jhypercomplex.algebras.identities

Class AlgebraicFunctions

java.lang.Object

jhypercomplex.algebras.identities.AlgebraicFunctions

public abstract class AlgebraicFunctions
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
AlgebraicFunctions()

Method Summary

Methods

Modifier and Type	Method and Description
static BinaryAlgebra	getAntiAssociator(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c)
static BinaryAlgebra	getAntiCommutator(BinaryAlgebra a, BinaryAlgebra b)
static BinaryAlgebra	getAssociationType(int nr, BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d) There are 5 ways to put the brackets in a product of 4 Algebras.
static BinaryAlgebra	getAssociationType(int nr, BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d, BinaryAlgebra e) There are 14 ways to put the brackets in a product of 5 Algebras.
static BinaryAlgebra	getAssociationType(int nr, BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d, BinaryAlgebra e, boolean commutator) "Switch"-method, that allows for an invocation of the association type method or alternatively the commutator association type method.
static BinaryAlgebra	getAssociationType(int nr, BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d, boolean commutator) "Switch"-method, that allows for an invocation of the association type method or alternatively the commutator association type method.
static java.lang.String	getAssociationTypeAsString(int nr, java.lang.String a, java.lang.String b, java.lang.String c, java.lang.String d)
static java.lang.String	getAssociationTypeAsString(int nr, java.lang.String a, java.lang.String b, java.lang.String c, java.lang.String d, java.lang.String e)
static BinaryAlgebra	getAssociationTypeCommutator(int nr, BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d) There are 5 ways to nest 3 commutators of 4 Algebras, namely nr 1 [[[a,b],c],d] 2 [a,[b,[c,d]]] 3 [[a,[b,c]],d] 4 [a,[[b,c],d]] 5 [[a,b],[c,d]]
static BinaryAlgebra	getAssociationTypeCommutator(int nr, BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d, BinaryAlgebra e) There are 14 ways to nest 4 commutators of 5 Algebras, namely nr 1 [a,[b,[c,[d,e]]]] 3 [a,[b,[[c,d],e]]] 8 [a,[[b,c],[d,e]]] 5 [a,[[b,[c,d]],e]] 9 [a,[[[b,c],d],e]] 2 [[a,b],[c,[d,e]]] 4 [[a,b],[[c,d],e]] 11 [[a,[b,c]],[d,e]] 6 [[a,[b,[c,d]]],e] 10 [[a,[[b,c],d]],e] 13 [[[a,b],c],[d,e]] 7 [[[a,b],[c,d]],e] 12 [[[a,[b,c]],d],e] 14 [[[[a,b],c],d],e]
static java.lang.String	getAssociationTypeCommutatorAsString(int nr, java.lang.String a, java.lang.String b, java.lang.String c, java.lang.String d)
static java.lang.String	getAssociationTypeCommutatorAsString(int nr, java.lang.String a, java.lang.String b, java.lang.String c, java.lang.String d, java.lang.String e)
static BinaryAlgebra	getAssociator(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c)
static BinaryAlgebra	getBMQuaternator1(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d) First quaternator as defined by Murray R.
static BinaryAlgebra	getBMQuaternator2(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d) Second quaternator as defined by Murray R.
static BinaryAlgebra	getBMQuaternatorIdentity1(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d)
static BinaryAlgebra	getBMQuaternatorIdentity2(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d)
static BinaryAlgebra	getBMQuaternatorIdentity3(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d)
static BinaryAlgebra	getBMQuaternatorIdentity4(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d)
static BinaryAlgebra	getCommuAssociator(BinaryAlgebra h1, BinaryAlgebra h2, BinaryAlgebra h3, BinaryAlgebra h4)
static BinaryAlgebra	getCommutator(BinaryAlgebra a, BinaryAlgebra b)
static BinaryAlgebra	getCommutator(BinaryAlgebra a, BinaryAlgebra b, boolean collect_terms)
static BinaryAlgebra	getCommutatorBMQuaternator1(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d) First quaternator as defined by Murray R.
static BinaryAlgebra	getCommutatorJacobian(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c)
static BinaryAlgebra	getCommutatorSaglian(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d)
static BinaryAlgebra	getFirstHetztelPeresian(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d, BinaryAlgebra e)
static BinaryAlgebra	getJacobian(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c)
static BinaryAlgebra	getJordanAssociator(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c)
static BinaryAlgebra	getKleinfeldFunction(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d)
static BinaryAlgebra	getQuaternator(BinaryAlgebra h1, BinaryAlgebra h2, BinaryAlgebra h3, BinaryAlgebra h4)
static BinaryAlgebra	getQuaternator1(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d)
static BinaryAlgebra	getQuaternator2(BinaryAlgebra h1, BinaryAlgebra h2, BinaryAlgebra h3, BinaryAlgebra h4)
static BinaryAlgebra	getSabininator(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d)
static BinaryAlgebra	getSabininQuaternator(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d)
static BinaryAlgebra	getSecondHetztelPeresian(BinaryAlgebra a, BinaryAlgebra b, BinaryAlgebra c, BinaryAlgebra d)

