Geometric aspects of a deformation of the standard addition on integer lattices

Let $\mathfrak A_n$ be the set of all those vectors of the standard lattice $\mathbb Z^n$ whose coordinates are pairwise incomparable modulo $n$. In this paper, we analyze the group structure on $\mathfrak A_n$ that arises from the construction of a deformation of multiplication described by V. M. Buchstaber. We present a geometric realization of this group in the ambient space $\mathbb R^n\supset\mathbb Z^n$ and find its generators and relations.