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Initial algebras and terminal coalgebras in many-sorted sets

Published online by Cambridge University Press:  25 March 2011

JIŘÍ ADÁMEK  and
VĚRA TRNKOVÁ
JIŘÍ ADÁMEK
Affiliation:
Institut für Theoretische Informatik, Technische Universität Braunschweig, Germany Email: J.Adamek@tu-bs.de
VĚRA TRNKOVÁ
Affiliation:
Mathematical Institute, Charles University Prague, Prague, Czech Republic Email: Trnkova@karlin.mff.cuni.cz
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Abstract

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We prove that the iterative construction of initial algebras converges for endofunctors F of many-sorted sets whenever F has an initial algebra. In the case of one-sorted sets, the convergence takes n steps where n is either an infinite regular cardinal or is at most 3. Dually, the existence of a many-sorted terminal coalgebra implies that the iterative construction of a terminal coalgebra converges. Moreover, every endofunctor with a fixed-point pair larger than the number of sorts is proved to have a terminal coalgebra. As demonstrated by James Worell, the number of steps here need not be a cardinal even in the case of a single sort: it is ω + ω for the finite power-set functor. The above results do not hold for related categories, such as graphs: we present non-constructive initial algebras and terminal coalgebras.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Special Issue 2: Coalgebraic Logic , April 2011 , pp. 481 - 509
Copyright
Copyright © Cambridge University Press 2011

References

Adámek, J. (1974) Free algebras and automata realizations in the language of categories. Comment. Math. Univ. Carolinae 14 589–602.Google Scholar
Adámek, J., Herrlich, H. and Strecker, G. E. (2009) Abstract and concrete categories, Dover.Google Scholar
Adámek, J. and Koubek, V. (1995) On the greatest fixed point of a set functor. Theoret. Comput. Sci. 15 57–75.CrossRefGoogle Scholar
Adámek, J., Koubek, V. and Pohlová, V. (1972) Colimits in the generalized algebraic categories. Acta Univ. Carolinae 13 311–324.Google Scholar
Adámek, J. and Reiterman, J. (1994) Banach's fixed point theorem as a base for data-type equations. Appl. Categ. Struct. 2 77–90.CrossRefGoogle Scholar
Adámek, J. and Rosický, J. (1994) Locally presentable and accessible categories, Cambridge University Press.CrossRefGoogle Scholar
Adámek, J. and Trnková, V. (1990) Automata and algebras in categories, Kluwer.Google Scholar
America, P. and Rutten, J. (1989) Solving reflexive domain equations in a category of complete metric spaces. J. Comp. Syst. Sciences 39 343–375.CrossRefGoogle Scholar
Barr, M. (1993) Terminal coalgebras in well-founded set theory. Theoret. Comput. Sci. 114 299–315.CrossRefGoogle Scholar
Jech, T. (1978) Set theory, Academic Press.Google Scholar
Koubek, V. (1971) Set functors. Comment. Math. Univ. Carolinae 12 175–195.Google Scholar
Koubek, V. and Reiterman, J. (1979) Categorical constructions of free algebras, colimits and completions of partial algebras. J. Pure Appl. Algebra 14 195–231.CrossRefGoogle Scholar
Milius, S. (2005) Completely iterative algebras and completely iterative monads. Inform. and Comput. 196 1–41.CrossRefGoogle Scholar
Pultr, A. and Trnková, V. (1980) Combinatorial, algebraic and topological representations of groups, semigroups and categories, North Holland and Academia Praha.Google Scholar
Reiterman, J. (1976) Categorical algebraic constructions, Ph.D. Thesis (in Czech), Charles University Prague.Google Scholar
Rutten, J. (2000) Universal coalgebra: a theory of systems. Theoret. Comput. Sci 249 3–80.CrossRefGoogle Scholar
Smyth, M. B. and Plotkin, G. E. (1982) The category-theoretic solution of recursive domain equations. SIAM J. Comput. 11 761–783.CrossRefGoogle Scholar
Tarski, A. (1929) Sur la décomposition des ensembles en sous-ensemble prèsque disjoint. Fund. Math. 12 205–215.CrossRefGoogle Scholar
Trnková, V. (1969) Some properties of set functors. Comment. Math. Univ. Carolinae 10 323–352.Google Scholar
Trnková, V., Adámek, J., Koubek, V. and Reiterman, J. (1975) Free algebras, input process and free monads. Comment. Math. Univ. Carolinae 16 339–351.Google Scholar
Worrell, J. (2005) On the final sequence of a finitary set functor. Theoret. Compu. Sci. 338 184–199.CrossRefGoogle Scholar