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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input gonshor.input}
+\author{The Axiom Team}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{License}
+<<license>>=
+--Copyright The Numerical Algorithms Group Limited 1991.
+@
+<<*>>=
+<<license>>
+
+-------------  Some examples of algebras in genetics  -------------
+
+-- Literature:
+-- [WB] A. Woerz-Busekros: Algebras in Genetics, LNB 36,
+-- Springer-Verlag, Berlin etc. 1980.
+
+
+
+---------------  Commutative, non-associative algebras  --
+
+
+
+-- A Gonshor genetic algebra ([WB], p. 41-42) of dimension 4:
+-- =========================================================
+
+)clear all
+
+-- The coefficient ring:
+R := FRAC POLY INT
+
+-- The following multiplication constants may be chosen arbitrarily
+-- (notice that we write ckij for c_(i,j)^k):
+
+(c100, c101, _
+c200, c201, c202, c211, _
+c300, c301, c302, c303, c311, c312, c322) : R
+
+----------------------------------------------------------------
+c100 :=  1 ;     c101 := -1 ;
+----------------------------------------------------------------
+c200 :=  0 ;     c201 :=  1 ;     c202 := -1 ;
+                 c211 :=  2 ;
+----------------------------------------------------------------
+c300 :=  1 ;     c301 :=  0 ;     c302 := -1 ;     c303 :=  1 ;
+                 c311 :=  1 ;     c312 :=  0 ;
+                                  c322 :=  2 ;
+----------------------------------------------------------------
+
+-- The matrices of the multiplication constants:
+
+gonshor : List SquareMatrix(4,R) :=
+  [matrix [ [1, 0, 0, 0], [0, 0, 0, 0],_
+            [0, 0, 0, 0], [0, 0, 0, 0] ],_
+   matrix [ [c100, c101, 0, 0], [c101, 0, 0, 0],_
+            [0, 0, 0, 0], [0, 0, 0, 0] ],_
+   matrix [ [c200, c201, c202, 0], [c201, c211, 0, 0],_
+            [c202, 0, 0, 0], [0, 0, 0, 0] ],_
+   matrix [ [c300, c301, c302, c303], [c301, c311, c312, 0],_
+            [c302, c312, c322, 0], [c303, 0, 0, 0] ] ] ;
+
+
+basisSymbols : List Symbol := [subscript(e,[i]) for i in 0..3]
+
+GonshorGenetic := ALGSC(R, 4, basisSymbols, gonshor)
+
+commutative?()$GonshorGenetic
+associative?()$GonshorGenetic
+
+-- The canonical basis:
+e0 : GonshorGenetic := [1, 0, 0, 0] :: Vector R ;
+e1 : GonshorGenetic := [0, 1, 0, 0] :: Vector R ;
+e2 : GonshorGenetic := [0, 0, 1, 0] :: Vector R ;
+e3 : GonshorGenetic := [0, 0, 0, 1] :: Vector R ;
+
+
+-- A generic element of the algebra:
+x  : GonshorGenetic := x0*e0 + x1*e1 + x2*e2 + x3*e3
+
+-- The matrix of the left multiplication with x :
+Lx := leftRegularRepresentation x
+
+-- leftRegularRepresentationt 8 : GonshorGenetic -> R be the weight homomorphism
+-- defined by 8(e0) := 1 and 8(ei) := 0 for i = 1,2,3 .
+-- The coefficients of the characteristic polynomial
+-- of Lx depend only on 8(x) = x0 :
+p := characteristicPolynomial(Lx,Y)
+
+-- The left minimal polynomial of x divides Y * p(Y) :
+leftMinimalPolynomial x
+
+
+)clear prop A a b c r s
+A := GonshorGenetic
+a := x
+b := (1/4)*e1 + (1/5)*e2 + (3/20)*e3 + (2/5)*e0
+c := (1/3)*e1 + (1/7)*e2 + (8/21)*e3 + (1/7)*e0
+r  : R := r
+s  : R := s
+
+b*c
+(b*c)*b
+b*(c*b)
+
+-- A: Algebra
+-- a,b,c : A
+-- r,s : R
+
+)clear prop AP
+AP := ALGPKG(R,A)
+
+r*a
+a*r
+
+a*b
+b*c
+
+12 * c
+(-3) * a
+
+d  :=  a ** 12
+-d
+
+a + b
+d-c
+
+(a*(a*a) = leftPower(a,3)) :: Boolean
+(a ** 11 =  (a**8 * a**2) * a) :: Boolean
+(a ** 11 =  a**8 * (a**2 * a)) :: Boolean
+
+
+zero := 0$A
+zero : A := 0
+
+alternative?()$A
+antiCommutative?()$A
+associative?()$A
+commutative?()$A
+
+commutator(a,b)
+antiCommutator(a,b)
+associator(a,b,c)
+basis()$A
+
+n := rank()$A
+v : Vector R := [i for i in 1..n]
+g : A := represents  v
+
+coordinates a
+coordinates [a,b]
+
+a.3
+
+flexible?()$A
+leftAlternative?()$A
+
+rightAlternative?()$A
+sB := someBasis()$A
+zero? a
+associatorDependence()$A
+--conditionsForIdempotents()$A
+jacobiIdentity?()$A
+jordanAlgebra?()$A
+jordanAdmissible?()$A
+lieAdmissible?()$A
+--conditionsForIdempotents sB
+b2 := [reduce(+,[sB.i for i in 1..k]) for k in 1..n]
+coordinates  (a ,b2 :: Vector A)
+coordinates  ([a,b] ,bb := (b2 :: Vector A))
+leftMinimalPolynomial a
+leftPower (a,10)
+rightPower(a,10)
+leftRegularRepresentation a
+leftRegularRepresentation (a,bb)
+leftUnit()$A
+represents (v,bb)
+rightMinimalPolynomial a
+rightRegularRepresentation a
+rightRegularRepresentation (a,bb)
+rightUnit()$A
+structuralConstants()$A
+structuralConstants(bb)
+unit()$A
+
+
+-- functions from ALGPKG
+
+biRank  a
+leftRank a
+doubleRank a
+rightRank a
+weakBiRank a
+
+
+basisOfCenter()$AP
+basisOfLeftNucleus()$AP
+basisOfNucleus()$AP
+basisOfRightNucleus()$AP
+basisOfCentroid()$AP
+basisOfCommutingElements()$AP
+basisOfLeftNucloid()$AP
+basisOfMiddleNucleus()$AP
+basisOfRightNucloid()$AP
+
+
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}