Methods inherited from class java.lang.Object

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

Constructor Detail

AlgebraicFunctions

public AlgebraicFunctions()

Method Detail

getCommutator

public static BinaryAlgebra getCommutator(BinaryAlgebra a,
                          BinaryAlgebra b)
                                   throws java.lang.Exception

Throws:

java.lang.Exception

getCommutator

public static BinaryAlgebra getCommutator(BinaryAlgebra a,
                          BinaryAlgebra b,
                          boolean collect_terms)
                                   throws java.lang.Exception

Throws:

java.lang.Exception

getAntiCommutator

public static BinaryAlgebra getAntiCommutator(BinaryAlgebra a,
                              BinaryAlgebra b)
                                       throws java.lang.Exception

Throws:

java.lang.Exception

getAssociator

public static BinaryAlgebra getAssociator(BinaryAlgebra a,
                          BinaryAlgebra b,
                          BinaryAlgebra c)
                                   throws java.lang.Exception

Parameters:

a - BinaryAlgebra 1

b - BinaryAlgebra 2

c - BinaryAlgebra 3

Returns:

Associator: [a,b,c] != (a*b)*c - a*(b*c).

Throws:

java.lang.Exception

getAntiAssociator

public static BinaryAlgebra getAntiAssociator(BinaryAlgebra a,
                              BinaryAlgebra b,
                              BinaryAlgebra c)
                                       throws java.lang.Exception

Parameters:

a - BinaryAlgebra 1

b - BinaryAlgebra 2

c - BinaryAlgebra 3

Returns:

Anti-Associator: (a,b,c) != (a*b)*c + a*(b*c).

Throws:

java.lang.Exception

getAssociationType

public static BinaryAlgebra getAssociationType(int nr,
                               BinaryAlgebra a,
                               BinaryAlgebra b,
                               BinaryAlgebra c,
                               BinaryAlgebra d,
                               boolean commutator)
                                        throws java.lang.Exception

"Switch"-method, that allows for an invocation of the association type method or alternatively the commutator association type method.

Parameters:

commutator - 'true' Commutator association type method. 'false' Associtaion type method.

Throws:

java.lang.Exception

getAssociationType

public static BinaryAlgebra getAssociationType(int nr,
                               BinaryAlgebra a,
                               BinaryAlgebra b,
                               BinaryAlgebra c,
                               BinaryAlgebra d)
                                        throws java.lang.Exception

There are 5 ways to put the brackets in a product of 4 Algebras. These 5 so called association types of order 4 are: nr 1 ((ab)c)d 2 a((bc)d) 3 (ab)(cd) 4 (a(bc))d 5 a(b(cd))

Parameters:

nr - Number of association type.

a - 1-st BinaryAlgebra of product.

b - 2-nd BinaryAlgebra of product.

c - 3-rd BinaryAlgebra of product.

d - 4-th BinaryAlgebra of product.

