I seek a Matlab or easily-to-translate function that lists all monotonic binary functions. These are sometimes called Dedekind functions.
What is a binary function? Let $\mathbf{u} = [u_{1}, u_{2}, \cdots u_{N}]$ be a binary valued vector, in which $u_{k} = 0 \text{ or } 1$. A monotonic Boolean function requires $f\left( \mathbf{u} \right) \ge f\left( \mathbf{u^{\prime}} \right)$ if $u_{k} \ge u_{k}^{\prime}$ for each $k$ between $1$ and $N$. There are $2^{M}$ functions, where $M=2^{N}$. Much fewer are monotonic. In the case of $N=2$, there are $16$ different functions, but $6$ of these are monotonic. I pasted an image of the table. In this case, $r_{1}$, $r_{2}$, $r_{4}$, $r_{4}$, $r_{8}$, and $r_{16}$ are monotonic. The term sensor input indicates an input vector $\mathbf{u}=[u_{1}, u_{2}]$ and the term "Global fusion rules" labels the function output.
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If you specify the inputs $u$ (in your notation) which are minimal, i.e., $f(u)=1,$ and for all $u'<u, f(u')=0,$ you have completely specified a monotone boolean function (MBF). Simply put, all the inputs $v$ such that $v\geq u$ also satisfy $f(v)=1.$
Such inputs have to be parts of an antichain, i.e., they are incomparable in the boolean algebra of binary strings. For example $u=(1,0,1,0)$ and $u'=(1,1,0,0)$ are incomparable since neither $u'\leq u$ nor $u\leq u'$ holds.
See the beginning of the following paper by Stephen and Yusun for a more detailed description to get started.
This reduces your question to the listing of all antichains. See the question Enumerating all antichains in a finite poset.