The weak order and Bruhat order are well-known posets on permutations, the weak order being a lattice which is contained in the Bruhat order.
Recently a new distributive lattice on permutations called „middle order“ was discovered, containing the weak order and contained in the Bruhat order.
We generalize this result by constructing C_{n-1} distributive lattices on permutations of size n with the same property, using a simple bijection between permutations and ideals of posets.
We then show these lattices are the only distributive lattices between weak and Bruhat orders, and we also consider generalizations of middle orders in other Coxeter groups.
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