Returns:

Association type.

Throws:

java.lang.Exception

getAssociationType

public static BinaryAlgebra getAssociationType(int nr,
                               BinaryAlgebra a,
                               BinaryAlgebra b,
                               BinaryAlgebra c,
                               BinaryAlgebra d,
                               BinaryAlgebra e,
                               boolean commutator)
                                        throws java.lang.Exception

"Switch"-method, that allows for an invocation of the association type method or alternatively the commutator association type method.

Parameters:

commutator - 'true' Commutator associtaion type method. 'false' Aassocitaion type method.

Throws:

java.lang.Exception

getAssociationType

public static BinaryAlgebra getAssociationType(int nr,
                               BinaryAlgebra a,
                               BinaryAlgebra b,
                               BinaryAlgebra c,
                               BinaryAlgebra d,
                               BinaryAlgebra e)
                                        throws java.lang.Exception

There are 14 ways to put the brackets in a product of 5 Algebras. These 14 so called association types of order 5 are: nr 1 a(b(c(de))) 3 a(b((cd)e)) 8 a((bc)(de)) 5 a((b(cd))e) 9 a(((bc)d)e) 2 (ab)(c(de)) 4 (ab)((cd)e) 11 (a(bc))(de) 6 (a(b(cd)))e 10 (a((bc)d))e 13 ((ab)c)(de) 7 ((ab)(cd))e 12 ((a(bc))d)e 14 (((ab)c)d)e

Parameters:

nr - Number of association type.

a - 1-st BinaryAlgebra of product.

b - 2-nd BinaryAlgebra of product.

c - 3-rd BinaryAlgebra of product.

d - 4-th BinaryAlgebra of product.

e - 5-th BinaryAlgebra of product.

Returns:

Association type.

Throws:

java.lang.Exception

getAssociationTypeAsString

public static java.lang.String getAssociationTypeAsString(int nr,
                                          java.lang.String a,
                                          java.lang.String b,
                                          java.lang.String c,
                                          java.lang.String d)
                                                   throws java.lang.Exception

Throws:

java.lang.Exception

getAssociationTypeAsString

public static java.lang.String getAssociationTypeAsString(int nr,
                                          java.lang.String a,
                                          java.lang.String b,
                                          java.lang.String c,
                                          java.lang.String d,
                                          java.lang.String e)
                                                   throws java.lang.Exception

Throws:

java.lang.Exception

getAssociationTypeCommutator

public static BinaryAlgebra getAssociationTypeCommutator(int nr,
                                         BinaryAlgebra a,
                                         BinaryAlgebra b,
                                         BinaryAlgebra c,
                                         BinaryAlgebra d)
                                                  throws java.lang.Exception

There are 5 ways to nest 3 commutators of 4 Algebras, namely nr 1 [[[a,b],c],d] 2 [a,[b,[c,d]]] 3 [[a,[b,c]],d] 4 [a,[[b,c],d]] 5 [[a,b],[c,d]]

Parameters:

nr - Number of triple commutator

a - 1-st BinaryAlgebra of product.

b - 2-nd BinaryAlgebra of product.

c - 3-rd BinaryAlgebra of product.

d - 4-th BinaryAlgebra of product.

Returns:

Triple commutator.

Throws:

java.lang.Exception

getAssociationTypeCommutator

public static BinaryAlgebra getAssociationTypeCommutator(int nr,
                                         BinaryAlgebra a,
                                         BinaryAlgebra b,
                                         BinaryAlgebra c,
                                         BinaryAlgebra d,
                                         BinaryAlgebra e)
                                                  throws java.lang.Exception

