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We define noncrossing partitions of a marked surface without punctures(interior marked points). We show that the natural partial order on noncrossingpartitions is a graded lattice and describe its rank function topologically.Lower intervals in the lattice are isomorphic to products of noncrossingpartition lattices of other surfaces. We similarly define noncrossingpartitions of a symmetric marked surface with double points and prove some ofthe analogous results. The combination of symmetry and double points plays arole that one might have expected to be played by punctures.

Noncrossing partitions of a marked surface