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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
in nite) 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 in nite. In 1970's the ADJ group have
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