org.uacalc.alg.conlat

Class CongruenceLattice

java.lang.Object

org.uacalc.alg.conlat.CongruenceLattice

All Implemented Interfaces:

Algebra, Lattice, Order

public class CongruenceLattice
extends java.lang.Object
implements Lattice

A class to represent the congruence lattice of a SmallAlgebra; this is, an algebra with universe the integers from 0 to n-1. This uses the very fast algorithms from my unpublished paper Computing Congruences Efficiently which is available at http://www.math.hawaii.edu/~ralph/papers.html.

Version:

$Id$

Author:

Ralph Freese

Field Summary

Fields

Modifier and Type	Field and Description
static int	MAX_DRAWABLE_INPUT_SIZE
static int	MAX_DRAWABLE_SIZE
static ProgressReport	monitor

Fields inherited from interface org.uacalc.alg.Algebra

CARDINALITY_COUNTABLE, CARDINALITY_COUNTABLY_INFINITE, CARDINALITY_FINITE, CARDINALITY_INFINITE, CARDINALITY_UNKNOWN

Constructor Summary

Constructors

Constructor and Description
CongruenceLattice(SmallAlgebra alg)

Method Summary

Modifier and Type	Method and Description
java.util.List<Partition>	atoms()
java.util.List<CentralityData>	calcCentrality(BinaryRelation S, BinaryRelation T) Given tolerances (or congruences) S and T, this tests, for every congruence delta of A, if S centralizes T mod delta.
java.util.List<CentralityData>	calcCentrality(BinaryRelation S, BinaryRelation T, ProgressReport report)
int	cardinality()
SubProductElement	centralityFailure(BinaryRelation S, BinaryRelation T, Partition delta) Find a witness to the failure of centrality, or null if there is none.
SubProductElement	centralityFailure(BinaryRelation S, BinaryRelation T, Partition delta, SubProductAlgebra mats)
boolean	centralizes(BinaryRelation S, BinaryRelation T, Partition delta)
boolean	centralizes(BinaryRelation S, BinaryRelation T, Partition delta, SubProductAlgebra mats)
Partition	Cg_Old(Partition part, ProgressReport report)
Partition	Cg(int a, int b)
Partition	Cg(int a, int b, ProgressReport report) The congruence generated by ints a and b.
Partition	Cg(java.lang.Object a, java.lang.Object b) The congruence generated by Object a and b.
Partition	Cg(Partition part) The smallest congruence containing part.
Partition	Cg(Partition part, ProgressReport report) The smallest congruence containing part.
java.util.List<Partition>	coatoms() This is not implemented yet.
Partition	commutator(BinaryRelation S, BinaryRelation T) Calculate the commutator of S and T, which are tolerances or congruences.
Partition	commutator(BinaryRelation S, BinaryRelation T, java.util.List<CentralityData> centralityList, ProgressReport report)
java.util.List<Partition>	complements(Partition par) Returns a list of the complements of par.
CongruenceLattice	con() Get Con(Con(A)).
java.util.List<Operation>	constantOperations() This gives a list of the operations of arity 0, which is a little different from the constants.
Partition	findCoatomAbove(Partition theta) find a coaton of the congruence lattice greater than or equal to theta.
Partition	findJoinIrred(Partition a, Partition b) This finds a join irreducible congruence which is minimal with respect to being below b and not below a.
java.util.List<Partition>	findL3Generators() Find congruences alpha, beta, gamma which generate L3 of Jonsson-Rival notation; see Fig 2.2 of Jipsen-Rose.
java.util.List<Partition>	findLPJ10Generators()
