Summary: Terminal Coalgebras and Free Iterative
Theories
JiŸr'i Ad'amek 1 and Stefan Milius
Institute of Theoretical Computer Science,
Technical University of Braunschweig,
Germany
Abstract
Every finitary endofunctor H of Set can be represented via a finitary signature \Sigma
and a collection of equations called ``basic''. We describe a terminal coalgebra of H
as the terminal \Sigma­coalgebra (of all \Sigma­trees) modulo the congruence of applying the
basic equations potentially infinitely often. As an application we describe a free
iterative theory on H (in the sense of Calvin Elgot) as the theory of all rational
\Sigma­trees modulo the analogous congruence. This yields a number of new examples
of iterative theories, e.g., the theory of all strongly extensional, rational, finitely
branching trees, free on the finite power­set functor, or the theory of all binary,
rational unordered trees, free on one commutative binary operation.
Key words: terminal coalgebra, rational tree, iterative theory, basic equation
1 Introduction
It is well­known that for any finitary signature \Sigma an initial \Sigma­algebra I \Sigma is
the algebra of all finite \Sigma­trees, and a terminal \Sigma­coalgebra T \Sigma is the algebra

Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig

Collections: Computer Technologies and Information Sciences