Let $X$ be a $l$-space, i.e. a locally compact totally disconnected hausdorff space, which is not compact. Then $P = \{K : K \subseteq U \text{ compact-open}\}$ 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.

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.

Let $X$ be a $l$-space, i.e. a locally compact totally disconnected hausdorff space, which is not compact. Then $P = \{K : K \subseteq U \text{ compact-open}\}$ is a basis for the topology. Regard $P$ as a partial order with respect to "$\subseteq$".

Question: Which preorders are isomorphic to preorders 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.

Let $X$ be a $l$-space, i.e. a locally compact totally disconnected hausdorff space, which is not compact. Then $P = \{K : K \subseteq U \text{ compact-open}\}$ 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.