There are 14 ways to nest 4 commutators of 5 Algebras, namely nr 1 [a,[b,[c,[d,e]]]] 3 [a,[b,[[c,d],e]]] 8 [a,[[b,c],[d,e]]] 5 [a,[[b,[c,d]],e]] 9 [a,[[[b,c],d],e]] 2 [[a,b],[c,[d,e]]] 4 [[a,b],[[c,d],e]] 11 [[a,[b,c]],[d,e]] 6 [[a,[b,[c,d]]],e] 10 [[a,[[b,c],d]],e] 13 [[[a,b],c],[d,e]] 7 [[[a,b],[c,d]],e] 12 [[[a,[b,c]],d],e] 14 [[[[a,b],c],d],e]

Parameters:

nr - Number of quadruple commutator

a - 1-st BinaryAlgebra of product.

b - 2-nd BinaryAlgebra of product.

c - 3-rd BinaryAlgebra of product.

d - 4-th BinaryAlgebra of product.

Returns:

Quadruple commutator.

Throws:

java.lang.Exception

getAssociationTypeCommutatorAsString

public static java.lang.String getAssociationTypeCommutatorAsString(int nr,
                                                    java.lang.String a,
                                                    java.lang.String b,
                                                    java.lang.String c,
                                                    java.lang.String d)
                                                             throws java.lang.Exception

Throws:

java.lang.Exception

getAssociationTypeCommutatorAsString

public static java.lang.String getAssociationTypeCommutatorAsString(int nr,
                                                    java.lang.String a,
                                                    java.lang.String b,
                                                    java.lang.String c,
                                                    java.lang.String d,
                                                    java.lang.String e)
                                                             throws java.lang.Exception

Throws:

java.lang.Exception

getJacobian

public static BinaryAlgebra getJacobian(BinaryAlgebra a,
                        BinaryAlgebra b,
                        BinaryAlgebra c)
                                 throws java.lang.Exception

Parameters:

a - BinaryAlgebra 1.

b - BinaryAlgebra 2.

c - BinaryAlgebra 3.

Returns:

(a*b)*c + (b*c)*a + (c*a)*b

Throws:

java.lang.Exception

getCommutatorJacobian

public static BinaryAlgebra getCommutatorJacobian(BinaryAlgebra a,
                                  BinaryAlgebra b,
                                  BinaryAlgebra c)
                                           throws java.lang.Exception

Parameters:

a - BinaryAlgebra 1.

b - BinaryAlgebra 2.

c - BinaryAlgebra 3.

Returns:

[[a,b],c] + [[b,c],a] + [[c,a],b]

Throws:

java.lang.Exception

getJordanAssociator

public static BinaryAlgebra getJordanAssociator(BinaryAlgebra a,
                                BinaryAlgebra b,
                                BinaryAlgebra c)
                                         throws java.lang.Exception

Parameters:

a - BinaryAlgebra 1

b - BinaryAlgebra 2

c - BinaryAlgebra 3

Returns:

Jordan Associator defined by: {{a,b},c}- {a,{b,c}}

Throws:

java.lang.Exception

getCommutatorSaglian

public static BinaryAlgebra getCommutatorSaglian(BinaryAlgebra a,
                                 BinaryAlgebra b,
                                 BinaryAlgebra c,
                                 BinaryAlgebra d)
                                          throws java.lang.Exception

Parameters:

a - BinaryAlgebra 1.

b - BinaryAlgebra 2.

c - BinaryAlgebra 3.

d - BinaryAlgebra 3.

Returns:

[[[a,b],c],d] + [[[b,c],d],a] + [[[c,d],a],b] + [[[d,a],b],c] + [[b,d],[a,c]]

Throws:

java.lang.Exception

getFirstHetztelPeresian

public static BinaryAlgebra getFirstHetztelPeresian(BinaryAlgebra a,
                                    BinaryAlgebra b,
                                    BinaryAlgebra c,
                                    BinaryAlgebra d,
                                    BinaryAlgebra e)
                                             throws java.lang.Exception

Parameters:

a - BinaryAlgebra 1

b - BinaryAlgebra 2

c - BinaryAlgebra 3

d - BinaryAlgebra 4

e - BinaryAlgebra 5

Returns:

sum_alt{b,c,e} ([a,(c,d,e),b] + [a,b,(c,d,e)]) = sum_alt{b,c,e} ([a,{{c,d},e} - {c,{d,e}},b] + [a,b,{{c,d},e} - {c,{d,e}}]) with sum_alt{b,c,e} the alternating sum over all permutations of b,c,e and (c,d,e) the Jordan associator.

