$(1)\hspace{4pt}$ Let $\left\langle X,\leq\right\rangle$ be a poset, and let $Y\subseteq X$ with $Y\ne\emptyset$. I’m trying to prove that $Y$ is centered—i.e., $Y$ is a filterbase on $X$— if and only if any finite subset $Z\subseteq Y$ has a lower bound. I’m not at all sure how to do this, so I would appreciate any sugggestions.
$(2)\hspace{4pt}$ I’m trying to find an example of a poset $\left\langle X,\leq\right\rangle$ such that $\exists Y\subseteq X$ with $Y$ linked—i.e., pairwise compatible—but $Y$ not centered. Again, I really don’t know how to do this, and I would appreciate any suggestions.
HINT:
(2) Let $\langle X,\le\rangle=\langle\wp(\{0,1,2\}),\subseteq\rangle$, and take $Y$ to be the right $3$-element subset of $X$.