Group subsets¶

RepLAB has FiniteSet as a base for all sets of elements coming originally from a finite group. Note that a FiniteGroup itself is also a FiniteSet.

FiniteSet: Base class for all subsets of a finite group.

Such subsets have a decomposition which speeds up many algorithms in RepLAB.

SetProduct: Decomposition of a FiniteSet as a cartesian product of sets.

A finite group is partitioned into conjugacy classes.

ConjugacyClasses: Set of all conjugacy classes of a finite group.
ConjugacyClass: Conjugacy class of a finite group.

A finite group can also be partitioned into cosets by one (or two!) of its subgroups.

Cosets: Base class for sets of cosets of a finite group.
Coset: Base class for all cosets.
CosetTable: Describes the left action of a group on its cosets by a subgroup.
LeftCosets, RightCosets, NormalCosets: Sets of left/right/normal cosets, obtained using a subgroup of a finite group
LeftCoset, RightCoset, NormalCoset: Describes an element of the above
DoubleCosets: Sets of double cosets, obtained using two subgroups of a finite group
DoubleCoset: Describes an element of the above

FiniteSet¶

class replab.FiniteSet¶

Bases: replab.Domain

Describes a subset of a finite group

Examples of finite subsets include:

left cosets (LeftCoset),
right cosets (RightCoset),
normal cosets (NormalCoset),
double cosets (DoubleCoset),
conjugacy classes (ConjugacyClass),
finite groups (FiniteGroup).

If the finite set is not empty (nElements is nonzero), the set has a distinguished representative element. For permutation groups, this representative element is the minimal element under lexicographic ordering.

Class members list

Properties

type – Type of the contained elements

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Tests finite sets for equality
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Elements

contains – Tests whether this set contains the given element
elements – Returns a cell array containing all the elements of this set
elementsSequence – Returns a sequence corresponding to this set
nElements – Returns the size of this set
representative – Returns the minimal element of this set under the type ordering
setProduct – Returns a description of this set as a product of sets

Relations to other sets

hasSameTypeAs – Returns if this finite set has the same type as the given finite set

Image under isomorphism

imap – Returns the image of this finite set under an isomorphism

General

mtimes –
multiply –
shortType –

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

mtimes()¶: No documentation

multiply()¶: No documentation

ne(self, rhs)¶: Documentation in replab.Obj.ne()

sample(self)¶: Documentation in replab.Domain.sample()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

shortType()¶: No documentation

Own members

representative_¶

Minimal member of this set under the type ordering. If the set is empty, value is undefined.

Type: element of type

type¶

Type of the contained elements

Type: FiniteGroupType

contains(self, el)¶

Tests whether this set contains the given element

The element must be part of type, as in the example below: only permutations with the same domain size should be tested using contains.

Example

>>> G = replab.PermutationGroup.of([2 3 4 1]);
>>> G.contains([4 3 2 1])
    0

Parameters: el (element of type) – Element to test for membership
Returns: True if this set contains el and false otherwise
Return type: logical

elements(self)¶

Returns a cell array containing all the elements of this set

The sequence is sorted according to the total ordering defined by type.

Note: if the number of elements is bigger than replab.globals.maxElements, an error is thrown.

Returns: Elements
Return type: cell(1,*) of finite set elements

elementsSequence(self)¶

Returns a sequence corresponding to this set

The sequence is sorted according to the total ordering defined by type.

Returns: An enumeration of the set elements
Return type: Sequence

hasSameTypeAs(self, rhs)¶

Returns if this finite set has the same type as the given finite set

In particular, it means that the contains method of one set can be called with elements of the other set.

