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Edit detail for SandBoxObserverAsIdempotent2 revision 3 of 18

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Editor: Bill Page
Time: 2013/04/22 16:20:19 GMT+0
Note: display version clarification

added:

*Note: Please do not be confused by the use of the word "*axiom*" the upper
righthand  corner of the generated output below. This page was actually
generated by:*

added:
\end{axiom}

Generators
\begin{axiom}

added:
\end{axiom}

Representation
\begin{axiom}

Obs(2) is a 4 dimensional Frobenius Algebra

Note: Please do not be confused by the use of the word "axiom" the upper righthand corner of the generated output below. This page was actually generated by:

axiom
)version
Value = "FriCAS 2012-10-18 compiled at Monday March 25, 2013 at 20:50:21 "

Generators

axiom
)set output abbreviate on
V := OrderedVariableList [p,q]

\label{eq1}\hbox{\axiomType{OVAR}\ } ([ p , q ])(1)
Type: TYPE
axiom
M := FreeMonoid V

\label{eq2}\hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ]))(2)
Type: TYPE
axiom
gens:List M := enumerate()$V

\label{eq3}\left[ p , \: q \right](3)
Type: LIST(FMONOID(OVAR([p,q])))

Representation

axiom
divisible := Record(lm: M,rm: M)

\label{eq4}\mbox{\rm \hbox{\axiomType{Record}\ } (lm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ])) , rm : \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ])))}(4)
Type: TYPE
axiom
leftDiv(k:Union(divisible,"failed")):M == (k::divisible).lm
Function declaration leftDiv : Union(Record(lm: FMONOID(OVAR([p,q])) ,rm: FMONOID(OVAR([p,q]))),"failed") -> FMONOID(OVAR([p,q])) has been added to workspace.
Type: VOID
axiom
rightDiv(k:Union(divisible,"failed")):M == (k::divisible).rm
Function declaration rightDiv : Union(Record(lm: FMONOID(OVAR([p,q]) ),rm: FMONOID(OVAR([p,q]))),"failed") -> FMONOID(OVAR([p,q])) has been added to workspace.
Type: VOID
axiom
K := FRAC POLY INT

\label{eq5}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ }))(5)
Type: TYPE
axiom
MK := FreeModule(K,M)

\label{eq6}FM (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{POLY}\ } (\hbox{\axiomType{INT}\ })) , \hbox{\axiomType{FMONOID}\ } (\hbox{\axiomType{OVAR}\ } ([ p , q ])))(6)
Type: TYPE
axiom
coeff(x:MK):K == leadingCoefficient(x)
Function declaration coeff : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q])) ) -> FRAC(POLY(INT)) has been added to workspace.
Type: VOID
axiom
base(x:MK):M == leadingMonomial(x)
Function declaration base : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q]))) -> FMONOID(OVAR([p,q])) has been added to workspace.
Type: VOID
axiom
m(x:M):Symbol == subscript('m,[retract(x)::Symbol])
Function declaration m : FMONOID(OVAR([p,q])) -> SYMBOL has been added to workspace.
Type: VOID
axiom
γ(x:M,y:M):Symbol == subscript('γ,[concat(string retract x, string retract y)::Symbol])
Function declaration γ : (FMONOID(OVAR([p,q])),FMONOID(OVAR([p,q]))) -> SYMBOL has been added to workspace.
Type: VOID

Basis

axiom
basis := concat(gens,concat [[i*j for j in gens | i~=j] for i in gens])

\label{eq7}\left[ p , \: q , \:{p \  q}, \:{q \  p}\right](7)
Type: LIST(FMONOID(OVAR([p,q])))

Idempotent

axiom
rule1(ij:MK):MK ==
  for k in gens repeat
    kk := divide(base(ij),k*k)
    if kk case divisible then
      ij:=monom(leftDiv(kk)*k*rightDiv(kk), coeff(ij)*m(k)*γ(k,k))
  return(ij)
Function declaration rule1 : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q])) ) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q]))) has been added to workspace.
Type: VOID

