The Dedekind number is the count of monotone boolean function from $\mathbf{2}^d$ to $\{0,1\}$. Does anyone know any research about its extension, from $[e_1]\times\cdots\times[e_d]$ to $\{0,1\}$? Here $[n]$ means $\{1,2,\ldots,n\}$.
Terminology:
A monotone boolean function $f$ is such that $f(x)\geq f(y)$ whenever $x\geq y$, where $f(x)\in\{0,1\}$ for all $x$.
There are two equivalent ways to describe the poset $P$ that $x$ and $y$ live in:
(1) $P=[e_1]\times\cdots\times[e_d]$, where $x\geq y$ if $x_i\geq y_i$ for all $i\in [d]$. Note that $\mathbf{2}^d$ corresponds to $e_1=\cdots=e_d=2$ up to offset.
(2) $P$ is the set of divisors of $N=p_1^{e_1-1}\cdots p_d^{e_d-1}$, where $p_1,\ldots,p_d$ are distinct primes and relation $\geq$ is overwritten by "is a multiple of". That is, the monotone boolean function $f$ should be such that $f(x)\geq f(y)$ if $x$ is divisible by $y$.