I need the following object in my research:
A proper subset of a poset $\{x_k\}\subset P$ with $|\{x_k\}|>1$ is reducible to a point (a RAP) when for every $y\in P-\{x_k\}$, either $y\leq\{x_k\}$, $\{x_k\}\leq y$, or $\{x_k\}$ and $y$ are incomparable.
I have a hard time believing that no one has studied or defined such an object before in the poset literature, but I have not found it in Stanley or any other source. Does anyone know if this definition exists in the literature and if so where?