B.2.3 Global orderings
----------------------

For all these orderings: 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


    \phantom{$x^\alpha < x^\beta \Leftrightarrow $}$ \deg(x^\alpha) =
    \deg(x^\beta)$ and $\exists\ 1 \le i \le n: \alpha_n = \beta_n,
    \ldots, \alpha_{i+1} = \beta_{i+1}, \alpha_i > \beta_i.$
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


    \phantom{ $x^\alpha < x^\beta \Leftrightarrow $} $\deg(x^\alpha) =
    \deg(x^\beta)$ and $\exists\ 1 \le i \le n:\alpha_1 = \beta_1,
    \ldots, \alpha_{i-1} = \beta_{i-1}, \alpha_i < \beta_i.$
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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&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,
generated by <a href="http://www.gnu.org/software/texinfo/"><i>texi2html</i></a>.
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