  We study congruences of the semigroup
$\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ of monotone
injective partial selfmaps of the set of
$L_n\times_{\operatorname{lex}}\mathbb{Z}$ having co-finite domain and image,
where $L_n\times_{\operatorname{lex}}\mathbb{Z}$ is the lexicographic product
of $n$-elements chain and the set of integers with the usual linear order. The
structure of the sublattice of congruences on
$\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ which contain in
the least group congruence is described.

Group Theory