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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.