  The theme of symbolic computation in algebraic categories has become of
utmost importance in the last decade since it enables the automatic modeling of
modern algebra theories. On this theoretical background, the present paper
reveals the utility of the parameterized categorical approach by deriving a
multivariate polynomial category (over various coefficient domains), which is
used by our Mathematica implementation of Buchberger's algorithms for
determining the Groebner basis. These implementations are designed according to
domain and category parameterization principles underlining their advantages:
operation protection, inheritance, generality, easy extendibility. In
particular, such an extension of Mathematica, a widely used symbolic
computation system, with a new type system has a certain practical importance.
The approach we propose for Mathematica is inspired from D. Gruntz and M.
Monagan's work in Gauss, for Maple.
