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

IfKis a field, let the ringR′consist of finite sums of homogeneous elements inR = K[[x1,x2,x3,…]]. Then,R′contains , 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 to itself; assume that > is such an admissible order on . We show that we can define leading power products, with respect to >, of elements inR′, and thus the initial ideal gr(I) of an arbitrary idealI R′. IfIis what we call a locally finitely generated ideal, then we show that gr(I) is also locally finitely generated; this implies thatIhas a finite truncated Gröbner basis up to any total degree. We give an example of a finitely generated homogeneous ideal which has a non-finitely generated initial ideal with respect to the lexicographic initial order >lexon .