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B.2.3 Global orderings

For all these orderings, we have Loc $K[x]$ = $K[x]$

lp:
lexicographical ordering:
$x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
\alpha_1 = \beta_1, \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i <
\beta_i$.
rp:
reverse lexicographical ordering:
$x^\alpha < x^\beta \Leftrightarrow \exists\; 1 \le i \le n :
\alpha_n = \beta_n,
\ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i < \beta_i.$
dp:
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 $\exists\ 1 \le i \le n: \alpha_n = \beta_n,
\ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
Dp:
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 $\exists\ 1 \le i \le n:\alpha_1 = \beta_1,
\ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
wp:
weighted reverse lexicographical ordering:
let $w_1, \ldots, w_n$ be positive integers. Then ${\tt wp}(w_1, \ldots,
w_n)$ is defined as dp but with $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$
Wp:
weighted lexicographical ordering:
let $w_1, \ldots, w_n$ be positive integers. Then ${\tt Wp}(w_1, \ldots,
w_n)$ is defined as Dp but with $\deg(x^\alpha) = w_1 \alpha_1 + \cdots + w_n\alpha_n.$

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