Let $K$ be an abstract simplicial complex on vertices $x_1,\ldots,x_n$, then there is the familiar construction of the face ideal $I_K=\langle x_{i_1}\cdots x_{i_r} | \{x_{i_1},\ldots,x_{i_{r}}\}\not\in K\rangle$ generated by the monomials corresponding to the non-faces of $K$, and the Stanley-Reisner ring $R_K = k[x_1,\ldots x_n]/I_K$. (Yes, abuse of notation: both the vertices and variables have the same name.)
We can of course also write $I_K = \bigcap P_i$, where $P_i$ are the minimal primes, aka. the defining ideals of the maximal faces of $K$; let us introduce some notation and say that $P_i$ corresponds to the maximal face $K_i$, etc.
I stumbled upon the following "duality" operation: If $P_i = \langle x_{j}|j\in K_i\rangle$, let $P_i^* :=\langle x_{j}|j\not\in K_i\rangle$ (in other words replace each minimal prime of $I_K$ by the prime ideal generated by the variables not in that ideal), and let $I_K^* :=\bigcap_i P_i^*$.
$I_K^*$ is also a face ideal, and $(I_K^*)^*=I_K$, so it seems reasonable to call this operation a duality; it vaguely resembles Alexander duality but is mostly quite different. What I want to know is
Q1: Is this construction known?
and
Q2: if so, what is it called?
