On the sorting complexity of Cartesian products of partially ordered sets

We study the complexity $L(M_n)$ of algorithms for sorting the partially ordered set $M_n$, which is isomorphic to the Cartesian product
$$ K_1\times\ldots\times K_n, $$
where all $K_i$ are taken from some finite family, have a unique maximum element, and are prime with respect to the Cartesian product. For the set $\{M_n\}$, as $n\to\infty$, we obtain the estimates
\begin{align*} L(M_n)&\gtrsim|M_n|\log_{2}|M_n|, \\ L(M_n)&\lesssim(|K_1|+\ldots+|K_n|)|M_n|. \end{align*}
Besides, we prove that for a partially ordered set with one maximum element the Cartesian decomposition into prime factors is unique up to a permutation of the factors.