java.util.List<Partition>	findLXXGenerators()
Partition	findMeetIrred(Partition a, Partition b) This finds a meet irreducible congruence which is maximal with respect to being above a and not above b.
java.util.List<Partition>	findPrincipalChain() Find a chain where each element is the join of the previous one and a principal congruence.
Partition	findUpperCover(Partition congr) Find some upper cover of congr.
IntArray	generatingPair(Partition part) Find a pair [a, b] (as an IntArray) which generates part.
SmallAlgebra	getAlgebra() Get the algebra this is the congruence lattice of.
BasicLattice	getBasicLattice()
BasicLattice	getBasicLattice(boolean makeIfNull) Get the BasicLattice used primarily for drawing.
javax.swing.JTable	getConTable()
java.lang.String	getDescription()
org.latdraw.diagram.Diagram	getDiagram()
java.util.Map<Partition,Subtrace>	getJoinIrredToSubtraceMap()
int	getMakeUniverseK()
ProgressReport	getMonitor()
java.lang.String	getName()
Operation	getOperation(OperationSymbol sym) Get the operation correspond to a symbol or null if the symbol is not part of the similarityType.
java.util.Map<OperationSymbol,Operation>	getOperationsMap()
Partition[]	getPermutabilityLevelWitnesses()
int	getSizeComputed()
TypeFinder	getTypeFinder()
java.util.List<Partition>	getUniverseList()
java.util.List<IntArray>	idempotentPolynomials() Find the nonconstant, idempotent member of Pol1(A).
int	inputSize() The sum of the cardinality of the algebra raised to the arity of the operations.
java.util.List<Partition>	irredundantMeetDecomposition() Find a set of meet irreducible irredundantly meeting to zero.
java.util.List<Partition>	irredundantMeetDecompositionOld()
boolean	isDistributive()
boolean	isDrawable()
boolean	isIdempotent() Test if all of the operations are idempotent.
boolean	isSimilarTo(Algebra alg)
boolean	isSmallerThan(int size)
boolean	isTotal() This will fail only if there are some OperationWithDefaultValue's which are not total.
boolean	isUnary()
java.util.Iterator<Partition>	iterator() returns the iterator of the universe.
java.lang.Object	join(java.util.List args)
java.lang.Object	join(java.lang.Object a, java.lang.Object b)
boolean	joinIrreducible(Partition part)
java.util.List<Partition>	joinIrreducibles() A list of the join irreducible congruences sorted by rank (in the partition lattice).
java.util.List<Partition>	joinIrreducibles(ProgressReport report) A list of the join irreducibles sorted by rank (in the partition lattice); constructed if necessary.
boolean	joinIrreduciblesFound()
boolean	joinPrime(Partition beta) Test if beta is join prime.
boolean	leq(java.lang.Object a, java.lang.Object b) The order relation.
Partition	lowerStar(Partition beta) If beta is join irreducible, this gives its lower cover; otherwise null.
static void	main(java.lang.String[] args)
void	makeAtoms()
java.util.List<Partition>	makeIrredundantMeet(java.util.List<Partition> list)
void	makeJoinIrreducibles(ProgressReport report)
void	makeOperationTables() Make operation tables to speed up the evaluation of operations at the cost using more space.
void	makePrincipals(ProgressReport report)
void	makeUniverse()
void	makeUniverse(int size)
void	makeUniverse(int maxSize, ProgressReport report) Construct the universe.
void	makeUniverse(ProgressReport report)