Throws:

java.lang.Exception

getSecondHetztelPeresian

public static BinaryAlgebra getSecondHetztelPeresian(BinaryAlgebra a,
                                     BinaryAlgebra b,
                                     BinaryAlgebra c,
                                     BinaryAlgebra d)
                                              throws java.lang.Exception

Parameters:

a - BinaryAlgebra 1

b - BinaryAlgebra 2

c - BinaryAlgebra 3

d - BinaryAlgebra 4

Returns:

[a(ab)-(ba)a,c,d] + [{[a,b],c},a,d] - [[a,b],{a,c},d]

Throws:

java.lang.Exception

getCommuAssociator

public static BinaryAlgebra getCommuAssociator(BinaryAlgebra h1,
                               BinaryAlgebra h2,
                               BinaryAlgebra h3,
                               BinaryAlgebra h4)
                                        throws java.lang.Exception

Parameters:

h1 - BinaryAlgebra 1

h2 - Hypernumner 2

h3 - BinaryAlgebra 3

h4 - BinaryAlgebra 4

Returns:

Commu-Associator: [A,B,C,D] = [[A,B],C,D]-[[A,B,C],D]+[[D,B],C,A]-[[D,B,C],A]

Throws:

java.lang.Exception

getKleinfeldFunction

public static BinaryAlgebra getKleinfeldFunction(BinaryAlgebra a,
                                 BinaryAlgebra b,
                                 BinaryAlgebra c,
                                 BinaryAlgebra d)
                                          throws java.lang.Exception

Parameters:

a - Algebra.

b - Algebra.

c - Algebra.

d - Algebra.

Returns:

[ab,c,d] - b[a,c,d] - [b,c,d]a

Throws:

java.lang.Exception

getSabininQuaternator

public static BinaryAlgebra getSabininQuaternator(BinaryAlgebra a,
                                  BinaryAlgebra b,
                                  BinaryAlgebra c,
                                  BinaryAlgebra d)
                                           throws java.lang.Exception

Parameters:

a - Algebra.

b - Algebra.

c - Algebra.

d - Algebra.

Returns:

Sabinin quaternator: p2,1 = [ab,c,d] - a[b,c,d] - b[a,c,d] (There is yet another Sabinin quaternator, p1,2).

Throws:

java.lang.Exception

getQuaternator

public static BinaryAlgebra getQuaternator(BinaryAlgebra h1,
                           BinaryAlgebra h2,
                           BinaryAlgebra h3,
                           BinaryAlgebra h4)
                                    throws java.lang.Exception

Parameters:

h1 - BinaryAlgebra 1

h2 - BinaryAlgebra 2

h3 - BinaryAlgebra 3

h4 - BinaryAlgebra 4

Returns:

Quaternator 1: // TODO ...

Throws:

java.lang.Exception

getQuaternator1

public static BinaryAlgebra getQuaternator1(BinaryAlgebra a,
                            BinaryAlgebra b,
                            BinaryAlgebra c,
                            BinaryAlgebra d)
                                     throws java.lang.Exception

Parameters:

a - BinaryAlgebra 1

b - Hypernumner 2

c - BinaryAlgebra 3

d - BinaryAlgebra 4

Returns:

Quaternator 1: [a,b,c,d]_1 != ((ab)c)d - 2(ab)(cd) + a(b(cd))

Throws:

java.lang.Exception

getQuaternator2

public static BinaryAlgebra getQuaternator2(BinaryAlgebra h1,
                            BinaryAlgebra h2,
                            BinaryAlgebra h3,
                            BinaryAlgebra h4)
                                     throws java.lang.Exception

