jhypercomplex.algebras.identities

Class AlgebraicProperties

java.lang.Object

jhypercomplex.algebras.identities.AlgebraicProperties

public abstract class AlgebraicProperties
extends java.lang.Object

Copyright © 2005-2015 by Markus Maute. All rights reserved.

Author:

Markus Maute (http:///www.jalgebra.com)

Constructor Summary

Constructors

Constructor and Description
AlgebraicProperties()

Method Summary

Methods

Modifier and Type	Method and Description
static java.util.TreeMap<java.lang.String,java.lang.Boolean>	getAssociationTypeIdentitiesCheckList(java.lang.Class hypernumber, int order, int rank, int n_comps, boolean commutator) Method that wraps (bunches) the checking of association type identities.
static boolean	isAdamsIdentitiesForAlternativeBinaryAlgebrasSatisfied(java.lang.Class<?> hypernumber, int n_comps)
static boolean	isAdamsIdentitySatisfied(java.lang.Class<?> hypernumber, int n_comps)
static boolean	isAlternative(BinaryAlgebra h1, BinaryAlgebra h2) Alternativity means right- and left-alternative.
static boolean	isAlternativeBinaryAlgebraIdentitySatisfied(java.lang.Class hypernumber, int nr_of_identity, int n_comps) A collection of identities satisfied by alternative BinaryAlgebras.
static boolean	isAssociative(java.lang.Class hypernumber, int n_comps) A(BC) = (AB)C
static boolean	isAssociatorIdentity1Satisfied(java.lang.Class<?> algebra, int n_comps)
static boolean	isAssociatorTotallyAntisymmeric(java.lang.Class<?> hypernumber, int n_comps)
static boolean	isBMQuaternatorIdentitySatisfied(java.lang.Class<?> hypernumber, int nr, int n_comps)
static boolean	isBolLoop(java.lang.Class hypernumber, int n_comps)
static boolean	isBrukLoop(java.lang.Class hypernumber, int n_comps)
static boolean	isCLoop(java.lang.Class hypernumber, int n_comps, boolean set_first_comp)
static boolean	isCommutatorAdamsIdentitySatisfied(java.lang.Class<?> hypernumber, int n_comps)
static boolean	isCommutatorCLoop(java.lang.Class hypernumber, int n_comps)
static boolean	isCommutatorIdentity1Satisfied(java.lang.Class<?> algebra, int n_comps)
static boolean	isCommutatorMalcevIdentitySatisfied(java.lang.Class<?> hypernumber, int n_comps)
static boolean	isCommutatorSagleIdentitiesSatisfied(java.lang.Class<?> hypernumber, int n_comps, boolean COMMUTATOR) The commutator form of the Sagle identity is given by one of the following 4 equivalent expressions: [A,[B,[C,D]]] + [B,[C,[D,A]]] + [C,[D,[A,B]]] + [D,[A,[B,C]]] + [[B,D],[A,C]] = 0 [[[A,B],C],D] + [[[B,C],D],A] + [[[C,D],A],B] + [[[D,A],B],C] + [[B,D],[A,C]] = 0 [A,[[B,C],D]] + [B,[[C,D],A]] + [C,[[D,A],D]] + [D,[[A,B],C]] + [[A,C],[B,D]] = 0 [[A,[B,C]],D] + [[B,[C,D]],A] + [[C,[D,A]],B] + [[D,[A,B]],C] + [[A,C],[B,D]] = 0 All 4 versions are checked.
static boolean	isDerivation(java.lang.Class<?> hypernumber, int n_comps) A derivative is a linear map D that satisfies the "Leibnitz rule", i.e.
static boolean	isExtraLoop(java.lang.Class hypernumber, int n_comps, boolean set_first_comp) A(B(CA)) = ((AB)C)A
