Cofree coalgebras and multivariable recursiveness

✍ Scribed by Michiel Hazewinkel

• Book ID: 104152652
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 359 KB
• Volume: 183
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(03)00013-6

✦ Synopsis

For coalgebras over ÿelds, 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 ÿelds). In particular, this paper contains a notion of recursiveness that exactly ÿts. For the case of a ÿnite number of noncommuting variables over a ÿeld, it is the same as Sch utzenberger 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 ÿnitely generated free module over a ring is the 'zero dual' of the free algebra over that module. A ÿnal application is a faithful representation theorem for coalgebras, that is representing a coalgebra as a subcoalgebra of a matrix-like coalgebra.