Stasheff polytope as a sublattice of permutohedron

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An assosiahedron Kⁿ, known also as Stasheff polytope, is a multifaceted combinatorial object, which, in particular, can be realized as a convex hull of certain points in Rⁿ, forming (n-1)-dimensional polytope. A permutahedron Pⁿ is a polytope of dimension (n-1) in Rⁿ with vertices forming various permutations of n-element set. There exist well-known orderings of vertices of Pⁿ and Kⁿ that make these objects into lattices: the first known as permutation lattices, and the latter as Tamari lattices. We establish that the vertices of Kⁿ can be naturally associated with particular vertices of Pⁿ in such a way that the corresponding lattice operations are preserved. In lattices terms, Tamari lattices are sublattices of permutation lattices. More generally, this defines the application of associative law as a special form of permutation.