Parameters: rhs (replab.FiniteSet) – Other finite set
Returns: True if the groups have compatible types
Return type: logical

imap(self, f)¶

Returns the image of this finite set under an isomorphism

Parameters: f (FiniteIsomorphism) – Isomorphism such that this finite set is contained in f.source
Returns: This finite set mapped under f, expressed as a subset of f.image
Return type: FiniteSet

isequal(self, rhs)¶

Tests finite sets for equality

Alternative syntax to eq

Example

>>> G = replab.PermutationGroup.of([2 3 1], [2 1 3]);
>>> H = replab.PermutationGroup.of([2 1 3], [1 3 2]);
>>> isequal(G, H)
    1

nElements(self)¶

Returns the size of this set

Returns: Set cardinality
Return type: vpi

representative(self)¶

Returns the minimal element of this set under the type ordering

Returns: Canonical representative of this set
Return type: element

setProduct(self)¶

Returns a description of this set as a product of sets

Returns: Generic description of the elements of this set
Return type: SetProduct

SetProduct¶

class replab.SetProduct(monoid, sets, identityFirst)¶

Bases: replab.Domain

Describes a multiset of elements obtained by reduction of the elements of a cartesian product

Let T1, ..., Tn be finite subsets of a monoid G. The set product of such T is the multiset of elements of the form x = t1 t2 ... tn iterating over each ti in each Ti. The binary operation is given by the composition of G.

Thus, the multiset has |T1| * |T2| * ... * |Tn| elements, possibly with repetition.

RepLAB uses SetProduct in two areas:

to describe a unique decomposition of a finite group, in which case the multiset is actually a set whose cardinality is the group order; we additionally require that the first element of each T{i} is the identity;
to describe the left cosets of the connected component in a compact Lie group.

Class members list

Properties

identityFirst – True if each element T contains the identity as its first element
monoid – Monoid providing the binary operation
sets – Stores the sets Ti as {T1 T2 ... Tn}

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

General

SetProduct –
identity – Constructs a SetProduct containing only the identity of a monoid
imap – Maps this multiset under an isomorphism

Inherited elements

SetProduct()¶: No documentation

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

sample(self)¶: Documentation in replab.Domain.sample()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

Own members

identityFirst¶

True if each element T contains the identity as its first element

Type: logical

monoid¶

Monoid providing the binary operation

Type: replab.Monoid

sets¶

Stores the sets Ti as {T1 T2 ... Tn}

Type: cell(1,*) of cell(1,*) of elements

static identity(monoid)¶

Constructs a SetProduct containing only the identity of a monoid

Parameters: monoid (Monoid) – Monoid to take the identity from
Returns: A singleton SetProduct
Return type: SetProduct

imap(self, isomorphism)¶

Maps this multiset under an isomorphism

Parameters: isomorphism (Isomorphism) – Isomorphism with its containing all elements of this SetProduct
Returns: Multiset with elements in isomorphism.target
Return type: SetProduct

ConjugacyClasses¶

class replab.ConjugacyClasses(group, classes)¶

Bases: replab.Obj

Stores information about the conjugacy classes of a group

This class can compute power maps.

Class members list

Properties

classes – Conjugacy classes
group – Group whose conjugacy classes we describe

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

General

ConjugacyClasses –

Properties

centralizerSizes – Returns the sizes of the centralizers
classElementOrders – Returns the element order for each of the conjugacy classes
classRepresentatives – Returns the canonical representatives of the conjugacy classes
classSizes – Returns the sizes of the conjugacy classes
nClasses – Returns the number of conjugacy classes in the group

Find conjugacy classes

classIndexOf – Finds the conjugacy class where a given element is located
indicesOfClasses – Returns the indices of the given conjugacy classes in this list

Transformations

imap – Maps the conjugacy classes under an isomorphism
sorted –

Subgroup construction

normalSubgroupClasses – Returns the normal subgroup consisting of the conjugacy classes whose positions are given

Power maps

powerMap – Returns the power map corresponding to the given exponent
powerMapDefaultPrimes – Returns the list of primes that are necessary to reconstruct any power map
powerMapMatrix – Returns an adjacency-like matrix that describes (incomplete) relationships between conjugacy classes
powerMaps – Returns or computes&stores the power map corresponding to the given exponent

Inherited elements

ConjugacyClasses()¶: No documentation

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

sorted()¶: No documentation

Own members

classes¶

Conjugacy classes

Type: cell(1,nC) of ConjugacyClass

group¶

Group whose conjugacy classes we describe

Type: FiniteGroup

primePowerMap¶

Power maps (computed as needed)

