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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.

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.

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 that there is a duality between locally compact totally disconnected 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.

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.

If I am not mistaken there is Stone duality between

• locally compact 0-dimensional Hausdorff spaces, i.e., $l$-spaces, and
• generalized Boolean algebras.

I am having trouble locating a reference for this "fact", so please provide one if you know of it (or tell me I am wrong). The duality assigns to each $l$-space $X$ the lattice of its compact open subsets. So the answer ought to be that your posets are precisely the generalized Boolean algebras. We just have to hunt down the relevant literature.

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 that there is a duality between locally compact totally disconnected 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.