Reduction

axiom
rule2(ij:MK):MK ==
  for i in gens repeat
    for j in gens | j ~= i repeat
      for k in gens | k ~= j repeat
        ijk:=divide(base(ij),i*j*k)
        if ijk case divisible then
          ij := monom(leftDiv(ijk)*i*k*rightDiv(ijk), coeff(ij)*m(j)*γ(i,j)*γ(j,k)/γ(i,k))
return(ij)
Function declaration rule2 : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q])) ) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q]))) has been added to workspace.
Type: VOID

An endomorphism on the K-Module is defined by the fixed point of applied rules

axiom
mod(ij:MK):MK ==
  ijFix:MK := 1
  while ijFix~=ij repeat
    ijFix := ij
    ij := rule1(ij)
    ij := rule2(ij)
  return(ij)
Function declaration mod : FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q]))) has been added to workspace.
Type: VOID

Matrix

Algebra is the free algebra product modulo the fixed point

axiom
MT := [[mod(i*j) for j in basis] for i in basis]
axiom
Compiling function base with type FM(FRAC(POLY(INT)),FMONOID(OVAR([p
      ,q]))) -> FMONOID(OVAR([p,q]))
axiom
Compiling function leftDiv with type Union(Record(lm: FMONOID(OVAR([
      p,q])),rm: FMONOID(OVAR([p,q]))),"failed") -> FMONOID(OVAR([p,q])
      )
axiom
Compiling function rightDiv with type Union(Record(lm: FMONOID(OVAR(
      [p,q])),rm: FMONOID(OVAR([p,q]))),"failed") -> FMONOID(OVAR([p,q]
      ))
axiom
Compiling function coeff with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q]))) -> FRAC(POLY(INT))
axiom
Compiling function m with type FMONOID(OVAR([p,q])) -> SYMBOL
axiom
Compiling function γ with type (FMONOID(OVAR([p,q])),FMONOID(OVAR([p
      ,q]))) -> SYMBOL
axiom
Compiling function rule1 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q])))
axiom
Compiling function rule2 with type FM(FRAC(POLY(INT)),FMONOID(OVAR([
      p,q]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q])))
axiom
Compiling function mod with type FM(FRAC(POLY(INT)),FMONOID(OVAR([p,
      q]))) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q])))

\label{eq8}\begin{array}{@{}l}
\displaystyle
\left[{\left[{{m_{p}}\ {��_{pp}}\  p}, \:{p \  q}, \:{{m_{p}}\ {��_{pp}}\  p \  q}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}\  p}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{q \  p}, \:{{m_{q}}\ {��_{qq}}\  q}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}\  q}, \:{{m_{q}}\ {��_{qq}}\  q \  p}\right]}, \: \right.
\
\
\displaystyle
\left.{
\begin{array}{@{}l}
\displaystyle
\left[{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}\  p}, \:{{m_{q}}\ {��_{qq}}\  p \  q}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}\  p \  q}, \: \right.
\
\
\displaystyle
\left.{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}\  p}\right] 
(8)
Type: LIST(LIST(FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q])))))

Structure Constants

axiom
R:=FRAC DMP(concat [[m(i) for i in gens],concat [[γ(j,i) for i in gens] for j in gens]], INT)

\label{eq9}\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ <em> 01 mp , </em> 01 mq , * 01 �� pp , * 01 �� pq , * 01 �� qp , * 01 �� qq ] , \hbox{\axiomType{INT}\ }))(9)
Type: TYPE
axiom
mat3(y:M):List List R == map(z+->map(x+->coefficient(x,y)::FRAC POLY INT,z),MT)
Function declaration mat3 : FMONOID(OVAR([p,q])) -> LIST(LIST(FRAC( DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))) has been added to workspace.
Type: VOID
axiom
ss:=map(mat3, basis)
axiom
Compiling function mat3 with type FMONOID(OVAR([p,q])) -> LIST(LIST(
      FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT))))

\label{eq10}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\left[{\left[{{m_{p}}\ {��_{pp}}}, \: 0, \: 0, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}\right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \: 0, \: 0, \:{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}\right]}, \:{\left[ 0, \: 0, \: 0, \: 0 \right]}\right] (10)
Type: LIST(LIST(LIST(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))))

