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

If K is a eld, let the ring R 0 consist of nite sums of homogeneous elements in R = Kx 1 ; x 2 ; x 3 ; : : : ]]. Then, R 0 contains M, the free semi-group on the countable set of variables fx 1 ; x 2 ; x 3 ; : : :g. In this paper, we generalize the notion of admissible order from nitely generated sub-monoids of M to M itself; assume that > is such an admissible order on M. We show that we can deene leading power products, with respect to >, of elements in R 0 , and thus the initial ideal gr(I) of an arbitrary ideal I R 0. If I is what we call a locally nitely generated ideal, then we show that gr(I) is also locally nitely generated; this implies that I has a nite truncated Grr obner basis up to any total degree. We give an example of a nitely generated homogeneous ideal which has a non-nitely generated initial ideal with respect to the lexicographic initial order > lex on M.