Let
![$ I\subseteq K[x]$](img4.png)
be an ideal,
![$ K$](img5.png)
a field,
![$ x=(x_1,\ldots, x_n)$](img6.png)
and
![$ >$](img7.png)
a monomial ordering (not necessarily a well-ordering). Let
![$ L(I)$](img8.png)
be the leading ideal of
![$ I$](img9.png)
with respect to
![$ >$](img7.png)
. A standard basis (in case of a well-ordering a Gröbner basis)
![$ G=\{f_1,\ldots ,f_m\} $](img10.png)
is defined by the property that
![$ L(I)$](img8.png)
is generated by the leading monomials
![$ LM(f_1), \ldots, LM(f_m)$](img11.png)
of
![$ G$](img12.png)
. Usually one considers a standard basis associated to the uniquely determined minimal system of monomials generating
![$ L(I),$](img13.png)
a minimal standard basis. In case of border bases, Janet bases or Pommaret bases (usually only defined for well-orderings but we generalize the concept to any ordering) the underlying standard bases have as leading monomials special (non minimal) generators of
![$ L(I).$](img14.png)
We describe an algorithm which computes these bases using a minimal standard bases and the corresponding special generators of
![$ L(I)$](img8.png)
. We have implemented these algorithms in S
INGULAR (cf. [DGPS16]) including modular and parallel implementations and give timings to compare them (cf.[KP17]). We also discuss the verification of the modular algorithm to compute border bases and Janet bases.