Algebra

axiom
cats(m:M):Symbol==concat(map(x+->string(x.gen::Symbol),factors m))::Symbol
Function declaration cats : FMONOID(OVAR([p,q])) -> SYMBOL has been added to workspace.
Type: VOID
axiom
A:=AlgebraGivenByStructuralConstants(R,#(basis)::PI,map(cats,basis),ss::Vector(Matrix R))
axiom
Compiling function cats with type FMONOID(OVAR([p,q])) -> SYMBOL

\label{eq11}\hbox{\axiomType{ALGSC}\ } (\hbox{\axiomType{FRAC}\ } (\hbox{\axiomType{DMP}\ } ([ <em> 01 mp , </em> 01 mq , * 01 �� pp , * 01 �� pq , * 01 �� qp , * 01 �� qq ] , \hbox{\axiomType{INT}\ })) , 4, [ p , q , pq , qp ] , [ \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } , \hbox{\axiomType{MATRIX}\ } ])(11)
Type: TYPE
axiom
alternative?()$A
algebra satisfies 2*associator(a,b,b) = 0 = 2*associator(a,a,b) = 0

\label{eq12} \mbox{\rm true} (12)
Type: BOOLEAN
axiom
antiAssociative?()$A
algebra is not anti-associative

\label{eq13} \mbox{\rm false} (13)
Type: BOOLEAN
axiom
antiCommutative?()$A
algebra is not anti-commutative

\label{eq14} \mbox{\rm false} (14)
Type: BOOLEAN
axiom
associative?()$A
algebra is associative

\label{eq15} \mbox{\rm true} (15)
Type: BOOLEAN
axiom
commutative?()$A
algebra is not commutative

\label{eq16} \mbox{\rm false} (16)
Type: BOOLEAN
axiom
flexible?()$A
algebra is flexible

\label{eq17} \mbox{\rm true} (17)
Type: BOOLEAN
axiom
jacobiIdentity?()$A
Jacobi identity does not hold

\label{eq18} \mbox{\rm false} (18)
Type: BOOLEAN
axiom
jordanAdmissible?()$A
algebra is not Jordan admissible

\label{eq19} \mbox{\rm false} (19)
Type: BOOLEAN
axiom
jordanAlgebra?()$A
algebra is not commutative this is not a Jordan algebra

\label{eq20} \mbox{\rm false} (20)
Type: BOOLEAN
axiom
leftAlternative?()$A
algebra is left alternative

\label{eq21} \mbox{\rm true} (21)
Type: BOOLEAN
axiom
lieAdmissible?()$A
algebra is Lie admissible

\label{eq22} \mbox{\rm true} (22)
Type: BOOLEAN
axiom
lieAlgebra?()$A
algebra is not anti-commutative this is not a Lie algebra

\label{eq23} \mbox{\rm false} (23)
Type: BOOLEAN
axiom
powerAssociative?()$A
Internal Error The function powerAssociative? with signature hashcode is missing from domain AlgebraGivenByStructuralConstants (Fraction (DistributedMultivariatePolynomial ((*01m p) (*01m q) (*01γ pp) (*01γ pq) (*01γ qp) (*01γ qq)) (Integer))) 4(p q pq qp)UNPRINTABLE

Check Multiplication

axiom
AB := entries basis()$A

\label{eq24}\left[ p , \: q , \: pq , \: qp \right](24)
Type: LIST(ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX]))
axiom
A2MK(z:A):MK==reduce(+,map((x:R,y:M):MK+->(x::K)*y,coordinates(z),basis))
Function declaration A2MK : ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γ pq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX, MATRIX]) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q]))) has been added to workspace.
Type: VOID
axiom
test(MT=map(x+->map(A2MK,x),[[i*j for j in AB] for i in AB]))
axiom
Compiling function A2MK with type ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp
      ,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,
      MATRIX]) -> FM(FRAC(POLY(INT)),FMONOID(OVAR([p,q])))

\label{eq25} \mbox{\rm true} (25)
Type: BOOLEAN

Trace

axiom
[rightTrace(i)$A for i in AB]

\label{eq26}\left[{2 \ {m_{p}}\ {��_{pp}}}, \:{2 \ {m_{q}}\ {��_{qq}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}\right](26)
Type: LIST(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))
axiom
[leftTrace(i)$A for i in AB]

