  We discuss the structure of the Motzkin algebra $M_k(D)$ by introducing a
sequence of idempotents and the basic construction. We show that $\cup_{k\geq
1}M_k(D)$ admits a factor trace if and only if $D\in \{2\cos(\pi/n)+1|n\geq
3\}\cup [3,\infty)$ and higher commutants of these factors depend on $D$. Then
a family of irreducible bimodules over the factors are constructed. A tensor
category with $A_n$ fusion rule is obtained from these bimodules.
