Geodesic
On the potential divisibility of matrices over distributive lattices
Diskretnaya Matematika, Tome 22 (2010) no. 2, pp. 148-159.

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We consider matrices of arbitrary sizes (including infinite matrices) over a distributive lattice $L$ and prove that if $L=2^X$ is a lattice of all subsets of a set $X$, then the potential divisibility of matrices (from the left or from the right) of one of the matrices by the other matrix is equivalent to the usual divisibility. In particular, in the semigroup of square matrices over the lattice $2^X$ the Green relation $\mathscr L$ coincides with the generalised Green relation $\mathscr L^*$. 
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I. B. Kozhukhov; V. A. Yaroshevich. On the potential divisibility of matrices over distributive lattices. Diskretnaya Matematika, Tome 22 (2010) no. 2, pp. 148-159. http://geodesic.mathdoc.fr/item/DM_2010_22_2_a10/

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