\label{eq27}\left[{2 \ {m_{p}}\ {��_{pp}}}, \:{2 \ {m_{q}}\ {��_{qq}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{2 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}\right](27)
Type: LIST(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))
axiom
trace(i)==rightTrace(i) / #gens
Type: VOID
axiom
[trace(i) for i in AB]
axiom
Compiling function trace with type ALGSC(FRAC(DMP([*01mp,*01mq,*01γ
      pp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,
      MATRIX,MATRIX]) -> FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,
      *01γqq],INT))

\label{eq28}\left[{{m_{p}}\ {��_{pp}}}, \:{{m_{q}}\ {��_{qq}}}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}, \:{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}\right](28)
Type: LIST(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))

Center

axiom
C:=basisOfCenter()$AlgebraPackage(R,A); # C

\label{eq29}1(29)
Type: PI
axiom
c:=C(1)

\label{eq30}qp + pq -{{m_{p}}\ {��_{pp}}\  q}-{{m_{q}}\ {��_{qq}}\  p}(30)
Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX])
axiom
[c*i-i*c for i in AB]

\label{eq31}\left[ 0, \: 0, \: 0, \: 0 \right](31)
Type: LIST(ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX]))
axiom
test(c*c=c)

\label{eq32} \mbox{\rm false} (32)
Type: BOOLEAN

Unit

axiom
rightTrace(c)

\label{eq33}-{4 \ {m_{p}}\ {m_{q}}\ {��_{pp}}\ {��_{qq}}}+{4 \ {m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}(33)
Type: FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT))
axiom
n := #basis / rightTrace(c) * c

\label{eq34}\begin{array}{@{}l}
\displaystyle
-{{1 \over{{{m_{p}}\ {m_{q}}\ {��_{pp}}\ {��_{qq}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}}}\  qp}- 
\
\
\displaystyle
{{1 \over{{{m_{p}}\ {m_{q}}\ {��_{pp}}\ {��_{qq}}}-{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}}}\  pq}+{{{��_{pp}}\over{{{m_{q}}\ {��_{pp}}\ {��_{qq}}}-{{m_{q}}\ {��_{pq}}\ {��_{qp}}}}}\  q}+ 
\
\
\displaystyle
{{{��_{qq}}\over{{{m_{p}}\ {��_{pp}}\ {��_{qq}}}-{{m_{p}}\ {��_{pq}}\ {��_{qp}}}}}\  p}
(34)
Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX])
axiom
trace(n)

\label{eq35}2(35)
Type: FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT))
axiom
test(n*n=n)

\label{eq36} \mbox{\rm true} (36)
Type: BOOLEAN

axiom
test(n=unit()$A)

\label{eq37} \mbox{\rm true} (37)
Type: BOOLEAN
axiom
f:=gcd map(x+->denom x,coordinates(n))

\label{eq38}{{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}(38)
Type: DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)
axiom
ff:= matrix [[γ(i,j)::R for j in gens] for i in gens]

\label{eq39}\left[ 
\begin{array}{cc}
{��_{pp}}&{��_{pq}}
\
{��_{qp}}&{��_{qq}}
(39)
Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))
axiom
test(f = determinant(ff))

\label{eq40} \mbox{\rm true} (40)
Type: BOOLEAN
axiom
(f*n)::OutputForm / f::OutputForm

\label{eq41}{-{{1 \over{{m_{p}}\ {m_{q}}}}\  qp}-{{1 \over{{m_{p}}\ {m_{q}}}}\  pq}+{{{��_{pp}}\over{m_{q}}}\  q}+{{{��_{qq}}\over{m_{p}}}\  p}}\over{{{��_{pp}}\ {��_{qq}}}-{{��_{pq}}\ {��_{qp}}}}(41)
Type: OUTFORM

Antisymmetric γ

axiom
eqAnti:List Equation EXPR INT := concat [[(i>j => γ(i,j)=γ(i,j);i=j =>γ(i,j)=0;γ(i,j)=-γ(j,i)) for j in gens] for i in gens]

