 
Summary: ITERATION MONADS
PETER ACZEL, JI
R I AD
AMEK, AND JI
R I VELEBIL
Abstract. It has already been noticed by C. Elgot and his collaborators that the algebra of (nite and
innite) trees is completely iterative, i.e., every system of ideal recursive equations has a unique solution.
We prove that this is a special case of a very general coalgebraic phenomenon: suppose that an endofunctor
H of an abstract category A is \iterative", i.e., that it has the property that for every object X in A a nal
coalgebra for H( ) + X exists. Then these nal coalgebras, TX, form a monad on A, called the iteration
monad of H. And every ideal equation e : X ! T (X + Y ) has a unique solution e y : X ! TY .
We also present a more general view substituting the category [A; A] of all endofunctors of A by a
monoidal category B: an object H in B is called iterative if the endofunctor
H
( ) + I of B has a nal
coalgebra. This coalgebra is, then, a monoid in B, called the iteration monoid of H. And the assignment of
an iteration monoid to all objects forms a monoid in [B; B].
1. Introduction
There are various algebraic approaches to the formalization of computations of data through a given
program, taking into account that such computations are potentially innite. In 1970's the ADJ group have
