 
Summary: FREE ITERATIVE THEORIES: A COALGEBRAIC VIEW
JI
R I AD
AMEK, STEFAN MILIUS, AND JI
R I VELEBIL
Abstract. Every nitary endofunctor of Set is proved to generate a free iterative theory in the sense of
Elgot. This is based on coalgebras, specically on parametric corecursion, and the proof is presented for
categories more general than just Set.
1. Introduction
Iterative algebraic theories have been introduced by Calvin C. Elgot in the 1970's as a concept serving
to study computation (on Turing machines, say) at a level abstracting from the nature of external memory.
The main example presented by Elgot is the theory of regular trees, i.e., innite trees which are solutions of
systems of nitary
at iterative equations. Or, equivalently, which posses only nitely many subtrees. He
and his coauthors have later proved that this theory is a free iterative theory of a given (nitary) signature,
see [EBT].
The purpose of the present paper is to generalize Elgot's result from signatures (=polynomial endofunctors
of the category of sets) to nitary endofunctors of Set and some \setlike" categories, e. g. the category of
posets. Using a very general Solution Theorem, developed in previous work, which shows by coalgebraic
methods how iterative equations can be solved in categories, we prove that nitary endofunctors generate
free iterative theories (=nitary monads), called rational monads. We construct the rational monad in two
