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B.2.4 Local orderings

For ls, ds, Ds and, if the weights are positive integers, also for ws and Ws, we have Loc $K[x]$ = $K[x]_{(x)}$, the localization of $K[x]$at the maximal ideal $(x_1, ..., x_n)$.

ls:
negative lexicographical ordering:
$x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
\alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i >
\beta_i$.
ds:
negative degree reverse lexicographical ordering:
let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) > \deg(x^\beta)$ or
$ \deg(x^\alpha) =
\deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n = \beta_n,
\ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
Ds:
negative degree lexicographical ordering:
let $\deg(x^\alpha) = \alpha_1 + \cdots + \alpha_n,$ then $x^\alpha < x^\beta \Leftrightarrow \deg(x^\alpha) > \deg(x^\beta)$ or
$ \deg(x^\alpha) =
\deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 = \beta_1,
\ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
ws:
(general) weighted reverse lexicographical ordering:
${\tt ws}(w_1, \ldots, w_n),\; w_1$ a nonzero integer, $w_2,\ldots,w_n$ any integer (including 0), is defined as ds but with $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
Ws:
(general) weighted lexicographical ordering:
${\tt Ws}(w_1, \ldots, w_n),\; w_1$ a nonzero integer, $w_2,\ldots,w_n$ any integer (including 0), is defined as Ds but with $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$


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