static boolean	isFirstHentzelPeresiIdentitySatisfied(java.lang.Class<?> hypernumber, int n_comps) The first Hentzel-Peresi identity is defined as follows: sigma{b,c,e} ([a,(c,d,e),b] + [a,b,(c,d,e)]) = sigma{b,c,e} ([a,{{c,d},e} - {c,{d,e}},b] + [a,b,{{c,d},e} - {c,{d,e}}]) with sigma{b,c,e} the cyclic permutations over b,c,e and (c,d,e) the Jordan associator.
static boolean	isFlexible(java.lang.Class hypernumber, int n_comps) A(BA) = (AB)A
static boolean	isInLeftNucleus(BinaryAlgebra h)
static boolean	isInMiddleNucleus(BinaryAlgebra h)
static boolean	isInNucleus(BinaryAlgebra h)
static boolean	isInRightNucleus(BinaryAlgebra h)
static boolean	isInversePropertiesSatisfied(java.lang.Class<?> hypernumber, int n_comps) A^–1(AB)=B=(BA)A^–1
static boolean	isJordanIdentitySatisfied(java.lang.Class<?> hypernumber, int n_comps) (A^2B)A - A^2(BA) = 0 <=> [A^2,B,A] = 0 Equivalent identities are A^2(AB) =A(A^2B) (BA)A^2 = (BA^2)A (AB)A^2 = (A^2B)A which are also checked.
static boolean	isJordansTripleIdentitySatisfied(java.lang.Class<?> BinaryAlgebra, int n_comps) Cycl_Sum_{ ([A, [B, A, C], D] + [B, [A, A, C], D] - [B, A, [C, A, D]]) = 0
static boolean	isLCLoop(java.lang.Class hypernumber, int n_comps, boolean set_first_comp)
static boolean	isLeftAlternative(BinaryAlgebra h1, BinaryAlgebra h2) Left-alternativity is defined by means of the associator identity [h1,h1,h2] = 0.
static boolean	isLeftBolLoop(java.lang.Class hypernumber, int n_comps)
static boolean	isLeftMoufangLoop(java.lang.Class hypernumber, int n_comps) ((AB)A)C=A(B(AC))
static boolean	IsLeftNuclearSquareLoop(java.lang.Class hypernumber, int n_comps)
static boolean	isLie3BinaryAlgebra(java.lang.Class<?> hypernumber, int n_comps)
static boolean	isLinearizedJordanIdentitySatisfied(java.lang.Class<?> hypernumber, int n_comps)
static boolean	isMiddleMoufangLoop(java.lang.Class hypernumber, int n_comps) (AB)(CA) = (A(BC))A = A((BC)A)
static boolean	IsMiddleNuclearSquareLoop(java.lang.Class hypernumber, int n_comps)
static boolean	isMoufangLoop(java.lang.Class hypernumber, int n_comps)
static boolean	isNormMultiplicative(java.lang.Class<?> hypernumber, int n_comps)
static boolean	isPolarizedJordanIdentitySatisfied(java.lang.Class<?> hypernumber, int n_comps)
static boolean	isRCLoop(java.lang.Class hypernumber, int n_comps, boolean set_first_comp)
static boolean	isRightAlternative(BinaryAlgebra h1, BinaryAlgebra h2) Right-alternativity is defined by means of the associator identity [h1,h2,h2] = 0.
static boolean	isRightBolLoop(java.lang.Class hypernumber, int n_comps)
static boolean	isRightMoufangLoop(java.lang.Class hypernumber, int n_comps) A(B(CB))=((AB)C)B
static boolean	IsRightNuclearSquareLoop(java.lang.Class hypernumber, int n_comps)
static boolean	isSecondHentzelPeresiIdentitySatisfied(java.lang.Class<?> hypernumber, int n_comps) The second Hentzel-Peresi identity for a flexible BinaryAlgebra is defined as follows: [A(AB)-(BA)A,C,D] + [{[A,B],C},A,D] - [[A,B],{A,C},D] = 0
static boolean	isTakhtajanIdentitySatisfied(java.lang.Class<?> hypernumber, int n_comps)
static boolean	isWeneIdentitySatisfied(java.lang.Class<?> hypernumber, int n_comps)
static void	setNumericalMode(boolean is_numerical)
static void	testFourCommutator(java.lang.Class<?> hypernumber, int n_comps)

