Gröbner Bases and Normal Forms in a Subring of the Power Series Ring on Countably Many Variables

If K is a field, let the ring R′ consist of finite sums of homogeneous elements in R = K[[x1, x2, x3, . . .]]. Then, R′ contains M, the free semi-group on the countable set of variables {x1, x2, x3, . . .}. In this paper, we generalize the notion of admissible order from finitely generated sub-monoids of M to M itself; assume that > is such an admissible order on M. We show that we can define leading power products, with respect to >,