Polyomino coloring and complex numbers

Usually polyominoes are represented as subsets of the lattice Z2Z2. In this paper we study a representation of polyominoes by Gaussian integers. Polyomino {(x1,y1),(x2,y2),…,(xs,ys)}⊂Z2{(x1,y1),(x2,y2),…,(xs,ys)}⊂Z2 is represented by the set {(x1+iy1),(x2+iy2),…,(xs+iys)}⊂Z[i].{(x1+iy1),(x2+iy2),…,(xs+iys)}⊂Z[i]. Then we consider functions of type f:P→Gf:P→G from the set PP of all polyominoes to an abelian group GG, given by f(x,y)≡(x+iy)m(modv), where vv is prime in Z[i],1≤m