Methods inherited from class java.lang.Object

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

Constructor Detail

AlgebraicProperties

public AlgebraicProperties()

Method Detail

setNumericalMode

public static void setNumericalMode(boolean is_numerical)

Parameters:

is_numerical - 'true': calculations are carried out numerically. 'false': calculations are carried out numerically.

getAssociationTypeIdentitiesCheckList

public static java.util.TreeMap<java.lang.String,java.lang.Boolean> getAssociationTypeIdentitiesCheckList(java.lang.Class hypernumber,
                                                                                          int order,
                                                                                          int rank,
                                                                                          int n_comps,
                                                                                          boolean commutator)
                                                                                                   throws java.lang.Exception

Method that wraps (bunches) the checking of association type identities. It runs through a list of identities specified and checks them. For each identity a flag is written to a map, stating if the identity is satisfied or not. An association type identity of order n and rank r, designated (n,r), is is understood to contain n elements of which r are different. Example: (4,2): A(B(CA)) = (AB)(CA) There are 6 possibilities to put a pair of equal elements in a 4-tuple, thus the method checks these 6 identities.

Parameters:

hypernumber - Hypernumber.

order - Order > 2.

rank - Rank > 1 and <= order.

n_comps - Number of components set.

commutator - 'true': Commutator association type identity. 'false': Association type identity, based on the native BinaryAlgebraic product.

Returns:

map (,) --> true/false

Throws:

java.lang.Exception

isAssociative

public static boolean isAssociative(java.lang.Class hypernumber,
                    int n_comps)
                             throws java.lang.Exception

A(BC) = (AB)C

Throws:

java.lang.Exception

isFlexible

public static boolean isFlexible(java.lang.Class hypernumber,
                 int n_comps)
                          throws java.lang.Exception

A(BA) = (AB)A

Throws:

java.lang.Exception

isExtraLoop

public static boolean isExtraLoop(java.lang.Class hypernumber,
                  int n_comps,
                  boolean set_first_comp)
                           throws java.lang.Exception

A(B(CA)) = ((AB)C)A

Throws:

java.lang.Exception

isMoufangLoop

public static boolean isMoufangLoop(java.lang.Class hypernumber,
                    int n_comps)
                             throws java.lang.Exception

Throws:

java.lang.Exception

isLeftMoufangLoop

public static boolean isLeftMoufangLoop(java.lang.Class hypernumber,
                        int n_comps)
                                 throws java.lang.Exception

((AB)A)C=A(B(AC))

Throws:

java.lang.Exception

isRightMoufangLoop

public static boolean isRightMoufangLoop(java.lang.Class hypernumber,
                         int n_comps)
                                  throws java.lang.Exception

A(B(CB))=((AB)C)B

Throws:

java.lang.Exception

isMiddleMoufangLoop

public static boolean isMiddleMoufangLoop(java.lang.Class hypernumber,
                          int n_comps)
                                   throws java.lang.Exception

(AB)(CA) = (A(BC))A = A((BC)A)

Throws:

java.lang.Exception

isCLoop

public static boolean isCLoop(java.lang.Class hypernumber,
              int n_comps,
              boolean set_first_comp)
                       throws java.lang.Exception

Throws:

java.lang.Exception

isCommutatorCLoop

public static boolean isCommutatorCLoop(java.lang.Class hypernumber,
                        int n_comps)
                                 throws java.lang.Exception

Throws:

java.lang.Exception

isLCLoop

public static boolean isLCLoop(java.lang.Class hypernumber,
               int n_comps,
               boolean set_first_comp)
                        throws java.lang.Exception