\label{eq42}\left[{{��_{pp}}= 0}, \:{{��_{pq}}={��_{pq}}}, \:{{��_{qp}}= -{��_{pq}}}, \:{{��_{qq}}= 0}\right](42)
Type: LIST(EQ(EXPR(INT)))
axiom
anti(x:R):R == subst(x::EXPR INT, eqAnti)::FRAC POLY INT
Function declaration anti : FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γ qp,*01γqq],INT)) -> FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp, *01γqq],INT)) has been added to workspace.
Type: VOID
axiom
(anti(f)*map(anti ,coordinates(n))::A)::OutputForm / anti(f)::OutputForm
axiom
Compiling function anti with type FRAC(DMP([*01mp,*01mq,*01γpp,*01γ
      pq,*01γqp,*01γqq],INT)) -> FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,
      *01γqp,*01γqq],INT))

\label{eq43}{-{{1 \over{{m_{p}}\ {m_{q}}}}\  qp}-{{1 \over{{m_{p}}\ {m_{q}}}}\  pq}}\over{{��_{pq}}^{2}}(43)
Type: OUTFORM

Momentum

axiom
P:=reduce(+,concat [[(1/γ(basis(i),basis(j)))::R*AB(i)*AB(j) for j in 1..size()$V] for i in 1..size()$V])

\label{eq44}{{1 \over{��_{qp}}}\  qp}+{{1 \over{��_{pq}}}\  pq}+{{m_{q}}\  q}+{{m_{p}}\  p}(44)
Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX])
axiom
trace(P)

\label{eq45}{{{m_{p}}^{2}}\ {��_{pp}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}}+{{m_{p}}\ {m_{q}}\ {��_{qp}}}+{{{m_{q}}^{2}}\ {��_{qq}}}(45)
Type: FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT))
axiom
c:=1/trace(P)*P

\label{eq46}\begin{array}{@{}l}
\displaystyle
{{1 \over{{{{m_{p}}^{2}}\ {��_{pp}}\ {��_{qp}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}+{{m_{p}}\ {m_{q}}\ {{��_{qp}}^{2}}}+{{{m_{q}}^{2}}\ {��_{qp}}\ {��_{qq}}}}}\  qp}+ 
\
\
\displaystyle
{{1 \over{{{{m_{p}}^{2}}\ {��_{pp}}\ {��_{pq}}}+{{m_{p}}\ {m_{q}}\ {{��_{pq}}^{2}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}+{{{m_{q}}^{2}}\ {��_{pq}}\ {��_{qq}}}}}\  pq}+ 
\
\
\displaystyle
{{{m_{q}}\over{{{{m_{p}}^{2}}\ {��_{pp}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}}+{{m_{p}}\ {m_{q}}\ {��_{qp}}}+{{{m_{q}}^{2}}\ {��_{qq}}}}}\  q}+ 
\
\
\displaystyle
{{{m_{p}}\over{{{{m_{p}}^{2}}\ {��_{pp}}}+{{m_{p}}\ {m_{q}}\ {��_{pq}}}+{{m_{p}}\ {m_{q}}\ {��_{qp}}}+{{{m_{q}}^{2}}\ {��_{qq}}}}}\  p}
(46)
Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX])
axiom
c*c-c

\label{eq47}0(47)
Type: ALGSC(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)),4,[p,q,pq,qp],[MATRIX,MATRIX,MATRIX,MATRIX])
axiom
trace(c)

\label{eq48}1(48)
Type: FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT))

Scalar Product

axiom
S := matrix [[trace(x*y) for y in AB] for x in AB]

\label{eq49}\left[ 
\begin{array}{cccc}
{{{m_{p}}^{2}}\ {{��_{pp}}^{2}}}&{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}
\
{{m_{p}}\ {m_{q}}\ {��_{pq}}\ {��_{qp}}}&{{{m_{q}}^{2}}\ {{��_{qq}}^{2}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}
\
{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}
\
{{{m_{p}}^{2}}\ {m_{q}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}}&{{m_{p}}\ {{m_{q}}^{2}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {��_{pp}}\ {��_{pq}}\ {��_{qp}}\ {��_{qq}}}&{{{m_{p}}^{2}}\ {{m_{q}}^{2}}\ {{��_{pq}}^{2}}\ {{��_{qp}}^{2}}}
(49)
Type: MATRIX(FRAC(DMP([*01mp,*01mq,*01γpp,*01γpq,*01γqp,*01γqq],INT)))