Let $I \in R = F[x_1,\dots,x_n]$ be a pure difference ideal. The universal Gröbner basis of $I$, denoted $\mathcal{U}(I)$, is the union of all of the reduced Gröbner bases of $I$. For various well-founded reasons, it is the convention in the literature to identify a binomial with its negative when dealing with the universal Gröbner basis $\mathcal{U}(I)$.

However, set union does not operate up to multiplication by $-1$. If $I = \langle x-y \rangle \subseteq F[x,y]$, then $I$ has two reduced Gröbner bases ${x-y}$ and ${y-x}$ and $\mathcal{U}(I)=\{x-y, \ y-x\}$. Whilst it is true that any one of $x-y$ and $y-x$ constitutes a basis for $I$, we would lose information about $I$ if we threw either of them away when defining $\mathcal{U}(I)$. In some cases of interest, the existence of $b \in \mathcal{U}(I)$ does not imply that $-b \in \mathcal{U}(I)$.

For a defined binomial ideal I, the output of the command universalGroebnerBasis I in the interface for Gfan in Macaulay2 does not distinguish between a binomial and its negative (one can test this by computing $\mathcal{U}(I)$ for the ideal $I$ above). If one needs to have this distinction for some reason when computing the universal Gröbner basis with Macaulay2, there is an easy workaround.

Once the package “gfanInterface” is loaded in a Macaulay2 session, and a binomial ideal I is defined, execute the commands:

(inL,L) = gfan I
unique flatten L

This will return the universal Gröbner basis $\mathcal{U}(I)$ of $I$ in the strict sense i.e. where $\mathcal{U}(I)$ is the union of all of the reduced Gröbner bases of $I$, where no element $-b$ is thrown away simply on account of $b$ already being a member of $\mathcal{U}(I)$.