  Given a simple and connected graph $G=(V,E)$, and a positive integer $k$, a
set $S\subseteq V$ is said to be a $k$-metric generator for $G$, if for any
pair of different vertices $u,v\in V$, there exist at least $k$ vertices
$w_1,w_2,\ldots,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$, for every $i\in
\{1,\ldots,k\}$, where $d_G(x,y)$ denotes the distance between $x$ and $y$. The
minimum cardinality of a $k$-metric generator is the $k$-metric dimension of
$G$. A set $S\subseteq V$ is a $k$-adjacency generator for $G$ if any two
different vertices $x,y\in V(G)$ satisfy $|((N_G(x)\triangledown
N_G(y))\cup\{x,y\})\cap S|\ge k$, where $N_G(x)\triangledown N_G(y)$ is the
symmetric difference of the neighborhoods of $x$ and $y$. The minimum
cardinality of any $k$-adjacency generator is the $k$-adjacency dimension of
$G$. In this article we obtain tight bounds and closed formulae for the
$k$-metric dimension of the lexicographic product of graphs in terms of the
$k$-adjacency dimension of the factor graphs.
