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


    \phantom{ $x^\alpha < x^\beta \Leftrightarrow$}$ \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 


    \phantom{ $ x^\alpha < x^\beta \Leftrightarrow$}$ \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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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; User manual for <a href="http://www.singular.uni-kl.de/"><i>Singular</i></a> version 2-0-4, October 2002,
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