SubProductAlgebra	matrices(BinaryRelation S, BinaryRelation T, ProgressReport report) The set M(S,T) as in Kearnes-Kiss and other places.
java.lang.Object	meet(java.util.List args)
java.lang.Object	meet(java.lang.Object a, java.lang.Object b)
boolean	meetIrreducible(Partition part)
java.util.List<Partition>	meetIrreducibles() An optional operation returning the list of meet irreducible elements.
static java.util.List<Partition>	minimalElements(java.util.List<Partition> parList) Find the min elements of a topologically sorted list.
boolean	monitoring()
Partition	one()
java.util.List	operations()
int	permutabilityLevel() Get the permutabiltiy level of this lattice (not the whole variety).
java.util.List<Partition>	principals() A list of the principal congruences sorted by rank (in the partition lattice).
java.util.List<Partition>	principals(ProgressReport report) A list of the principal congruences sorted by rank (in the partition lattice).
void	setConTable(javax.swing.JTable table)
void	setDescription(java.lang.String desc)
void	setMonitor(ProgressReport m)
void	setName(java.lang.String v)
SimilarityType	similarityType()
void	sortByRank(java.util.List<Partition> lst) Sort by rank (in the partition lattice).
void	stopMakeUniverse()
Partition	strongRectangularityCommutator(BinaryRelation S, BinaryRelation T) Calculate the strong rectangular commutator of S and T, which are tolerances or congruences.
Partition	strongRectangularityCommutator(BinaryRelation S, BinaryRelation T, java.util.List<CentralityData> centralityList, ProgressReport report)
SubProductElement	strongRectangularityFailure(BinaryRelation S, BinaryRelation T, Partition delta) Find a witness to the failure of strong rectangularity, or null if there is none.
SubProductElement	strongRectangularityFailure(BinaryRelation S, BinaryRelation T, Partition delta, SubProductAlgebra mats)
SubalgebraLattice	sub()
Subtrace	subtrace(Partition beta) Find a subtrace for beta over its lower cover.
Subtrace	subtrace(Partition beta, Partition alpha) Find a subtrace for beta over alpha.
BinaryRelation	Tg(int a, int b) The tolerance generated by int a and b.
int	type(Partition beta)
int	type(Partition beta, Partition alpha)
int	type(Partition beta, Partition alpha, ProgressReport report) Find the type for beta over alpha.
int	typeJI(Partition beta, ProgressReport report) Find the type of beta over its lower cover.
java.util.Set<java.lang.Integer>	typeSet()
java.util.Set<java.lang.Integer>	typeSet(ProgressReport report)
java.util.Set<Partition>	universe() We use java.util.Set to hold the universe of the algebra.
java.util.Set<Partition>	universe(ProgressReport report)
boolean	universeFound()
void	updateSimilarityType() Not allowed, of course.
java.util.Map<Partition,java.util.List<Partition>>	upperCoversMap()
SubProductElement	weakCentralityFailure(BinaryRelation S, BinaryRelation T, Partition delta) Find a witness to the failure of weak centrality, or null if there is none.
SubProductElement	weakCentralityFailure(BinaryRelation S, BinaryRelation T, Partition delta, SubProductAlgebra mats)
Partition	weakCommutator(BinaryRelation S, BinaryRelation T) Calculate the weak commutator of S and T, which are tolerances or congruences.
Partition	weakCommutator(BinaryRelation S, BinaryRelation T, java.util.List<CentralityData> centralityList, ProgressReport report)
Partition	zero()