Throws:

java.lang.Exception

isRCLoop

public static boolean isRCLoop(java.lang.Class hypernumber,
               int n_comps,
               boolean set_first_comp)
                        throws java.lang.Exception

Throws:

java.lang.Exception

IsLeftNuclearSquareLoop

public static boolean IsLeftNuclearSquareLoop(java.lang.Class hypernumber,
                              int n_comps)
                                       throws java.lang.Exception

Throws:

java.lang.Exception

IsMiddleNuclearSquareLoop

public static boolean IsMiddleNuclearSquareLoop(java.lang.Class hypernumber,
                                int n_comps)
                                         throws java.lang.Exception

Throws:

java.lang.Exception

IsRightNuclearSquareLoop

public static boolean IsRightNuclearSquareLoop(java.lang.Class hypernumber,
                               int n_comps)
                                        throws java.lang.Exception

Throws:

java.lang.Exception

isLeftBolLoop

public static boolean isLeftBolLoop(java.lang.Class hypernumber,
                    int n_comps)
                             throws java.lang.Exception

Throws:

java.lang.Exception

isRightBolLoop

public static boolean isRightBolLoop(java.lang.Class hypernumber,
                     int n_comps)
                              throws java.lang.Exception

Throws:

java.lang.Exception

isBolLoop

public static boolean isBolLoop(java.lang.Class hypernumber,
                int n_comps)
                         throws java.lang.Exception

Throws:

java.lang.Exception

isBrukLoop

public static boolean isBrukLoop(java.lang.Class hypernumber,
                 int n_comps)
                          throws java.lang.Exception

Throws:

java.lang.Exception

isAlternativeBinaryAlgebraIdentitySatisfied

public static boolean isAlternativeBinaryAlgebraIdentitySatisfied(java.lang.Class hypernumber,
                                                  int nr_of_identity,
                                                  int n_comps)
                                                           throws java.lang.Exception

A collection of identities satisfied by alternative BinaryAlgebras.

Throws:

java.lang.Exception

isAlternative

public static boolean isAlternative(BinaryAlgebra h1,
                    BinaryAlgebra h2)
                             throws java.lang.Exception

Alternativity means right- and left-alternative.

Parameters:

h1 - Hypermummer 1.

h2 - Hypernummer 2.

Returns:

true if alternative, false otherwise.

Throws:

java.lang.Exception

isLeftAlternative

public static boolean isLeftAlternative(BinaryAlgebra h1,
                        BinaryAlgebra h2)
                                 throws java.lang.Exception

Left-alternativity is defined by means of the associator identity [h1,h1,h2] = 0.

Parameters:

h1 - Hypermummer 1.

h2 - Hypernummer 2.

Returns:

true if alternative, false otherwise.

Throws:

java.lang.Exception

isRightAlternative

public static boolean isRightAlternative(BinaryAlgebra h1,
                         BinaryAlgebra h2)
                                  throws java.lang.Exception

Right-alternativity is defined by means of the associator identity [h1,h2,h2] = 0.

Parameters:

h1 - Hypermummer 1.

h2 - Hypernummer 2.

Returns:

true if alternative, false otherwise.

Throws:

java.lang.Exception

isInLeftNucleus

public static boolean isInLeftNucleus(BinaryAlgebra h)
                               throws java.lang.Exception

Parameters:

h - Hyperumber

Returns:

'tue' if [h,A,B]] = 0 for any A,B 'false else

Throws:

java.lang.Exception

isInRightNucleus

public static boolean isInRightNucleus(BinaryAlgebra h)
                                throws java.lang.Exception

Parameters:

h - Hyperumber

Returns:

'tue' if [A,h,B]] = 0 for any A,B 'false else

Throws:

java.lang.Exception

isNormMultiplicative

public static boolean isNormMultiplicative(java.lang.Class<?> hypernumber,
                           int n_comps)
                                    throws java.lang.Exception

Parameters:

hypernumber -

n_comps -

Returns:

'true' if the norm N satisfies N(A)N(B) = N(AB), 'false' else.

