to this group. More... | |
void | addGroup (const MatrixInt &presentation) |
Adds the abelian group defined by the given presentation to this group. More... | |
void | addGroup (const AbelianGroup &group) |
Adds the given abelian group to this group. More... | |
unsigned | rank () const |
Returns the rank of the group. More... | |
unsigned | torsionRank (const Integer °ree) const |
Returns the rank in the group of the torsion term of given degree. More... | |
unsigned | torsionRank (unsigned long degree) const |
Returns the rank in the group of the torsion term of given degree. More... | |
size_t | countInvariantFactors () const |
Returns the number of invariant factors that describe the torsion elements of this group. More... | |
const Integer & | invariantFactor (size_t index) const |
Returns the given invariant factor describing the torsion elements of this group. More... | |
bool | isTrivial () const |
Determines whether this is the trivial (zero) group. More... | |
bool | isZ () const |
Determines whether this is the infinite cyclic group (Z). More... | |
bool | isFree (unsigned r) const |
Determines whether this is the free abelian group of the given rank. More... | |
bool | isZn (unsigned long n) const |
Determines whether this is the non-trivial cyclic group on the given number of elements. More... | |
bool | operator== (const AbelianGroup &other) const |
Determines whether this and the given abelian group are isomorphic. More... | |
bool | operator!= (const AbelianGroup &other) const |
Determines whether this and the given abelian group are non-isomorphic. More... | |
AbelianGroup & | operator= (const AbelianGroup &cloneMe)=default |
Sets this to be a clone of the given group. More... | |
void | writeXMLData (std::ostream &out) const |
Writes a chunk of XML containing this abelian group. More... | |
void | writeTextShort (std::ostream &out, bool utf8=false) const |
Writes a short text representation of this object to the given output stream. More... | |
void | writeTextLong (std::ostream &out) const |
A default implementation for detailed output. More... | |
std::string | str () const |
Returns a short text representation of this object. More... | |
std::string | utf8 () const |
Returns a short text representation of this object using unicode characters. More... | |
std::string | detail () const |
Returns a detailed text representation of this object. More... | |
Protected Member Functions | |
void | replaceTorsion (const MatrixInt &matrix) |
Replaces the torsion elements of this group with those in the abelian group represented by the given Smith normal form presentation matrix. More... | |
Protected Attributes | |
unsigned | rank_ |
The rank of the group (the number of Z components). More... | |
std::multiset< Integer > | invariantFactors |
The invariant factors d0,...,dn as described in the AbelianGroup notes. More... | |
Represents a finitely generated abelian group.
The torsion elements of the group are stored in terms of their invariant factors. For instance, Z_2+Z_3 will appear as Z_6, and Z_2+Z_2+Z_3 will appear as Z_2+Z_6.
In general the factors will appear as Z_d0+...+Z_dn, where the invariant factors di are all greater than 1 and satisfy d0|d1|...|dn. Note that this representation is unique.
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inline |
Creates a new trivial group.
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default |
Creates a clone of the given group.
cloneMe | the group to clone. |
Creates an abelian group as the homology of a chain complex.
M | the ‘right’ matrix in the chain complex; that is, the matrix that one takes the kernel of when computing homology. |
N | the ‘left’ matrix in the chain complex; that is, the matrix that one takes the image of when computing homology. |
Creates an abelian group as the homology of a chain complex, using mod-p coefficients.
M | the ‘right’ matrix in the chain complex; that is, the matrix that one takes the kernel of when computing homology. |
N | the ‘left’ matrix in the chain complex; that is, the matrix that one takes the image of when computing homology. |
p | the modulus, which may be any Integer. Zero is interpreted as a request for integer coefficents, which will give the same result as the AbelianGroup(const MatrixInt&, const MatrixInt&) constructor. |
void regina::AbelianGroup::addGroup | ( | const MatrixInt & | presentation | ) |
Adds the abelian group defined by the given presentation to this group.
Note that this routine might be slow since calculating the new invariant factors is not trivial.
presentation | a presentation matrix for the group to be added to this group, where each column represents a generator and each row a relation. |
void regina::AbelianGroup::addGroup | ( | const AbelianGroup & | group | ) |
Adds the given abelian group to this group.
Note that this routine might be slow since calculating the new invariant factors is not trivial.
group | the group to add to this one. |
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inline |
Increments the rank of the group by the given integer.
This integer may be positive, negative or zero.
extraRank | the extra rank to add; this defaults to 1. |
void regina::AbelianGroup::addTorsionElement | ( | const Integer & | degree, |
unsigned | mult = 1 |
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Adds the given torsion element to the group.
Note that this routine might be slow since calculating the new invariant factors is not trivial. If many different torsion elements are to be added, consider using addTorsionElements() instead so the invariant factors need only be calculated once.
In this routine we add a specified number of copies of Z_d, where d is some given degree.
degree | d, where we are adding copies of Z_d to the torsion. |
mult | the multiplicity m, where we are adding precisely m copies of Z_d; this defaults to 1. |
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inline |
Adds the given torsion element to the group.
Note that this routine might be slow since calculating the new invariant factors is not trivial. If many different torsion elements are to be added, consider using addTorsionElements() instead so the invariant factors need only be calculated once.
