I finally found a recursive possibility :

f [ x_ ,y_ , z__ ] = f [ f [x,y] , z ]

hope recursive aspect will be replaced at execution time by iterative computation...

Nice feature. I really appreciate the opulence of symbols. It make articles more readable and attractive.

Thanks for the tip.

This might also be interesting: Operators without Built In Meanings

In[1]:= CirclePlus[a, b] = b

Out[1]= b

In[2]:= a \[CirclePlus] b

Out[2]= b

(Note that [CirclePlus] can be entered as esc c+ esc and looks like a CirclePlus symbol in an actual notebook.)

Excellent !!! Inner[f,list1,list2,g] is exactly what I was searching for.

The last part now is : defining f and g with elegant writing. Let's consider the informations I have are an enumeration of all tuples X+Y=Z, could I define f and g by using some kind of pattern matching or some similar mechanism ? what is the most "readable" technique for such kind of definitions ?

Regards

Frédéric DIDIER

Thanks helping, but it doesnt help. I alrerady know Mathematica provide matrices product. I dont need to change matrix or vector product, but establishing "computing rules" between some elements of a set, and use it.

example :

E = { a,b,c }

plus : (a,a) -> a, (a,b) -> b, (b,a) -> b, ...

Multiply : (a,a) -> b, (a,b) -> a

And now, if I compute ( {{a, a}, {b, a} } ) ( {{a, b}, {a, a}} )

I expect the result is : {{b, a}, {a, b}}

and not only : {{a^2, a b}, {a b, a^2}}

I tried to search on google, but I dont have the good "key-words" to collect some interesting results.

I know it's something about finite groups, algebra or something like that, but i didnt found at now...

Please help.