Geodesic
On the sorting complexity of Cartesian products of partially ordered sets
Diskretnaya Matematika, Tome 13 (2001) no. 3, pp. 57-74

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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. 
@article{DM_2001_13_3_a3,
     author = {Yu. B. Nikitin},
     title = {On the sorting complexity of {Cartesian} products of partially ordered sets},
     journal = {Diskretnaya Matematika},
     pages = {57--74},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2001_13_3_a3/}
}
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Yu. B. Nikitin. On the sorting complexity of Cartesian products of partially ordered sets. Diskretnaya Matematika, Tome 13 (2001) no. 3, pp. 57-74. http://geodesic.mathdoc.fr/item/DM_2001_13_3_a3/