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Cofree coalgebras and multivariable recursiveness. (English) Zbl 1048.16022
Author’s abstract: 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 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.

MSC:
16W30Hopf algebras (associative rings and algebras) (MSC2000)
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References:
[1] Balcerzyk, S.: On groups of functions defined on bolean algebras. Fund. math. 50, 347-367 (1962) · Zbl 0101.26501
[2] J. Berstel, Séries rationelles. In: J. Berstel (Ed.), Séries formelles en variables non commutatives et applications, LITP, Paris, 1978, pp. 5--22.
[3] Block, R. E.; Leroux, P.: Generalized dual coalgebras of algebras with applications to cofree coalgebras. J. pure appl. Algebra 36, 15-21 (1985) · Zbl 0556.16005
[4] G. Carnovale, Le coalgebre sull’ anello degli interi, Master Thesis, Université di Roma’La Sapienza’, 1993.
[5] Dascalescu, S.; Nastasescu, C.; Raianu, S.: Hopf algebras. An introduction. (2001)
[6] P.C. Eklof, Whitehead modules. In: M. Hazewinkel (Ed.), Handbook of Algebra, Vol. 3, North-Holland, Amsterdam, 2003.
[7] P.C. Eklof, A.H. Mekler, Almost Free Modules. Set Theoretic Methods, North-Holland, Amsterdam, 1990. · Zbl 0718.20027
[8] Fliess, M.: Matrices de Hankel. J. math. Pures appl. 53, 197-224 (1974) · Zbl 0315.94051
[9] Fliess, M.: Fonctionelles causales non linéaires et indéterminées non commutatives. Bull. soc. Math. France 109, 3-40 (1981) · Zbl 0476.93021
[10] Fox, T. F.: The construction of cofree coalgebras. J. pure appl. Algebra 84, 191-198 (1993) · Zbl 0810.16038
[11] M. Hazewinkel (Ed.), Encyclopaedia of Mathematics, Vol. 7, Orb-Ray, Kluwer Academic Publisher, Dordrecht, 1991. · Zbl 0806.00006
[12] M. Hazewinkel, Letter to Giovanna Carnovale, 1993.
[13] Hazewinkel, M.: The algebra of quasi-symmetric functions is free over the integers. Adv. in math. 164, 283-300 (2001) · Zbl 0991.05102
[14] Hungerford, T. W.: Algebra. (1974) · Zbl 0293.12001
[15] Macdonald, I. G.: Symmetric functions and Hall polynomials. (1995) · Zbl 0824.05059
[16] Peterson, B.; Taft, E. J.: The Hopf algebra of linearly recursive sequences. Aequationes math. 20, 1-17 (1980) · Zbl 0434.16008
[17] Rouchaleou, Y.; Wyman, B. F.; Kalman, R. E.: Algebraic structure of linear dynamical systems. Iiirealization theory over a commutative ring. Proc. natl. Acad. sci. (USA) 69, 5404-5406 (1972)
[18] Schützenberger, M. P.: On the definition of a family of automata. Inform. and control 4, 215-270 (1961) · Zbl 0104.00702
[19] Schützenberger, M. P.: On a theorem of R jungen. Proc. amer. Math. soc. 13, 885-890 (1962) · Zbl 0107.03102
[20] Sontag, E. D.: Linear systems over commutative ringsa survey. Ricerche automat. 7, 1-34 (1976)
[21] Specker, E.: Additive gruppen von folgen ganzer zahlen. Portugal math. 9, 131-140 (1950) · Zbl 0041.36314
[22] Sweedler, M. E.: Hopf algebras. (1969)