Parameters:

h1 - BinaryAlgebra 1

h2 - Hypernumner 2

h3 - BinaryAlgebra 3

h4 - BinaryAlgebra 4

Returns:

Quaternator 2: [h1,h2,h3,h4]_2 != [h1, h2, h3] h4 + h1 [h2, h3, h4] - [h1, h2h3, h4]

Throws:

java.lang.Exception

getSabininator

public static BinaryAlgebra getSabininator(BinaryAlgebra a,
                           BinaryAlgebra b,
                           BinaryAlgebra c,
                           BinaryAlgebra d)
                                    throws java.lang.Exception

Throws:

java.lang.Exception

getBMQuaternator1

public static BinaryAlgebra getBMQuaternator1(BinaryAlgebra a,
                              BinaryAlgebra b,
                              BinaryAlgebra c,
                              BinaryAlgebra d)
                                       throws java.lang.Exception

First quaternator as defined by Murray R. Bremner and Sara Madariaga (denoted "Q1").

Parameters:

a - BinaryAlgebra 1

b - BinaryAlgebra 2

c - BinaryAlgebra 3

d - BinaryAlgebra 4

Returns:

Quaternator Q1(a,b,c,d) != [ab,c,d]-a[b,c,d]-[a,c,d]b.

Throws:

java.lang.Exception

getCommutatorBMQuaternator1

public static BinaryAlgebra getCommutatorBMQuaternator1(BinaryAlgebra a,
                                        BinaryAlgebra b,
                                        BinaryAlgebra c,
                                        BinaryAlgebra d)
                                                 throws java.lang.Exception

First quaternator as defined by Murray R. Bremner and Sara Madariaga (denoted "Q1") using the commutator product.

Parameters:

a - BinaryAlgebra 1

b - Hypernumner 2

c - BinaryAlgebra 3

d - BinaryAlgebra 4

Returns:

Quaternator Q1(a,b,c,d) != [[a,b],c,d]-[a,[b,c,d]]-[[a,c,d],b],

Throws:

java.lang.Exception

getBMQuaternator2

public static BinaryAlgebra getBMQuaternator2(BinaryAlgebra a,
                              BinaryAlgebra b,
                              BinaryAlgebra c,
                              BinaryAlgebra d)
                                       throws java.lang.Exception

Second quaternator as defined by Murray R. Bremner and Sara Madariaga (denoted "Q2").

Parameters:

a - BinaryAlgebra 1

b - Hypernumner 2

c - BinaryAlgebra 3

d - BinaryAlgebra 4

Returns:

Quaternator Q2(a,b,c,d) != [a,bc,d] - b[a,c,d]- [a,b,d]c

Throws:

java.lang.Exception

getBMQuaternatorIdentity1

public static BinaryAlgebra getBMQuaternatorIdentity1(BinaryAlgebra a,
                                      BinaryAlgebra b,
                                      BinaryAlgebra c,
                                      BinaryAlgebra d)
                                               throws java.lang.Exception

Throws:

java.lang.Exception

getBMQuaternatorIdentity2

public static BinaryAlgebra getBMQuaternatorIdentity2(BinaryAlgebra a,
                                      BinaryAlgebra b,
                                      BinaryAlgebra c,
                                      BinaryAlgebra d)
                                               throws java.lang.Exception

Throws:

java.lang.Exception

getBMQuaternatorIdentity3

public static BinaryAlgebra getBMQuaternatorIdentity3(BinaryAlgebra a,
                                      BinaryAlgebra b,
                                      BinaryAlgebra c,
                                      BinaryAlgebra d)
                                               throws java.lang.Exception

Throws:

java.lang.Exception

getBMQuaternatorIdentity4

public static BinaryAlgebra getBMQuaternatorIdentity4(BinaryAlgebra a,
                                      BinaryAlgebra b,
                                      BinaryAlgebra c,
                                      BinaryAlgebra d)
                                               throws java.lang.Exception

Throws:

java.lang.Exception