Regular chain

Definition

Let $k$ be a field and let $S = k[x_1, \ldots, x_n]$ be the polynomial ring with the variables ordered $x_1 < x_2 < \cdots < x_n$. Let $p \in S \setminus k$. Its main variable $\operatorname{mvar}(p)$ is the largest variable that occurs in $p$. Its initial $\operatorname{init}(p)$ is the leading coefficient of $p$ as a polynomial in $\operatorname{mvar}(p)$. A triangular set is a finite set $T$ of elements of $S \setminus k$ with pairwise distinct main variables. For a triangular set $T$ with product of initials $h_T$, the saturated ideal is the ideal quotient $\operatorname{sat}(T) = (T) : h_T^\infty = \set{ f \in S : h_T^m f \in (T) \text{ for some } m \ge 0 }$. A triangular set $T$ is a regular chain if either $T = \emptyset$, or $T = T' \cup \set{t}$ where:

1. $\operatorname{mvar}(t)$ is larger than the main variable of every element of $T'$.
2. $T'$ is a regular chain.
3. $\operatorname{init}(t)$ is not a zero divisor in the quotient ring $S/\operatorname{sat}(T')$.