@Thomas Richard 

https://drive.google.com/file/d/0B2D69u2pweEvQXJNdmpkOF9aWnc/edit?usp=sharing

no copy special in maple 12

@Thomas Richard 

is it maple 12 that do not have copy special?

@mehdi jafari 

why no solution after solved?

@Markiyan Hirnyk 

i success generated a correct cayley table 

how to convert to function ?

1,2,3,4,5,
2,2,3,4,5,
3,3,3,4,5,
4,4,4,4,5,
5,5,5,5,5,

@Markiyan Hirnyk 

this time i use (x,x) = 1

1,2,1,
2,1,2,
1,2,1,

1,2,3,4,1,
2,1,1,1,2,
3,1,1,2,3,
4,1,2,1,4,
1,2,3,4,1,

how about these?

if correct, how to convert to function?

@Markiyan Hirnyk 

(1,1) = 1

(2,2) = 2

(3,3) = 3

identity is valid , why not valid

@Markiyan Hirnyk 

i do not have version 17 and have version 15

and consider to buy maple version 17 if maple can transform cayley table into function

i just know cayley table is composition table, as i see the 3*3 table satisfy 3 axioms

i do not know why it is not cayley table

x,x = x is correct, symmetric is correct, transitive is also correct

@Markiyan Hirnyk 

 After finished dinner and test a few cases to test whether satisfy transitive

they are generated from my c# program which i use transitive, reflexive, and symmetric

it can be choose any integer n, n*n matrix, i choose n=3, and n = 5

1,2,1,

2,2,2,

1,2,3,

1,2,3,4,1,

2,2,1,1,2,

3,1,3,2,3,

4,1,2,4,4,

1,2,3,4,5,

@Markiyan Hirnyk 

i can get a function from another method using axioms

now i can get cayley table from axioms

need this final step to transform cayley table into function which satisfy all axioms which created cayley table

@Markiyan Hirnyk 

ignore my remark, which is my guess

return to the main question

this function input range is limited to less or equal to 6

is it possible convert to function in terms of x and y or x, y and z?

does you mean that finite group can not be a function in terms of x and y, x,y,z....etc

@Markiyan Hirnyk 

this function input range is limited to less or equal to 6

is it possible convert to any function in terms of x and y

6 x 6 matrix

does 6 mean that it can create 6 variables function, 5 variables function, .... 2 variables function, 1 variable function?

@ecterrab 

the original is from

page 707 of Partial Differential control theory Volume 2 by J.F. Pommaret

when using Robertz Daniel's Ore and Involutive packages

libname := libname, "D:\\homoalg";
with(Involutive):
with(OreModules):
Alg := DefineOreAlgebra(diff=[D,t], polynom=[t], comm=[a1,a2,a3,u]):
DD := evalm([[D, -1, 0, 0],[0, D, -1, 0],[1, 4, D + 3, -2*D^2 - 5*D - 7]]);
DDtilde := Involution(DD,Alg);
ApplyMatrix(DDtilde, [lambda1(t),lambda2(t),lambda3(t)],Alg)=evalm([[miu1(t)],[miu2(t)],[miu3(t)],[miu(t)]]);
E := Elimination(DDtilde, [lambda1,lambda2,lambda3],[miu1,miu2,miu3,miu],Alg);
XY := ApplyMatrix(E[1], [lambda1(t),lambda2(t),lambda3(t)], Alg)=ApplyMatrix(E[2],[miu1(t),miu2(t),miu3(t),miu(t)], Alg);

LiftAdjR := LiftOperators(DDtilde, Alg);

@ecterrab 

in book, it said checking injectivity of adjoint operator

then i searched Lift is for doing this

use Lift with OreModules seems can give hint 

@Adri van der Meer 

actually my purpose is to remove transitive function from a composite function

assume a function is created by composite identity function exp(x-y)

with transitive function

if i want to remove this transitive function from this composite function and the rest will be

identity function  exp(x-y)

if given any function which satisfy transitive property, then the goal is to remove a transitive function from this function

though i do not know how many composite in this function

may be A o B o C, if C is transitive function, then the rest is A o B

@ecterrab 

sorry for late reply, i forget this post

i may use wrong word eliminate

no matter use any method, how to result in 2*lambda1(t) - lambda2(t) + 2*lambda3(t) = 0;

from

a:= -(diff(lambda1(t), t))+lambda3(t);
b:= -lambda1(t)-(diff(lambda2(t), t))+4*lambda3(t);
c:= -lambda2(t)+3*lambda3(t)-(diff(lambda3(t), t));

i use Lift with OreModules

success to get -2/3, -1/3, -2/3*D + 1

it seems i near success