Type: integer(nP,nC)

primes¶

List of primes that divide the group order (computed as needed)

Type: integer(1,nP)

centralizerSizes(self)¶

Returns the sizes of the centralizers

Returns: Size of the centralizer for each conjugacy class
Return type: cell(1,*) of vpi

classElementOrders(self)¶

Returns the element order for each of the conjugacy classes

Returns: Elements orders
Return type: integer(1,*)

classIndexOf(self, g, varargin)¶

Finds the conjugacy class where a given element is located

Parameters

g (element of group) – Group element
isCanonical – Whether the given g is the class canonical representative, default: false

Kwtype isCanonical

logical, optional

Returns

Index of the class containing g

Return type

integer

classRepresentatives(self)¶

Returns the canonical representatives of the conjugacy classes

Returns: Representatives
Return type: cell(1,*) of ConjugacyClass

classSizes(self)¶

Returns the sizes of the conjugacy classes

Returns: Size of each conjugacy class
Return type: cell(1,*) of vpi

imap(self, f)¶

Maps the conjugacy classes under an isomorphism

Parameters: f (FiniteIsomorphism) – Isomorphism with self.group.isSubgroupOf(f.source)
Returns: The conjugacy classes mapped under f, expressed as a subset of f.image
Return type: ConjugacyClasses

indicesOfClasses(self, classes)¶

Returns the indices of the given conjugacy classes in this list

Parameters: classes (ConjugacyClasses) – Conjugacy classes of group
Returns: Indices of the given classes in these classes
Return type: integer(1,*)

nClasses(self)¶

Returns the number of conjugacy classes in the group

Returns: Number of classes
Return type: integer

normalSubgroupClasses(self, indices)¶

Returns the normal subgroup consisting of the conjugacy classes whose positions are given

Parameters: indices (integer(1,*)) – Indices of conjugacy classes
Returns: Normal subgroup representing that union
Return type: FiniteGroup

powerMap(self, n)¶

Returns the power map corresponding to the given exponent

We have m(i) = self.classIndexOf(self.group.composeN(self.classes{i}.representative, n)).

Parameters: p (integer) – Exponent
Returns: Index of the power of each conjugacy class
Return type: integer(1,*)

powerMapDefaultPrimes(self)¶

Returns the list of primes that are necessary to reconstruct any power map

Returns: Prime numbers that are less or equal to the largest class element order
Return type: integer(1,*)

powerMapMatrix(self)¶: Returns an adjacency-like matrix that describes (incomplete) relationships between conjugacy classes

powerMaps(self, exponents)¶

Returns or computes&stores the power map corresponding to the given exponent

Parameters: exponents (integer(1,*)) – Exponents
Returns: Index of the power of each conjugacy class
Return type: integer(length(exponents),*)

ConjugacyClass¶

class replab.ConjugacyClass¶

Bases: replab.FiniteSet

Describes a conjugacy class of a finite group

A conjugacy class containing the representative \(r \in G\) is the set \(\{g r g^{-1} : g \in G \}\).

The centralizer of \(r\) in \(G\) is the subgroup \(C_{G}(r) = \{ g r g^{-1} == r : g \in G \}\).

Thus, the left cosets \(G/C_{G}(r) = \{ g C_{G}(r) : g \in G \}\) are in one to one correspondence with the elements of the conjugacy class.