Throws:

java.lang.Exception

isInMiddleNucleus

public static boolean isInMiddleNucleus(BinaryAlgebra h)
                                 throws java.lang.Exception

Parameters:

h - Hyperumber

Returns:

'true' if [A,B,h]] = 0 for any A,B 'false else

Throws:

java.lang.Exception

isInNucleus

public static boolean isInNucleus(BinaryAlgebra h)
                           throws java.lang.Exception

Parameters:

h - Hypernumber.

Returns:

'true' if h is in the left-, middle and right-nucleus. 'false' else.

Throws:

java.lang.Exception

isInversePropertiesSatisfied

public static boolean isInversePropertiesSatisfied(java.lang.Class<?> hypernumber,
                                   int n_comps)
                                            throws java.lang.Exception

A^–1(AB)=B=(BA)A^–1

Throws:

java.lang.Exception

isCommutatorSagleIdentitiesSatisfied

public static boolean isCommutatorSagleIdentitiesSatisfied(java.lang.Class<?> hypernumber,
                                           int n_comps,
                                           boolean COMMUTATOR)
                                                    throws java.lang.Exception

The commutator form of the Sagle identity is given by one of the following 4 equivalent expressions: [A,[B,[C,D]]] + [B,[C,[D,A]]] + [C,[D,[A,B]]] + [D,[A,[B,C]]] + [[B,D],[A,C]] = 0 [[[A,B],C],D] + [[[B,C],D],A] + [[[C,D],A],B] + [[[D,A],B],C] + [[B,D],[A,C]] = 0 [A,[[B,C],D]] + [B,[[C,D],A]] + [C,[[D,A],D]] + [D,[[A,B],C]] + [[A,C],[B,D]] = 0 [[A,[B,C]],D] + [[B,[C,D]],A] + [[C,[D,A]],B] + [[D,[A,B]],C] + [[A,C],[B,D]] = 0 All 4 versions are checked.

Throws:

java.lang.Exception

testFourCommutator

public static void testFourCommutator(java.lang.Class<?> hypernumber,
                      int n_comps)
                               throws java.lang.Exception

Throws:

java.lang.Exception

isTakhtajanIdentitySatisfied

public static boolean isTakhtajanIdentitySatisfied(java.lang.Class<?> hypernumber,
                                   int n_comps)
                                            throws java.lang.Exception

Throws:

java.lang.Exception

isDerivation

public static boolean isDerivation(java.lang.Class<?> hypernumber,
                   int n_comps)
                            throws java.lang.Exception

A derivative is a linear map D that satisfies the "Leibnitz rule", i.e. D(AB) = D(A) B + A D(B)

Throws:

java.lang.Exception

isLie3BinaryAlgebra

public static boolean isLie3BinaryAlgebra(java.lang.Class<?> hypernumber,
                          int n_comps)
                                   throws java.lang.Exception

Throws:

java.lang.Exception

isAdamsIdentitySatisfied

public static boolean isAdamsIdentitySatisfied(java.lang.Class<?> hypernumber,
                               int n_comps)
                                        throws java.lang.Exception

Throws:

java.lang.Exception

isCommutatorAdamsIdentitySatisfied

public static boolean isCommutatorAdamsIdentitySatisfied(java.lang.Class<?> hypernumber,
                                         int n_comps)
                                                  throws java.lang.Exception

Throws:

java.lang.Exception

isAdamsIdentitiesForAlternativeBinaryAlgebrasSatisfied

public static boolean isAdamsIdentitiesForAlternativeBinaryAlgebrasSatisfied(java.lang.Class<?> hypernumber,
                                                             int n_comps)
                                                                      throws java.lang.Exception

Throws:

java.lang.Exception

isFirstHentzelPeresiIdentitySatisfied

public static boolean isFirstHentzelPeresiIdentitySatisfied(java.lang.Class<?> hypernumber,
                                            int n_comps)
                                                     throws java.lang.Exception

The first Hentzel-Peresi identity is defined as follows: sigma{b,c,e} ([a,(c,d,e),b] + [a,b,(c,d,e)]) = sigma{b,c,e} ([a,{{c,d},e} - {c,{d,e}},b] + [a,b,{{c,d},e} - {c,{d,e}}]) with sigma{b,c,e} the cyclic permutations over b,c,e and (c,d,e) the Jordan associator.

