On the existence of finite central groupoids of all possible ranks. I

The problem of determining the number of finite central groupoids (an algebraic system satisfying the identity (x · y) · (y ? z) = y) is equivalent to the problem of determining the number of solutions of the matrix equation A² = J, where A is a 0, 1 matrix and J is a matrix of 1's.The existence of solutions of A² = J of all ranks r, where $\text{n ? r ?}\text{(n}{}^2\text{+ 1)}\text{2}\text{]}$, and A is n² × n², is proven. Since these are the only possible values, the question of existence solutions of all possible ranks is completely answered. The techniques and proofs are of a constructive nature.