Let $\geq$ be the usual partial order over $\mathbb{R}^n$ (i.e. if $x,y\in\mathbb{R}^n$, $x\geq y$ iff $ x_i\geq y_i \forall i=1,\dots,n$).
Definition 1: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$. For $a,b\in \mathbb{R}^n$ denote the n-box $B=[a_1,b_1]\times[a_2,b_2]\times\dots\times[a_n,b_n]$. The vertices of the n-box $B$ are the vectors $c\in\mathbb{R}^n$ with $c_i$ equal to either $a_i$ or $b_i$. $f$ is said to be n-increasing if for every n-box $B$ $\sum sign(c)f(c)\geq 0$, where the sum is taken over all the vertices $c$ and $sign(c)=1$ if $c_i=a_i$ for and even number of $i$'s and $sign(c)=-1$ if $c_i=a_i$ for and odd number of $i$'s.
For example in $\mathbb{R}^2$ a 2-box is the rectangle $[x_1,x_2]\times[y_1,y_2]$, with $x_1\leq x_2$ and $y_1\leq y_2$, and $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is 2-increasing if for all the two boxes, $$ f(x_2,y_2)-f(x_1,y_2)-f(x_2,y_1)+f(x_1,y_1)\geq 0. $$
In $\mathbb{R}^3$ a 3-box is the cube $[x_1,x_2]\times[y_1,y_2]\times[z_1,z_2]$, with $x_1\leq x_2$, $y_1\leq y_2$ and $z_1\leq z_2$, and $f:\mathbb{R}^3\rightarrow \mathbb{R}$ is 3-increasing if for all the 3-boxes, $$ f(x_2,y_2,z_2)-f(x_2,y_2,z_1)-f(x_2,y_1,z_2)-f(x_1,y_2,z_2)\\ +f(x_2,y_1,z_1)+f(x_1,y_2,z_1)+f(x_1,y_1,z_2)-f(x_1,y_1,z_1)\geq 0. $$
Definition 2: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$. $f$ is said to be supermodular if $\forall x,y\in\mathbb{R}^n$, $f(x\vee y)+f(x\wedge y)\geq f(x)+f(y)$ (where the join and meet are considered w.r.t. the partial order $\geq$ of $\mathbb{R}^n$ defined above).
It is clear that in $\mathbb{R}^2$ supermodularity and n-increasingness are the same thing.
Now my question is the following: can someone provide and example that shows that they are different notions in $\mathbb{R}^n$ for $n>2$? Also references would be gratley acknowledged.