Throws:

java.lang.Exception

isAssociatorTotallyAntisymmeric

public static boolean isAssociatorTotallyAntisymmeric(java.lang.Class<?> hypernumber,
                                      int n_comps)
                                               throws java.lang.Exception

Throws:

java.lang.Exception

isSecondHentzelPeresiIdentitySatisfied

public static boolean isSecondHentzelPeresiIdentitySatisfied(java.lang.Class<?> hypernumber,
                                             int n_comps)
                                                      throws java.lang.Exception

The second Hentzel-Peresi identity for a flexible BinaryAlgebra is defined as follows: [A(AB)-(BA)A,C,D] + [{[A,B],C},A,D] - [[A,B],{A,C},D] = 0

Throws:

java.lang.Exception

isCommutatorMalcevIdentitySatisfied

public static boolean isCommutatorMalcevIdentitySatisfied(java.lang.Class<?> hypernumber,
                                          int n_comps)
                                                   throws java.lang.Exception

Throws:

java.lang.Exception

isWeneIdentitySatisfied

public static boolean isWeneIdentitySatisfied(java.lang.Class<?> hypernumber,
                              int n_comps)
                                       throws java.lang.Exception

Throws:

java.lang.Exception

isJordanIdentitySatisfied

public static boolean isJordanIdentitySatisfied(java.lang.Class<?> hypernumber,
                                int n_comps)
                                         throws java.lang.Exception

(A^2B)A - A^2(BA) = 0 <=> [A^2,B,A] = 0 Equivalent identities are A^2(AB) =A(A^2B) (BA)A^2 = (BA^2)A (AB)A^2 = (A^2B)A which are also checked.

Throws:

java.lang.Exception

isBMQuaternatorIdentitySatisfied

public static boolean isBMQuaternatorIdentitySatisfied(java.lang.Class<?> hypernumber,
                                       int nr,
                                       int n_comps)
                                                throws java.lang.Exception

Parameters:

hypernumber - Hypernumber.

nr - Number of identity (1, ...,4).

Returns:

'true' if Bremmner Qutaternator identity is satisfied.

Throws:

java.lang.Exception

isPolarizedJordanIdentitySatisfied

public static boolean isPolarizedJordanIdentitySatisfied(java.lang.Class<?> hypernumber,
                                         int n_comps)
                                                  throws java.lang.Exception

Throws:

java.lang.Exception

isLinearizedJordanIdentitySatisfied

public static boolean isLinearizedJordanIdentitySatisfied(java.lang.Class<?> hypernumber,
                                          int n_comps)
                                                   throws java.lang.Exception

Throws:

java.lang.Exception

isJordansTripleIdentitySatisfied

public static boolean isJordansTripleIdentitySatisfied(java.lang.Class<?> BinaryAlgebra,
                                       int n_comps)
                                                throws java.lang.Exception

Cycl_Sum_{ ([A, [B, A, C], D] + [B, [A, A, C], D] - [B, A, [C, A, D]]) = 0

Throws:

java.lang.Exception

isCommutatorIdentity1Satisfied

public static boolean isCommutatorIdentity1Satisfied(java.lang.Class<?> algebra,
                                     int n_comps)
                                              throws java.lang.Exception

Throws:

java.lang.Exception

isAssociatorIdentity1Satisfied

public static boolean isAssociatorIdentity1Satisfied(java.lang.Class<?> algebra,
                                     int n_comps)
                                              throws java.lang.Exception

Throws:

java.lang.Exception