Given a toric ideal, say $J$, in a polynomial ring $k[x\_1,...,x\_n]$ we can find a finite 
generating set for $J$. Is it possible, perhaps under additional assumptions on the structure of $J$, to give a finite minimal generating set for $J$ such that every subset of generators also generates a toric ideal. 


If not, are there any known counterexamples in the general case?


If yes, could you provide a reference? Does it generalize to lattice ideals?


Motivation: For particularly chosen, generating sets of toric ideals there exist subsets that also generate a toric ideal. For e.g. the 2xn determinantal ideal is toric. A generating set is given by the 2x2 minors of the defining matrix, say $M$. Then the ideal generated by those minors which correspond to a subset of the columns of the $M$ is also toric. 



Do subsets of generators of a toric ideal generate a toric ideal?
I believe this is a counterexample. Toric ideals are prime by definition (assuming that I am remembering correctly). Then I think that the ideal of the twisted cubic $C\subseteq \mathbb P^3$ ought to do it.


$J=\langle xz-y^2, xw-yz,wy-z^2\rangle$.


Any two of the three generators will intersect in the union of $C$ and a line. For instance, the vanishing of $I=\langle xz-y^2, xw-yz\rangle$ is $C\cup L$ where $L=V(x,y)$. So $I$ can't be prime.