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

monitor

public static ProgressReport monitor

MAX_DRAWABLE_SIZE

public static final int MAX_DRAWABLE_SIZE

See Also:

Constant Field Values

MAX_DRAWABLE_INPUT_SIZE

public static final int MAX_DRAWABLE_INPUT_SIZE

See Also:

Constant Field Values

Constructor Detail

CongruenceLattice

public CongruenceLattice(SmallAlgebra alg)

Method Detail

setMonitor

public void setMonitor(ProgressReport m)

Specified by:

setMonitor in interface Algebra

getMonitor

public ProgressReport getMonitor()

Specified by:

getMonitor in interface Algebra

monitoring

public final boolean monitoring()

Specified by:

monitoring in interface Algebra

isTotal

public boolean isTotal()

Description copied from interface: Algebra

This will fail only if there are some OperationWithDefaultValue's which are not total.

Specified by:

isTotal in interface Algebra

Returns:

getAlgebra

public SmallAlgebra getAlgebra()

Get the algebra this is the congruence lattice of.

Returns:

isUnary

public boolean isUnary()

Specified by:

isUnary in interface Algebra

getDescription

public java.lang.String getDescription()

Specified by:

getDescription in interface Algebra

setDescription

public void setDescription(java.lang.String desc)

Specified by:

setDescription in interface Algebra

isSmallerThan

public boolean isSmallerThan(int size)

isDrawable

public boolean isDrawable()

getBasicLattice

public BasicLattice getBasicLattice()

getBasicLattice

public BasicLattice getBasicLattice(boolean makeIfNull)

Get the BasicLattice used primarily for drawing.

Returns:

a BasicLattice view

getDiagram

public org.latdraw.diagram.Diagram getDiagram()

getConTable

public javax.swing.JTable getConTable()

setConTable

public void setConTable(javax.swing.JTable table)

con

public CongruenceLattice con()

Get Con(Con(A)).

Returns:

sub

public SubalgebraLattice sub()

principals

public java.util.List<Partition> principals()

A list of the principal congruences sorted by rank (in the partition lattice).

Returns:

principals

public java.util.List<Partition> principals(ProgressReport report)

A list of the principal congruences sorted by rank (in the partition lattice).

Parameters:

report - the ProgressReport

Returns:

cardinality

public int cardinality()

Specified by:

cardinality in interface Algebra

Returns:

the cardinality if possible, else a negative int

inputSize

public int inputSize()

Description copied from interface: Algebra

The sum of the cardinality of the algebra raised to the arity of the operations.

Specified by:

inputSize in interface Algebra

Returns:

the inputSize or -1 if it is not an int

universe

public java.util.Set<Partition> universe()

Description copied from interface: Algebra

We use java.util.Set to hold the universe of the algebra. In order to accomodate infinite algebras and algebras with unknown cardinality, we allow the following changes:

• size can return a negative value if the cardinality is infinite or unknown or is larger that the largest int. In this case we may want to specify some information about the cardinality in another method but we have not decided on this yet.
• it is allowed that the iterator method of Set throw an UnsupportedOperationException.
• when extending AbstractSet, the contains method should be overwritten.

Specified by:

universe in interface Algebra

universe

public java.util.Set<Partition> universe(ProgressReport report)

getUniverseList

public java.util.List<Partition> getUniverseList()

isSimilarTo

public boolean isSimilarTo(Algebra alg)

Specified by:

isSimilarTo in interface Algebra

similarityType

public SimilarityType similarityType()

Specified by:

similarityType in interface Algebra

updateSimilarityType

public void updateSimilarityType()

Not allowed, of course.

Specified by:

updateSimilarityType in interface Algebra

iterator

public java.util.Iterator<Partition> iterator()

Description copied from interface: Algebra

returns the iterator of the universe. Since we allow that to be optional, this may throw an UnsupportedOperationException.

Specified by:

iterator in interface Algebra

getName

public java.lang.String getName()

Specified by:

getName in interface Algebra

setName

public void setName(java.lang.String v)

Specified by:

setName in interface Algebra

joinIrreduciblesFound

public boolean joinIrreduciblesFound()

joinIrreducibles

public java.util.List<Partition> joinIrreducibles()

A list of the join irreducible congruences sorted by rank (in the partition lattice).

Specified by:

joinIrreducibles in interface Lattice

joinIrreducibles

public java.util.List<Partition> joinIrreducibles(ProgressReport report)

A list of the join irreducibles sorted by rank (in the partition lattice); constructed if necessary.

joinIrreducible

public boolean joinIrreducible(Partition part)

meetIrreducibles

public java.util.List<Partition> meetIrreducibles()

Description copied from interface: Lattice

An optional operation returning the list of meet irreducible elements.

Specified by:

meetIrreducibles in interface Lattice

meetIrreducible

public boolean meetIrreducible(Partition part)

atoms

public java.util.List<Partition> atoms()

Specified by:

atoms in interface Lattice

makeAtoms

public void makeAtoms()

coatoms

public java.util.List<Partition> coatoms()

This is not implemented yet.

