### Algebra of functions f:Z_3 -> R

I want to create an implementation of an algebra of functions with domain {0,1,2} and range in R. The sum and the product is the usual pointwise sum and product. I have the idea of represent it as a 3 dimensional vectors with the usual sum of vectors, bot I need implement a new product.
The implementation of this algebra will be used for create a polynomial system of equations that I need to solve, so the implementation should be compatible with the procedure indicated in the entry titled *Find algebraic solutions to system of polynomial equations*
If someone have any idea please help me.
Thanks in advance.