|  B.2.2 General definitions for orderings 
A monomial ordering (term ordering) on 
![$K[x_1,\ldots,x_n]$](sing_60.png) is
a total ordering  on the
set of monomials (power products)  which is compatible with the
natural semigroup structure, i.e.,  implies  for any  .
We do not require  to be  a wellordering. See the literature cited in  References. 
It is known that any monomial ordering can be represented by a matrix
 in  ,but, of course, only integer coefficients are of relevance in
practice. 
Global orderings are wellorderings (i.e.,   for each variable  ), local orderings satisfy  for each variable.   If some variables are ordered globally and others locally we
call it a mixed ordering.   Local or mixed orderings are not wellorderings. 
Let  be the ground field,  the
variables and  a monomial ordering, then Loc ![$K[x]$](sing_512.png) denotes the
localization of ![$K[x]$](sing_512.png) with respect to the multiplicatively closed set 
 Here,
  denotes the leading monomial of  , i.e., the biggest monomial of  with
respect to  .   The result of any computation which uses standard basis
computations has to be interpreted in Loc ![$K[x]$](sing_512.png) . 
Note that the definition of a ring includes the definition of its
monomial ordering (see
 Rings and orderings). SINGULAR offers the monomial orderings
described in the following sections.
 
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