Specified by:

coatoms in interface Lattice

join

public java.lang.Object join(java.lang.Object a,
                             java.lang.Object b)

Specified by:

join in interface Lattice

join

public java.lang.Object join(java.util.List args)

Specified by:

join in interface Lattice

meet

public java.lang.Object meet(java.lang.Object a,
                             java.lang.Object b)

Specified by:

meet in interface Lattice

meet

public java.lang.Object meet(java.util.List args)

Specified by:

meet in interface Lattice

leq

public boolean leq(java.lang.Object a,
                   java.lang.Object b)

Description copied from interface: Order

The order relation.

Specified by:

leq in interface Order

complements

public java.util.List<Partition> complements(Partition par)

Returns a list of the complements of par.

Parameters:

par -

Returns:

findL3Generators

public java.util.List<Partition> findL3Generators()

Find congruences alpha, beta, gamma which generate L3 of Jonsson-Rival notation; see Fig 2.2 of Jipsen-Rose.

Returns:

findLXXGenerators

public java.util.List<Partition> findLXXGenerators()

findLPJ10Generators

public java.util.List<Partition> findLPJ10Generators()

constantOperations

public java.util.List<Operation> constantOperations()

Description copied from interface: Algebra

This gives a list of the operations of arity 0, which is a little different from the constants.

Specified by:

constantOperations in interface Algebra

operations

public java.util.List operations()

Specified by:

operations in interface Algebra

getOperation

public Operation getOperation(OperationSymbol sym)

Description copied from interface: Algebra

Get the operation correspond to a symbol or null if the symbol is not part of the similarityType.

Specified by:

getOperation in interface Algebra

getOperationsMap

public java.util.Map<OperationSymbol,Operation> getOperationsMap()

Specified by:

getOperationsMap in interface Algebra

makePrincipals

public void makePrincipals(ProgressReport report)

universeFound

public boolean universeFound()

stopMakeUniverse

public void stopMakeUniverse()

getMakeUniverseK

public int getMakeUniverseK()

getSizeComputed

public int getSizeComputed()

makeUniverse

public void makeUniverse(ProgressReport report)

makeUniverse

public void makeUniverse(int size)

makeUniverse

public void makeUniverse()

makeUniverse

public void makeUniverse(int maxSize,
                         ProgressReport report)

Construct the universe. If this method is interupted, the whole calculation starts over. We might change that if there is enough demand.

joinPrime

public boolean joinPrime(Partition beta)

Test if beta is join prime.

isDistributive

public boolean isDistributive()

makeJoinIrreducibles

public void makeJoinIrreducibles(ProgressReport report)

sortByRank

public void sortByRank(java.util.List<Partition> lst)

Sort by rank (in the partition lattice). The rank in size() - numberOfBlocks().

Parameters:

lst -

lowerStar

public Partition lowerStar(Partition beta)

If beta is join irreducible, this gives its lower cover; otherwise null.

upperCoversMap

public java.util.Map<Partition,java.util.List<Partition>> upperCoversMap()

Cg

public Partition Cg(java.lang.Object a,
                    java.lang.Object b)

The congruence generated by Object a and b.

Parameters:

a - an Object

b - an Object

Returns:

Cg

public Partition Cg(int a,
                    int b)

Cg

public Partition Cg(int a,
                    int b,
                    ProgressReport report)

The congruence generated by ints a and b.

Parameters:

a - an int

b - an int

Returns:

Cg

public Partition Cg(Partition part)

The smallest congruence containing part.

Parameters:

part - the generating partition in canonical form

Returns:

Cg

public Partition Cg(Partition part,
                    ProgressReport report)

The smallest congruence containing part.

Parameters:

part - the generating partition in canonical form

Returns:

Cg_Old

public Partition Cg_Old(Partition part,
                        ProgressReport report)

Tg

public BinaryRelation Tg(int a,
                         int b)

The tolerance generated by int a and b.

