Matrix and matricial iteration theories, Part I

Matrix iteration theories are characterized by identities using theory operations as well as a star operation on T(n, n), for each n ⩾ 0. The initial matrix iteration theory is described explicitly. An extension theorem is proved which implies that if MatS is a matrix iteration theory, so is MatR where R is a semiring of formal power series over S. In Part II these results are extended to Elgot's matricial theories.