An $\omega _{2}$-minimal Boolean algebra

For every linear order $L$ we define a notion of $L$-minimal Boolean algebra and then give a consistent example of an $\omega _{2}$-minimal algebra. The Stone space $X$ of our algebra contains a point $\{*\}$ such that $X-\{*\}$ is an example of a countably tight, initially $\aleph _{1}$-compact, non-compact space. This answers a question of Dow and van Douwen.