Parameters:

a -

b -

Returns:

strongRectangularityCommutator

public Partition strongRectangularityCommutator(BinaryRelation S,
                                                BinaryRelation T)

Calculate the strong rectangular commutator of S and T, which are tolerances or congruences.

Parameters:

S - a tolerance or congruence

T - a tolerance or congruence

Returns:

a congruence

strongRectangularityCommutator

public Partition strongRectangularityCommutator(BinaryRelation S,
                                                BinaryRelation T,
                                                java.util.List<CentralityData> centralityList,
                                                ProgressReport report)

weakCommutator

public Partition weakCommutator(BinaryRelation S,
                                BinaryRelation T)

Calculate the weak commutator of S and T, which are tolerances or congruences.

Parameters:

S - a tolerance or congruence

T - a tolerance or congruence

Returns:

a congruence

weakCommutator

public Partition weakCommutator(BinaryRelation S,
                                BinaryRelation T,
                                java.util.List<CentralityData> centralityList,
                                ProgressReport report)

commutator

public Partition commutator(BinaryRelation S,
                            BinaryRelation T)

Calculate the commutator of S and T, which are tolerances or congruences.

Parameters:

S - a tolerance or congruence

T - a tolerance or congruence

Returns:

a congruence

commutator

public Partition commutator(BinaryRelation S,
                            BinaryRelation T,
                            java.util.List<CentralityData> centralityList,
                            ProgressReport report)

calcCentrality

public java.util.List<CentralityData> calcCentrality(BinaryRelation S,
                                                     BinaryRelation T)

Given tolerances (or congruences) S and T, this tests, for every congruence delta of A, if S centralizes T mod delta. It tests weak, ordinary and strong rectangular centrality.

Parameters:

S -

T -

Returns:

a list of CentralityData

calcCentrality

public java.util.List<CentralityData> calcCentrality(BinaryRelation S,
                                                     BinaryRelation T,
                                                     ProgressReport report)

strongRectangularityFailure

public SubProductElement strongRectangularityFailure(BinaryRelation S,
                                                     BinaryRelation T,
                                                     Partition delta)

Find a witness to the failure of strong rectangularity, or null if there is none.

Parameters:

S -

T -

delta -

Returns:

strongRectangularityFailure

public SubProductElement strongRectangularityFailure(BinaryRelation S,
                                                     BinaryRelation T,
                                                     Partition delta,
                                                     SubProductAlgebra mats)

weakCentralityFailure

public SubProductElement weakCentralityFailure(BinaryRelation S,
                                               BinaryRelation T,
                                               Partition delta)

Find a witness to the failure of weak centrality, or null if there is none.

Parameters:

S -

T -

delta -

Returns:

weakCentralityFailure

public SubProductElement weakCentralityFailure(BinaryRelation S,
                                               BinaryRelation T,
                                               Partition delta,
                                               SubProductAlgebra mats)

centralityFailure

public SubProductElement centralityFailure(BinaryRelation S,
                                           BinaryRelation T,
                                           Partition delta)

Find a witness to the failure of centrality, or null if there is none.

Parameters:

S -

T -

delta -

Returns:

centralityFailure

public SubProductElement centralityFailure(BinaryRelation S,
                                           BinaryRelation T,
                                           Partition delta,
                                           SubProductAlgebra mats)

centralizes

public boolean centralizes(BinaryRelation S,
                           BinaryRelation T,
                           Partition delta)

centralizes

public boolean centralizes(BinaryRelation S,
                           BinaryRelation T,
                           Partition delta,
                           SubProductAlgebra mats)

matrices

public SubProductAlgebra matrices(BinaryRelation S,
                                  BinaryRelation T,
                                  ProgressReport report)

The set M(S,T) as in Kearnes-Kiss and other places. Note our correspondence between 2x2 matrices and 4-tuples is (a00, a01, a10, a11), that is, first row then second row. This may differ from what Kiss uses sometimes.

