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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

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.

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

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