The geometry of a deformation of the standard addition on the integral lattice

Let ${\mathfrak{A}}_n$ be the subset of the standard integer lattice $\mathbb Z^n$,$\mathfrak A_n\subset\mathbb Z^n$whichisdefinedbythecondition$\mathfrak A_n=((a_1,…,a_n)\in\mathbb Z^n | a_i\not\equiv a_j\mod n, \forall i,j\in {1,… n})$\mathrm{.}Itisclearthatthestandardadditiononthelattice$\mathbb Z^n$doesnotinducethegroupstructureontheset$\mathfrak A_n$ since the componentwise sum of some two vectors may contain components which are equal modulo $n$. Our aim is to find a new associative multiplication on the lattice ${\mathbb{Z}}^n$ such that the induced multiplication on the set ${\mathfrak{A}}_n$ gives it the group structure. In this paper the group structure on the subset ${\mathfrak{A}}_n$ of the integer lattice ${\mathbb{Z}}^n$ is studied by means of the constructions of a deformation of a group multiplication. The geometric realization of this group in the enveloping space and its generators and relations between them are found. We begin with the main constructions and the results we need for them.