Parameters:

S - a binary relation but usually a tolerance or congruence

T -

Returns:

M(S,T)

typeSet

public java.util.Set<java.lang.Integer> typeSet()

typeSet

public java.util.Set<java.lang.Integer> typeSet(ProgressReport report)

type

public int type(Partition beta)

typeJI

public int typeJI(Partition beta,
                  ProgressReport report)

Find the type of beta over its lower cover. Beta is assumed to be join irreducible.

type

public int type(Partition beta,
                Partition alpha)

type

public int type(Partition beta,
                Partition alpha,
                ProgressReport report)

Find the type for beta over alpha. Beta is assumed to cover alpha.

subtrace

public Subtrace subtrace(Partition beta)

Find a subtrace for beta over its lower cover. Beta is assumed to be join irreducible.

subtrace

public Subtrace subtrace(Partition beta,
                         Partition alpha)

Find a subtrace for beta over alpha. Beta is assumed to cover alpha.

getJoinIrredToSubtraceMap

public java.util.Map<Partition,Subtrace> getJoinIrredToSubtraceMap()

getTypeFinder

public TypeFinder getTypeFinder()

generatingPair

public IntArray generatingPair(Partition part)

Find a pair [a, b] (as an IntArray) which generates part. Gives null if part is not principal.

findCoatomAbove

public Partition findCoatomAbove(Partition theta)

find a coaton of the congruence lattice greater than or equal to theta. This assumes theta is strictly less than one.

Parameters:

theta -

Returns:

findPrincipalChain

public java.util.List<Partition> findPrincipalChain()

Find a chain where each element is the join of the previous one and a principal congruence. The chain contains the least but not the greatest congruence.

Returns:

the chain

findUpperCover

public Partition findUpperCover(Partition congr)

Find some upper cover of congr.

Parameters:

congr -

Returns:

minimalElements

public static java.util.List<Partition> minimalElements(java.util.List<Partition> parList)

Find the min elements of a topologically sorted list.

Parameters:

parList - a topologically sorted list of partitions

Returns:

findMeetIrred

public Partition findMeetIrred(Partition a,
                               Partition b)

This finds a meet irreducible congruence which is maximal with respect to being above a and not above b. Note if a is covered by b then [a,b] transposes up to [m,m*].

findJoinIrred

public Partition findJoinIrred(Partition a,
                               Partition b)

This finds a join irreducible congruence which is minimal with respect to being below b and not below a. Note if a is covered by b then [a,b] transposes down to [j,j_*].

zero

public final Partition zero()

one

public final Partition one()

irredundantMeetDecomposition

public java.util.List<Partition> irredundantMeetDecomposition()

Find a set of meet irreducible irredundantly meeting to zero.

Returns:

irredundantMeetDecompositionOld

public java.util.List<Partition> irredundantMeetDecompositionOld()

makeIrredundantMeet

public java.util.List<Partition> makeIrredundantMeet(java.util.List<Partition> list)

permutabilityLevel

public int permutabilityLevel()

Get the permutabiltiy level of this lattice (not the whole variety). Store the actual partitions on permutabilityLevelWitnesses.

getPermutabilityLevelWitnesses

public Partition[] getPermutabilityLevelWitnesses()

idempotentPolynomials

public java.util.List<IntArray> idempotentPolynomials()

Find the nonconstant, idempotent member of Pol1(A).

Returns:

makeOperationTables

public void makeOperationTables()

Description copied from interface: Algebra

Make operation tables to speed up the evaluation of operations at the cost using more space.

Specified by:

makeOperationTables in interface Algebra

See Also:

Operation.makeTable

isIdempotent

public boolean isIdempotent()

Description copied from interface: Algebra

Test if all of the operations are idempotent.

Specified by:

isIdempotent in interface Algebra

main

public static void main(java.lang.String[] args)
                 throws java.lang.Exception

Throws:

java.lang.Exception