Cofree coalgebras and multivariable recursiveness

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For coalgebras over fields, there is a well-known construction which gives the cofree coalgebra over a vector space as a certain completion of the tensor coalgebra. In the case of a one-dimensional vector space this is the coalgebra of recursive sequences. In this paper, it is shown that similar ideas work in the multivariable case over rings (instead of fields). In particular, this paper contains a notion of recursiveness that exactly fits. For the case of a finite number of noncommuting variables over a field, it is the same as Schützenberger recognizability. There are applications to the question of the main theorem of coalgebras for coalgebras over rings. As should be the case, the cofree coalgebra over a finitely generated free module over a ring is the ‘zero dual’ of the free algebra over that module. A final application is a faithful representation theorem for coalgebras, that is representing a coalgebra as a subcoalgebra of a matrix-like coalgebra.