Class members list

Properties

group – Group containing this conjugacy class
type – Type of the contained elements

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Tests finite sets for equality
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Elements

contains – Tests whether this set contains the given element
elements – Returns a cell array containing all the elements of this set
elementsSequence – Returns a sequence corresponding to this set
nElements – Returns the size of this set
representative – Returns the minimal element of this set under the type ordering
setProduct – Returns a description of this set as a product of sets

Relations to other sets

hasSameTypeAs – Returns if this finite set has the same type as the given finite set

Image under isomorphism

imap – Maps this conjugacy class under an isomorphism

General

elementOrder – Returns the order of the elements in this conjugacy class
knownRepresentativeCentralizer – Returns whether the centralizer of representative is known
mtimes –
representativeCentralizer – Returns the centralizer of representative in group

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

contains(self, el)¶: Documentation in replab.FiniteSet.contains()

disp(self)¶: Documentation in replab.Str.disp()

elements(self)¶: Documentation in replab.FiniteSet.elements()

elementsSequence(self)¶: Documentation in replab.FiniteSet.elementsSequence()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

hasSameTypeAs(self, rhs)¶: Documentation in replab.FiniteSet.hasSameTypeAs()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(self, rhs)¶: Documentation in replab.FiniteSet.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

mtimes()¶: No documentation

nElements(self)¶: Documentation in replab.FiniteSet.nElements()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

representative(self)¶: Documentation in replab.FiniteSet.representative()

sample(self)¶: Documentation in replab.Domain.sample()

setProduct(self)¶: Documentation in replab.FiniteSet.setProduct()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

type¶: Documentation in replab.FiniteSet.type

Own members

group¶

Group containing this conjugacy class

Type: replab.FiniteGroup

elementOrder(self)¶

Returns the order of the elements in this conjugacy class

Returns: Element order
Return type: integer

imap(self, f)¶

Maps this conjugacy class under an isomorphism

Parameters: f (FiniteIsomorphism) – Isomorphism with self.group.isSubgroupOf(f.source)
Returns: The conjugacy class mapped under f, expressed as a subset of f.image
Return type: ConjugacyClass

knownRepresentativeCentralizer(self)¶

Returns whether the centralizer of representative is known

Returns: True if the centralizer has already been computed
Return type: logical

representativeCentralizer(self)¶

Returns the centralizer of representative in group

Returns: Representative centralizer
Return type: FiniteGroup

Cosets¶

class replab.Cosets¶

Bases: replab.Obj

Base class for the left/normal/right coset classes

This class is necessary because NormalCosets inherits from both LeftCosets or RightCosets, and thus they share the group and subgroup properties.

Class members list

Properties

group – Group
subgroup – Subgroup of group

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

General

nElements – Returns the number of cosets

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

Own members

group¶

Group

Type: FiniteGroup

subgroup¶

Subgroup of group

Type: FiniteGroup

nElements(self)¶

Returns the number of cosets

Returns: Number of cosets
Return type: integer

Coset¶

class replab.Coset¶

Bases: replab.FiniteSet

Base class for the left/normal/right coset classes

This class is necessary because NormalCoset inherits from both LeftCoset or RightCoset, and thus they share the group and subgroup properties.

Class members list

Properties

group – Group containing this coset
subgroup – Subgroup used to decompose group
type – Type of the contained elements

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Tests finite sets for equality
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Elements

contains – Tests whether this set contains the given element
elements – Returns a cell array containing all the elements of this set
elementsSequence – Returns a sequence corresponding to this set
nElements – Returns the size of this set
representative – Returns the minimal element of this set under the type ordering
setProduct – Returns a description of this set as a product of sets

Relations to other sets

hasSameTypeAs – Returns if this finite set has the same type as the given finite set

Image under isomorphism

imap – Returns the image of this finite set under an isomorphism

General

mtimes –

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

contains(self, el)¶: Documentation in replab.FiniteSet.contains()

disp(self)¶: Documentation in replab.Str.disp()

elements(self)¶: Documentation in replab.FiniteSet.elements()

elementsSequence(self)¶: Documentation in replab.FiniteSet.elementsSequence()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

hasSameTypeAs(self, rhs)¶: Documentation in replab.FiniteSet.hasSameTypeAs()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

imap(self, f)¶: Documentation in replab.FiniteSet.imap()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(self, rhs)¶: Documentation in replab.FiniteSet.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

mtimes()¶: No documentation

nElements(self)¶: Documentation in replab.FiniteSet.nElements()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

representative(self)¶: Documentation in replab.FiniteSet.representative()

sample(self)¶: Documentation in replab.Domain.sample()

setProduct(self)¶: Documentation in replab.FiniteSet.setProduct()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

type¶: Documentation in replab.FiniteSet.type

Own members

group¶

Group containing this coset

Type: FiniteGroup

subgroup¶

Subgroup used to decompose group

Type: FiniteGroup

CosetTable¶

class replab.CosetTable(generatorNames, internal)¶

Bases: replab.Str

Describes a coset table (experimental)

