T₄ — the Tamari lattice on four leaves

The Tamari lattice $T_4$ — using the convention that the subscript counts leaves — has five elements: the five rooted binary trees with four leaves, or equivalently the five ways to parenthesise the product $a \cdot b \cdot c \cdot d$. The cover relation is a single right rotation at some internal node:

$$((x \cdot y) \cdot z) \;\longmapsto\; (x \cdot (y \cdot z)).$$

As a lattice, $T_4$ on five elements is isomorphic to the pentagon $N_5$ — an instructive coincidence at the smallest non-trivial size. What distinguishes $T_4$ as an object of study is not its small incarnation but the family it belongs to: the Tamari lattices $T_n$ are the 1-skeleta of the associahedra, polytopes that organise the combinatorics of associativity, and they reappear throughout algebraic topology, category theory, and the theory of cluster algebras.

Where it appears

• The 1-skeleton of the Stasheff associahedron $K_{n+1}$ is the Hasse diagram of $T_n$ (leaves convention).
• Right rotations on binary trees show up in self-balancing search trees; the Tamari order makes that informal “rotate to the right” notion into a partial order.
• In categorical algebra, coherence morphisms for an associative operation are organised by the associahedron — so $T_n$ is the smallest combinatorial witness of “associativity up to homotopy.”

A larger Tamari lattice, $T_5$ on fourteen elements, is the 3-dimensional associahedron $K_5$; we draw the smaller cousin here for clarity.