Format of Original
Categorical Topology: Proceedings of the International Conference Held in Toledo, OH
If two compact Hausdorff spaces have elementarily equivalent lattices of closed sets, then their unital rings of continuous real-valued functions satisfy the same positive-universal sentences. We briefly indicate how this is proved (with details appearing elsewhere), and offer fairly strong evidence that the weakness of the conclusion is in some sense inevitable.
Bankston, Paul, "First Order Representations of Compact Hausdorff Spaces" (1984). Mathematics, Statistics and Computer Science Faculty Research and Publications. 228.