Example

>>> ct = replab.CosetTable.fromPresentation({'x' 'y'}, {'x^2','y^3','(x*y)^3'}, {'x*y'});
>>> ct.table.format
      | x  y  inv(x)  inv(y)
    --------------------------
    1 | 2  2     2       3
    2 | 1  3     1       1
    3 | 4  1     4       2
    4 | 3  4     3       4

Note

The action of words on cosets is a left action, contrary to the algorithms described in Holt. Thus, we implement the algorithms of Holt in a different class, which matches the pseudocode closely. This class, however, translates between those conventions.

Class members list

Properties

generatorNames – Names of the generators
internal – Internal coset table

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

General

CosetTable –
fromPresentation – Enumerate cosets
table – Returns the coset table as a prettyprintable table

Inherited elements

CosetTable()¶: No documentation

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

disp(self)¶: Documentation in replab.Str.disp()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

Own members

generatorNames¶

Names of the generators

Type: cell(1,*) of charstring

internal¶

Internal coset table

Type: replab.fp.CosetTable

static fromPresentation(generatorNames, relatorWords, subgroupGeneratorWords)¶

Enumerate cosets

Parameters

generatorNames (cell(1,*) of charstring) – Group generator names
relatorWords (cell(1,*) of charstring) – Group relators to parse
subgroupGeneratorWords (cell(1, *) of charstring) – Generators for a subgroup, expressed as words in the group generators

table(self)¶

Returns the coset table as a prettyprintable table

Returns: String table
Return type: replab.str.Table

LeftCosets¶

class replab.LeftCosets¶

Bases: replab.Cosets

Describes the set of left cosets of a finite group

Let \(H\) be a subgroup of a group \(G\). Then the left cosets are the sets \(g H = \{ g h : h \in H \}\). The set of such left cosets is often written \(G / H = \{ g H : g \in G \}\).

Class members list

Properties

group – Group
subgroup – Subgroup of group

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

General

cosetRepresentative – Returns the canonical coset representative corresponding to the given element
elements – Returns the set of left cosets as a cell array
leftAction – Returns, as a morphism, the action of the given group of its left cosets
nElements – Returns the number of cosets
transversal – Returns all the canonical representatives of cosets

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

group¶: Documentation in replab.Cosets.group

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

nElements(self)¶: Documentation in replab.Cosets.nElements()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

subgroup¶: Documentation in replab.Cosets.subgroup

Own members

cosetRepresentative(self, g)¶

Returns the canonical coset representative corresponding to the given element

If t = cosetRepresentative(g), then \(t^{-1} g = h \in H\), with decomposition \(g = t h\).

Moreover L.cosetRepresentative(g) == L.cosetRepresentative(compose(g, h)) for any \(h \in H\).

Finally, L.cosetRepresentative(g) == L.coset(g).representative.

Parameters: g (element of group) – Group element
Returns: Coset canonical representative
Return type: t (element of group)

elements(self)¶

Returns the set of left cosets as a cell array

Returns: Set of left cosets
Return type: cell(1,*) of replab.LeftCoset

leftAction(self)¶

Returns, as a morphism, the action of the given group of its left cosets

Returns: Morphism of this group into a permutation group
Return type: FiniteMorphism

transversal(self)¶

Returns all the canonical representatives of cosets

Returns: Transversal
Return type: cell(1, *) of group elements

RightCosets¶

class replab.RightCosets¶

Bases: replab.Cosets

Describes the set of right cosets of a nice finite group

Let \(H\) be a subgroup of a group \(G\). Then the right cosets are the sets \(H g = \{ h g : h \in H \}\). The set of such right cosets is often written \(H \\ G = \{ H g : g \in G \}\).

