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

Leslie E Shader
1974 Journal of combinatorial theory. Series A  
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 n < r < [(n2 + 1)/2], and A is n2 x rP, is proven. Since these are the only possible values, the question of existence solutions of all possible ranks is
more » ... letely answered. The techniques and proofs are of a constructive nature.
doi:10.1016/0097-3165(74)90047-8 fatcat:gjwkqmkonfahzm2n7kih3vag6q