By imposing certain weights $\mathbf{w}$ on the variables, say, of a polynomial ring $k[x_1,\ldots, x_n]$, I read that we may obtain the initial ideal $in_{\mathbf{w}}(I)$ of an ideal $I$ with respect to this weight.
Is it true that for any fixed weight $\mathbf{w}$ imposed on the variables, we have $in_{\mathbf{w}}(I)=in_{>}(I)$ for some monomial ordering $>$, where $>$ is thought to be lex or graded reverse lex, with the variables shuffled around?
I thought I read this somewhere but I cannot find the reference. Thanks for your time.
Edit: The initial ideal $in(I)$ of an ideal $I$ is thought to be an ideal generated by the leading terms of $f$, where $f\in I$.