Class members list

Properties

group – Group
subgroup – Subgroup of group

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

General

cosetRepresentative – Returns the coset representative corresponding to the given element
elements – Returns the set of right cosets as a cell array
leftCosetsBy –
mrdivide –
nElements – Returns the number of cosets
transversal – Returns all the canonical representatives of cosets

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

group¶: Documentation in replab.Cosets.group

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

leftCosetsBy()¶: No documentation

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

mrdivide()¶: No documentation

nElements(self)¶: Documentation in replab.Cosets.nElements()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

subgroup¶: Documentation in replab.Cosets.subgroup

Own members

cosetRepresentative(self, g)¶

Returns the coset representative corresponding to the given element

If t = cosetRepresentative(g), then \(g t^{-1} = h \in H\), with the decomposition \(g = h t\).

Moreover R.cosetRepresentative(g) == R.cosetRepresentative(compose(h, g)) for any \(h \in H\).

Finally, R.cosetRepresentative(g) == R.coset(g).representative.

Parameters: g (element of group) – Group element
Returns: Coset canonical representative
Return type: t (element of group)

elements(self)¶

Returns the set of right cosets as a cell array

Returns: Set of right cosets
Return type: cell(1,*) of replab.RightCoset

transversal(self)¶

Returns all the canonical representatives of cosets

Returns: Transversal
Return type: cell(1, *) of group elements

NormalCosets¶

class replab.NormalCosets¶

Bases: replab.LeftCosets, replab.RightCosets

Describes the set of normal cosets of a finite group

Let \(H\) be a normal subgroup of a group \(G\). Then the left cosets are the sets \(g H = \{ g h : h \in H \}\). Then the right cosets are the sets \(H g = \{ h g : h \in H \}\). Because the subgroup is normal, we have \(g H = H g\).

Class members list

Properties

group – Group
subgroup – Subgroup of group

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

General

cosetRepresentative – Returns the canonical coset representative corresponding to the given element
elements – Returns the set of normal cosets as a cell array
leftAction – Returns, as a morphism, the action of the given group of its left cosets
leftCosetsBy –
mrdivide –
nElements – Returns the number of cosets
transversal – Returns all the canonical representatives of cosets

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

cosetRepresentative(self, g)¶: Documentation in replab.LeftCosets.cosetRepresentative()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

group¶: Documentation in replab.Cosets.group

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

leftAction(self)¶: Documentation in replab.LeftCosets.leftAction()

leftCosetsBy()¶: No documentation

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

mrdivide()¶: No documentation

nElements(self)¶: Documentation in replab.Cosets.nElements()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

subgroup¶: Documentation in replab.Cosets.subgroup

transversal(self)¶: Documentation in replab.LeftCosets.transversal()

Own members

elements(self)¶

Returns the set of normal cosets as a cell array

Returns: Set of normal cosets
Return type: cell(1,*) of replab.NormalCoset

LeftCoset¶

class replab.LeftCoset¶

Bases: replab.Coset

Describes a left coset of a finite group

Let \(g \in G\) be a coset representative and \(H \le G\) a group. The left coset is \(g H = \{ g h : h \in H\}\).

Class members list

Properties

group – Group containing this coset
subgroup – Subgroup used to decompose group
type – Type of the contained elements

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Tests finite sets for equality
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Elements

contains – Tests whether this set contains the given element
elements – Returns a cell array containing all the elements of this set
elementsSequence – Returns a sequence corresponding to this set
nElements – Returns the size of this set
representative – Returns the minimal element of this set under the type ordering
setProduct – Returns a description of this set as a product of sets

Relations to other sets

hasSameTypeAs – Returns if this finite set has the same type as the given finite set

