Congruences on the monoid of monotone injective partial selfmaps of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ with co-finite domains and images

Abstract: 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.