shh.homolalg
Class QvspMorphism

Object
  QvspMorphism

All Implemented Interfaces:

ExactCategoryMorphism, Morphism

public class QvspMorphism

extends Object

implements ExactCategoryMorphism

A morphism of Qvsps. It is a linear map from the Qvsp given by getSource() to the Qvsp given by getTarget(), with matrix getMatrix().

While the morphism is between rational vector spaces, the present implementation uses matrices with integer entries. The method getMatrix() always returns a CSparse, and methods that work on QvspMorphisms may assume that all the non-zero matrix entries are in SparseEltZ. The return value of getMatrix() is always non-null in this class, but may be null for subclasses.

For a pair of morphisms, equals(java.lang.Object) works as expected, as long as the two sources and the two targets satisfy equals(java.lang.Object). However, hashCode() is not yet implemented.

Author:

Mark McConnell

Constructor Summary

QvspMorphism(Qvsp source, Qvsp target, CSparse matrix)
Constructor.

Method Summary

Morphism	compose(Morphism f) Returns the composition in the following diagram.
void	destroyAllButCokernel() Provides a more efficient way of computing the cokernel and related morphisms, at the price of destroying this morphism itself.
void	destroyAllButKernel() Provides a more efficient way of computing the kernel and related morphisms, at the price of destroying this morphism itself.
QvspMorphism	dual() If this is a QvspMorphism V → W with matrix A, then dual() returns a QvspMorphism W → V whose matrix is the transpose of A.
boolean	equals(Object o) Works as expected, as long as the two sources and the two targets satisfy equals(java.lang.Object).
ExactCategoryMorphism	getCoimage() See the comment on ExactCategoryMorphism.
ExactCategoryMorphism	getCoimSection() See the comment on ExactCategoryMorphism.
ExactCategoryMorphism	getCoimToIm() The bottom arrow (the isomorphism) in the comment on ExactCategoryMorphism.
ExactCategoryMorphism	getCokernel() See the comment on ExactCategoryMorphism.
ExactCategoryMorphism	getCokerSection() See the comment on ExactCategoryMorphism.
ExactCategoryMorphism	getImage() See the comment on ExactCategoryMorphism.
ExactCategoryMorphism	getImRetraction() See the comment on ExactCategoryMorphism.
Morphism	getInverse() Unsupported at this time (sorry).
ExactCategoryMorphism	getKernel() See the comment on ExactCategoryMorphism.
ExactCategoryMorphism	getKerRetraction() See the comment on ExactCategoryMorphism.
protected CSparse	getMatrix() See the comment on the class.
MPDQ	getMPDQ() Returns an MPDQ holding the Smith normal form for the underlying matrix.
String	getName() Returns the name set by setName(java.lang.String), or null if none has been set.
Obj	getSource() Returns the source (the domain) of this morphism.
Obj	getTarget() Returns the target (sometimes called the range or codomain) of this morphism.
int	hashCode() Unsupported at this time (sorry).
boolean	isCompositionZero(ExactCategoryMorphism f) Whether the composition of this and f (see the diagram for Morphism.compose(shh.homolalg.Morphism)) is a zero morphism.
boolean	isEpic() Whether this is an epimorphism.
boolean	isIsomorphism() Whether this has an inverse morphism.
boolean	isMonic() Whether this is a monomorphism.
boolean	isZero() Whether this is a zero morphism.
static void	main(String[] args) Tests the factorization in ExactCategoryMorphism.
int	rank() The rank of this morphism.
void	setName(String name) Sets the name.
String	torsion() Returns a pretty-print version of the elementary divisors of the underlying matrix over Z.
String	toString() Returns the name, if one has been set by setName(java.lang.String); otherwise, returns a printed representation, including the matrix if it is small enough.

Methods inherited from class Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Constructor Detail

QvspMorphism

public QvspMorphism(Qvsp source,
                    Qvsp target,
                    CSparse matrix)

Constructor. It is an error if any argument is null.

Method Detail

getMatrix

protected CSparse getMatrix()

See the comment on the class.

getName

public final String getName()

Returns the name set by setName(java.lang.String), or null if none has been set. See toString().

setName

public final void setName(String name)

Sets the name. See toString().

toString

public String toString()

Returns the name, if one has been set by setName(java.lang.String); otherwise, returns a printed representation, including the matrix if it is small enough.

getMPDQ

public MPDQ getMPDQ()

Returns an MPDQ holding the Smith normal form for the underlying matrix. Takes care to evaluate it only once. This is the time- and space-consuming step for large matrices.

getSource

public Obj getSource()

Returns the source (the domain) of this morphism. The return type is really Qvsp.

Specified by:

getSource in interface Morphism

getTarget

public Obj getTarget()

Returns the target (sometimes called the range or codomain) of this morphism. The return type is really Qvsp.

Specified by:

getTarget in interface Morphism

compose

public Morphism compose(Morphism f)

Description copied from interface: Morphism

Returns the composition in the following diagram.

       U <------- T <------- S
           this         f

       

The implementation must satisfy the axioms of a category. For instance, composition must be associative.

The implementation may throw an IllegalArgumentException if the source of this and the target of f are not equal in the sense of equals.

The implementation may throw an UnsupportedOperationException if the morphisms are not composable, e.g., if they're from different categories.

