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There is a poset constructed by combining in a certain way the usual order on the reals with any well-order on the reals (I can provide details if needed). I've heard it called the "Sierpinski Reals" and "The Leaning Tower of L'viv", but I do not see anything online when I search these names. Is this construction known by any other names?

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These slides on Suslin lattice mention a Sierpinski poset, as does Theory of Relations by Roland Fraïssé.

Edit: Also the term Sierpinski tower such as mentioned in Random Walks on a Fractal Solid.

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The "Sierpinski poset" defined here in Theory of Relations is indeed the one I was thinking of. Thank you. –  Austin Mohr Jan 20 '12 at 12:08

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