Image under isomorphism

imap – Returns the image of this finite set under an isomorphism

General

mtimes –

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

contains(self, el)¶: Documentation in replab.FiniteSet.contains()

disp(self)¶: Documentation in replab.Str.disp()

elements(self)¶: Documentation in replab.FiniteSet.elements()

elementsSequence(self)¶: Documentation in replab.FiniteSet.elementsSequence()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

group¶: Documentation in replab.Coset.group

hasSameTypeAs(self, rhs)¶: Documentation in replab.FiniteSet.hasSameTypeAs()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

imap(self, f)¶: Documentation in replab.FiniteSet.imap()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(self, rhs)¶: Documentation in replab.FiniteSet.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

mtimes()¶: No documentation

nElements(self)¶: Documentation in replab.FiniteSet.nElements()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

representative(self)¶: Documentation in replab.FiniteSet.representative()

sample(self)¶: Documentation in replab.Domain.sample()

setProduct(self)¶: Documentation in replab.FiniteSet.setProduct()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

subgroup¶: Documentation in replab.Coset.subgroup

type¶: Documentation in replab.FiniteSet.type

Own members

RightCoset¶

class replab.RightCoset¶

Bases: replab.Coset

Describes a right coset of a finite group

Let \(g \in G\) be a coset representative and \(H \le G\) a group. The right coset is \(H g = \{ h g : h \in H\}\).

Class members list

Properties

group – Group containing this coset
subgroup – Subgroup used to decompose group
type – Type of the contained elements

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Tests finite sets for equality
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Elements

contains – Tests whether this set contains the given element
elements – Returns a cell array containing all the elements of this set
elementsSequence – Returns a sequence corresponding to this set
nElements – Returns the size of this set
representative – Returns the minimal element of this set under the type ordering
setProduct – Returns a description of this set as a product of sets

Relations to other sets

hasSameTypeAs – Returns if this finite set has the same type as the given finite set

Image under isomorphism

imap – Returns the image of this finite set under an isomorphism

General

mtimes –

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

contains(self, el)¶: Documentation in replab.FiniteSet.contains()

disp(self)¶: Documentation in replab.Str.disp()

elements(self)¶: Documentation in replab.FiniteSet.elements()

elementsSequence(self)¶: Documentation in replab.FiniteSet.elementsSequence()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

group¶: Documentation in replab.Coset.group

hasSameTypeAs(self, rhs)¶: Documentation in replab.FiniteSet.hasSameTypeAs()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

imap(self, f)¶: Documentation in replab.FiniteSet.imap()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(self, rhs)¶: Documentation in replab.FiniteSet.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

mtimes()¶: No documentation

nElements(self)¶: Documentation in replab.FiniteSet.nElements()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

representative(self)¶: Documentation in replab.FiniteSet.representative()

sample(self)¶: Documentation in replab.Domain.sample()

setProduct(self)¶: Documentation in replab.FiniteSet.setProduct()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

subgroup¶: Documentation in replab.Coset.subgroup

type¶: Documentation in replab.FiniteSet.type

Own members

NormalCoset¶

class replab.NormalCoset¶

Bases: replab.LeftCoset, replab.RightCoset

Describes a coset in the normal subgroup of a finite group

It is a left and right coset at the same time.

See parent classes LeftCoset and RightCoset

Class members list

Properties

group – Group containing this coset
subgroup – Subgroup used to decompose group
type – Type of the contained elements

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Tests finite sets for equality
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Elements

contains – Tests whether this set contains the given element
elements – Returns a cell array containing all the elements of this set
elementsSequence – Returns a sequence corresponding to this set
nElements – Returns the size of this set
representative – Returns the minimal element of this set under the type ordering
setProduct – Returns a description of this set as a product of sets

Relations to other sets

hasSameTypeAs – Returns if this finite set has the same type as the given finite set