Specified by:

compose in interface Morphism

equals

public boolean equals(Object o)

Works as expected, as long as the two sources and the two targets satisfy equals(java.lang.Object). However, hashCode() is not yet implemented.

hashCode

public int hashCode()

Unsupported at this time (sorry).

Throws:

UnsupportedOperationException - Always.

isMonic

public boolean isMonic()

Description copied from interface: Morphism

Whether this is a monomorphism.

Specified by:

isMonic in interface Morphism

isEpic

public boolean isEpic()

Description copied from interface: Morphism

Whether this is an epimorphism.

Specified by:

isEpic in interface Morphism

isIsomorphism

public boolean isIsomorphism()

Description copied from interface: Morphism

Whether this has an inverse morphism. Compare Morphism.getInverse().

Specified by:

isIsomorphism in interface Morphism

getInverse

public Morphism getInverse()

Unsupported at this time (sorry).

Specified by:

getInverse in interface Morphism

Throws:

UnsupportedOperationException - Always.

isZero

public boolean isZero()

Description copied from interface: ExactCategoryMorphism

Whether this is a zero morphism.

Specified by:

isZero in interface ExactCategoryMorphism

isCompositionZero

public boolean isCompositionZero(ExactCategoryMorphism f)

Description copied from interface: ExactCategoryMorphism

Whether the composition of this and f (see the diagram for Morphism.compose(shh.homolalg.Morphism)) is a zero morphism. Is expected to be more efficient than just computing the composition and calling ExactCategoryMorphism.isZero() on it.

Specified by:

isCompositionZero in interface ExactCategoryMorphism

rank

public int rank()

The rank of this morphism.

torsion

public String torsion()

Returns a pretty-print version of the elementary divisors of the underlying matrix over Z.

getKernel

public ExactCategoryMorphism getKernel()

Description copied from interface: ExactCategoryMorphism

See the comment on ExactCategoryMorphism.

Specified by:

getKernel in interface ExactCategoryMorphism

getKerRetraction

public ExactCategoryMorphism getKerRetraction()

Description copied from interface: ExactCategoryMorphism

See the comment on ExactCategoryMorphism.

Specified by:

getKerRetraction in interface ExactCategoryMorphism

getCokernel

public ExactCategoryMorphism getCokernel()

Description copied from interface: ExactCategoryMorphism

See the comment on ExactCategoryMorphism.

Specified by:

getCokernel in interface ExactCategoryMorphism

getCokerSection

public ExactCategoryMorphism getCokerSection()

Description copied from interface: ExactCategoryMorphism

See the comment on ExactCategoryMorphism.

Specified by:

getCokerSection in interface ExactCategoryMorphism

getCoimage

public ExactCategoryMorphism getCoimage()

Description copied from interface: ExactCategoryMorphism

See the comment on ExactCategoryMorphism.

Specified by:

getCoimage in interface ExactCategoryMorphism

getCoimSection

public ExactCategoryMorphism getCoimSection()

Description copied from interface: ExactCategoryMorphism

See the comment on ExactCategoryMorphism.

Specified by:

getCoimSection in interface ExactCategoryMorphism

getImage

public ExactCategoryMorphism getImage()

Description copied from interface: ExactCategoryMorphism

See the comment on ExactCategoryMorphism.

Specified by:

getImage in interface ExactCategoryMorphism

getImRetraction

public ExactCategoryMorphism getImRetraction()

Description copied from interface: ExactCategoryMorphism

See the comment on ExactCategoryMorphism.

Specified by:

getImRetraction in interface ExactCategoryMorphism

getCoimToIm

public ExactCategoryMorphism getCoimToIm()

Description copied from interface: ExactCategoryMorphism

The bottom arrow (the isomorphism) in the comment on ExactCategoryMorphism.

Specified by:

getCoimToIm in interface ExactCategoryMorphism

dual

public QvspMorphism dual()

If this is a QvspMorphism V → W with matrix A, then dual() returns a QvspMorphism W → V whose matrix is the transpose of A.

Strictly speaking, the dual should be a morphism W^* → V^*. We identify V with V^* and W with W^* under the standard dot product.

main

public static void main(String[] args)

Tests the factorization in ExactCategoryMorphism.

Parameters:

args - m n entry0 entry1 ... We create an m-by-n matrix and feed the entries into it using CSparse.userInputMatrix(java.util.List). We then test the corresponding morphism. The entries must be ints.

destroyAllButKernel

public void destroyAllButKernel()

Provides a more efficient way of computing the kernel and related morphisms, at the price of destroying this morphism itself. After this method is called, getKernel(), getKerRetraction(), getCoimage(), getCoimSection() and rank() will return correct results. However, getMatrix() will be wrong, as will most compositions computed with this morphism. In fact, after calling this method, it's recommend that you grab the kernel and related morphisms if you need them, then throw this to the garbage collector.

If an MPDQ has already been created for this morphism, the present method does nothing.

destroyAllButCokernel

public void destroyAllButCokernel()

Provides a more efficient way of computing the cokernel and related morphisms, at the price of destroying this morphism itself. After this method is called, getCokernel(), getCokerSection(), getImage(), getImRetraction() and rank() will return correct results. However, getMatrix() will be wrong, as will most compositions computed with this morphism. In fact, after calling this method, it's recommend that you grab the cokernel and related morphisms if you need them, then throw this to the garbage collector.

If an MPDQ has already been created for this morphism, the present method does nothing.