 
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
