be the set consisting of the $n_j$ canonical basis vectors of $\mathbb E^{n_j}$, and set $X_j=T_{\mathbb S}B_j$. Moreover, let $Z$ be an $\mathbb E$-submodule of $\mathbb E^{n_1}\times\mathbb E^{n_2}$. Then $Z\cap(X_1\times X_2)$ is finitely generated as an algebra in .

A subcubic convex functor {#sec-SubcubicFunctor}
=========================

Recall the following definition from [@silva.sokolova:2011 p.309].

\[Ghat-def\] We introduce a functor ${{\ensuremath{\widehat F}}}\colon {{\text{\tt{PCA}}}}\to{{\text{\tt{PCA}}}}$.

1.  Let $X$ be a . Then $$\begin{aligned}
                {{\ensuremath{\widehat F}}}X=\Big\{ (o,\phi)\in[0,1] & \times X^A\mid 
                \\
                & \exists\,n_a\in\mathbb N{{.\kern3pt}}\exists\,p_{a,j}\in[0,1],x_{a,j}\in X\text{ for }j=1,\ldots,n_a,a\in A{{.\kern3pt}}\\
                & o+\sum_{a\in A}\sum_{j=1}^{n_a}p_{a,j}\leq 1,\ \phi(a)=\sum_{j=1}^{n_a}p_{a,j}x_{a,j} \Big\}.
            \end{aligned}$$

2.  Let $X,Y$ be s, and $f\colon X\to Y$ a convex map. Then ${{\ensuremath{\widehat F}}}f\colon {{\ensuremath{\widehat F}}}X\to{{\ensuremath{\widehat F}}}Y$ is the map ${{\ensuremath{\widehat F}}}f=\operatorname{id}_{[0,1]}\times(f\circ -)$.

For every $X$ we have ${{\ensuremath{\widehat F}}}X\subseteq{{\ensuremath{F_{\,{[0,1]}}}}}X$, and for every $f\colon X\to Y$ we have ${{\ensuremath{\widehat F}}}f=({{\ensuremath{F_{\,{[0,1]}}}}}f)|_{{{\ensuremath{\widehat F}}}X}$. For this reason, we think of ${{\ensuremath{\widehat F}}}$ as a *subcubic functor*.

The definition of [[$\widehat F$]{}]{} can be simplified.

\[Ghat-simple\] Let $X$ be a , then $$\begin{aligned}
        {{\ensuremath{\widehat F}}}X=\Big\{ (o,f)\in[0,1]\times X^A\mid\ & 
        \exists\,p_a\in[0,1],x_a\in X\text{ for }a\in A{{.\kern3pt}}\\
        & o+\sum_{a\in A}p_a\leq 1,\ f(a)=p_ax_a \Big\}.
    \end{aligned}$$

From this representation it is obvious that [[$\widehat F$]{}]{} is monotone in the sense that

-   If $X_1\subseteq X_2$, then ${{\ensuremath{\widehat F}}}X_1\subseteq{{\ensuremath{\widehat F}}}X_2$.

-   