zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

16W30Hopf algebras (associative rings and algebras) (MSC2000)
Full Text: DOI
[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)