TITLE: Partial orders arising from $l$-spaces
QUESTION [6 upvotes]: Let $X$ be a $l$-space, i.e. a locally compact totally disconnected hausdorff space, which is not compact. Then $P = \{K : K \subseteq X \text{ compact-open}\}$ is a basis for the topology. Regard $P$ as a partial order with respect to "$\subseteq$".
Question: Which partial orders are isomorphic to partial orders which arise from $l$-spaces as above? Note that they have finite infima and suprema and a smallest element, but not a maximal element. But I doubt that this is already the whole characterization.

REPLY [3 votes]: In

M. H. Stone. Applications of the Theory of Boolean Rings to General Topology. Transactions of the American Mathematical Society Vol. 41, No. 3 (May, 1937), pp. 375-481

Marshall Stone proved (see Theorem 4) that there is a duality between locally compact totally disconnected Hausdorff spaces and Boolean rings (possibly without unit). Boolean rings are in turn equivalent to generalized Boolean algebras. The duality assigns to each space the generalized Boolean algebra of the compact open subsets. So the answer to your question seems to be that such posets are precisely the generalized Boolean algebras.