A binary operation never satisfying the generalized associative law

Let x and y be positive integers and let a₁a₂...a_m and b₁b₂...b_n be the usual decimal notations of x and y. Note that a₁ ≠ 0 and b₁ ≠ 0. Then define x @ y = a₁a₂...a_mb₁b₂...b_n0. For example, 123 @ 45 = 123450.

Assume that natural numbers x₁, x₂, ..., x_N and a product x₁ @ x₂ @ ... @ x_N are given. Then, we can reconstruct the parenthesization as follows.

For example, assume that a₁ = 10, a₂ = 100, a₃ = 10, a₄ = 1000, a₅ = 10 and a₁ @ a₂ @ a₃ @ a₄ @ a₅ = 10100010100010000.

We can find the a_i's in the product as follows.

Replace the redundant 0's by @.

The last row is the reverse Polish notation of the product. This can be interpreted as (a₁ @ a₂) @ (a₃ @ (a₄ @ a₅)).