In this routine we add a specified number of copies of Z_d, where d is some given degree.
degree | d, where we are adding copies of Z_d to the torsion. |
mult | the multiplicity m, where we are adding precisely m copies of Z_d; this defaults to 1. |
void regina::AbelianGroup::addTorsionElements | ( | const std::multiset< Integer > & | torsion | ) |
Adds the given set of torsion elements to this group.
Note that this routine might be slow since calculating the new invariant factors is not trivial.
The torsion elements to add are described by a list of integers k1,...,km, where we are adding Z_k1,...,Z_km. Unlike invariant factors, the ki are not required to divide each other.
torsion | a list containing the torsion elements to add, as described above. |
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inline |
Returns the number of invariant factors that describe the torsion elements of this group.
See the AbelianGroup class notes for further details.
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inherited |
Returns a detailed text representation of this object.
This text may span many lines, and should provide the user with all the information they could want. It should be human-readable, should not contain extremely long lines (which cause problems for users reading the output in a terminal), and should end with a final newline. There are no restrictions on the underlying character set.
const Integer& regina::AbelianGroup::invariantFactor | ( | size_t | index | ) | const |
Returns the given invariant factor describing the torsion elements of this group.
See the AbelianGroup class notes for further details.
If the invariant factors are d0|d1|...|dn, this routine will return di where i is the value of parameter index.
index | the index of the invariant factor to return; this must be between 0 and countInvariantFactors()-1 inclusive. |
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inline |
Determines whether this is the free abelian group of the given rank.
r | the rank of the free abelian group that we are testing for. |
true
if and only if this is the free abelian group of rank r.
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Determines whether this is the trivial (zero) group.
true
if and only if this is the trivial group.
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Determines whether this is the infinite cyclic group (Z).
true
if and only if this is the infinite cyclic group.
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Determines whether this is the non-trivial cyclic group on the given number of elements.
As a special case, if n = 0 then this routine will test for the infinite cyclic group (i.e., it will behave the same as isZ()). If n = 1, then this routine will test for the trivial group (i.e., it will behave the same as isTrivial()).
n | the number of elements of the cyclic group in question. |
true
if and only if this is the cyclic group Z_n.
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inline |
Determines whether this and the given abelian group are non-isomorphic.
other | the group with which this should be compared. |
true
if and only if the two groups are non-isomorphic.
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default |
Sets this to be a clone of the given group.
cloneMe | the group to clone. |
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inline |
Determines whether this and the given abelian group are isomorphic.
other | the group with which this should be compared. |
true
if and only if the two groups are isomorphic.
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inline |
Returns the rank of the group.
This is the number of included copies of Z.
Equivalently, the rank is the maximum number of linearly independent elements, and it indicates the size of the largest free abelian subgroup. The rank effectively ignores all torsion elements.
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protected |
Replaces the torsion elements of this group with those in the abelian group represented by the given Smith normal form presentation matrix.
Any zero columns in the matrix will also be added to the rank as additional copies of Z. Note that preexisting torsion elements will be deleted, but preexisting rank will not.
matrix | a matrix containing the Smith normal form presentation matrix for the new torsion elements, where each column represents a generator and each row a relation. |
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inherited |
Returns a short text representation of this object.
This text should be human-readable, should fit on a single line, and should not end with a newline. Where possible, it should use plain ASCII characters.
__str__()
.unsigned regina::AbelianGroup::torsionRank | ( | const Integer & | degree | ) | const |
Returns the rank in the group of the torsion term of given degree.
If the given degree is d, this routine will return the largest m for which mZ_d is a subgroup of this group.
For instance, if this group is Z_6+Z_12, the torsion term of degree 2 has rank 2 (one occurrence in Z_6 and one in Z_12), and the torsion term of degree 4 has rank 1 (one occurrence in Z_12).
degree | the degree of the torsion term to query. |
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inline |
Returns the rank in the group of the torsion term of given degree.
If the given degree is d, this routine will return the largest m for which mZ_d is a subgroup of this group.
For instance, if this group is Z_6+Z_12, the torsion term of degree 2 has rank 2 (one occurrence in Z_6 and one in Z_12), and the torsion term of degree 4 has rank 1 (one occurrence in Z_12).
degree | the degree of the torsion term to query. |
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inherited |
Returns a short text representation of this object using unicode characters.
Like str(), this text should be human-readable, should fit on a single line, and should not end with a newline. In addition, it may use unicode characters to make the output more pleasant to read. This string will be encoded in UTF-8.
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inlineinherited |
A default implementation for detailed output.
This routine simply calls T::writeTextShort() and appends a final newline.
out | the output stream to which to write. |
void regina::AbelianGroup::writeTextShort | ( | std::ostream & | out, |
bool | utf8 = false |
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Writes a short text representation of this object to the given output stream.
The text representation will be of the form 3 Z + 4 Z_2 + Z_120
. The torsion elements will be written in terms of the invariant factors of the group, as described in the AbelianGroup notes.
out | the output stream to which to write. |
utf8 | if true , then richer unicode characters will be used to make the output more pleasant to read. In particular, the output will use subscript digits and the blackboard bold Z. |
void regina::AbelianGroup::writeXMLData | ( | std::ostream & | out | ) | const |
Writes a chunk of XML containing this abelian group.
out | the output stream to which the XML should be written. |
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protected |
The invariant factors d0,...,dn as described in the AbelianGroup notes.
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protected |
The rank of the group (the number of Z components).