Image under isomorphism

imap – Returns the image of this finite set under an isomorphism

General

mtimes –

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

contains(self, el)¶: Documentation in replab.FiniteSet.contains()

disp(self)¶: Documentation in replab.Str.disp()

elements(self)¶: Documentation in replab.FiniteSet.elements()

elementsSequence(self)¶: Documentation in replab.FiniteSet.elementsSequence()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

group¶: Documentation in replab.Coset.group

hasSameTypeAs(self, rhs)¶: Documentation in replab.FiniteSet.hasSameTypeAs()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

imap(self, f)¶: Documentation in replab.FiniteSet.imap()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(self, rhs)¶: Documentation in replab.FiniteSet.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

mtimes()¶: No documentation

nElements(self)¶: Documentation in replab.FiniteSet.nElements()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

representative(self)¶: Documentation in replab.FiniteSet.representative()

sample(self)¶: Documentation in replab.Domain.sample()

setProduct(self)¶: Documentation in replab.FiniteSet.setProduct()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

subgroup¶: Documentation in replab.Coset.subgroup

type¶: Documentation in replab.FiniteSet.type

Own members

DoubleCosets¶

class replab.DoubleCosets¶

Bases: replab.Domain

Describes the set of double cosets in a group

A double coset is a set of the form { H g K } = { h g k : h \in H, k \in K} for subgroups H and K of a group G

Class members list

Properties

group – Group
leftSubgroup – Subgroup of group
rightSubgroup – Subgroup of group

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

General

cosetRepresentative – Returns the canonical coset representative corresponding to the given element
elements – Returns the set of double cosets as a cell array
nElements – Returns the number of double cosets
transversal – Returns all the canonical representatives of cosets

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

sample(self)¶: Documentation in replab.Domain.sample()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

Own members

group¶

Group

Type: FiniteGroup

leftSubgroup¶

Subgroup of group

Type: FiniteGroup

rightSubgroup¶

Subgroup of group

Type: FiniteGroup

cosetRepresentative(self, g)¶

Returns the canonical coset representative corresponding to the given element

Parameters: g (element of group) – Group element
Returns: Double coset canonical representative
Return type: t (element of group)

elements(self)¶

Returns the set of double cosets as a cell array

Returns: Set of double cosets
Return type: cell(1,*) of replab.DoubleCoset

nElements(self)¶

Returns the number of double cosets

Returns: Number of double cosets
Return type: integer

transversal(self)¶

Returns all the canonical representatives of cosets

Returns: Transversal
Return type: cell(1, *) of group elements

DoubleCoset¶

class replab.DoubleCoset¶

Bases: replab.FiniteSet

Describes a double coset in a group

A double coset is a set of the form { H g K } = { h g k : h \in H, k \in K} for subgroups H and K of a group G

Class members list

Properties

group – Group containing this double coset
leftSubgroup – Group
rightSubgroup – Group
type – Type of the contained elements

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Tests finite sets for equality
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Elements

contains – Tests whether this set contains the given element
elements – Returns a cell array containing all the elements of this set
elementsSequence – Returns a sequence corresponding to this set
nElements – Returns the size of this set
representative – Returns the minimal element of this set under the type ordering
setProduct – Returns a description of this set as a product of sets

Relations to other sets

hasSameTypeAs – Returns if this finite set has the same type as the given finite set

Image under isomorphism

imap – Returns the image of this finite set under an isomorphism

General

mtimes –

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

contains(self, el)¶: Documentation in replab.FiniteSet.contains()

disp(self)¶: Documentation in replab.Str.disp()

elements(self)¶: Documentation in replab.FiniteSet.elements()

elementsSequence(self)¶: Documentation in replab.FiniteSet.elementsSequence()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

hasSameTypeAs(self, rhs)¶: Documentation in replab.FiniteSet.hasSameTypeAs()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

imap(self, f)¶: Documentation in replab.FiniteSet.imap()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(self, rhs)¶: Documentation in replab.FiniteSet.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

mtimes()¶: No documentation

nElements(self)¶: Documentation in replab.FiniteSet.nElements()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

representative(self)¶: Documentation in replab.FiniteSet.representative()

sample(self)¶: Documentation in replab.Domain.sample()

setProduct(self)¶: Documentation in replab.FiniteSet.setProduct()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

type¶: Documentation in replab.FiniteSet.type

Own members

group¶

Group containing this double coset

Type: FiniteGroup

leftSubgroup¶

Group

Type: FiniteGroup

rightSubgroup¶

Group

Type: FiniteGroup