The numerous internal symmetries are found in N-dimensional integer lattices (ZN). The relation of these symmetries with the new mathematical category – the so-called the Masks (or Neighborhoods) – is shown. A set of definitions for the Correct Masks and Perfect Masks is presented; an identity between the Correct and Perfect Masks is hypothesized. The relationship between the Perfection of the Mask and the new category named “Mathematical String” is shown. The Correctness of the several Masks in ZN (N=1,2) is proven and a simple method to find the Correctness for all other N is outlined. The hypothesis of high population density of Perfect Masks in integer lattices ZN with large N is stated.