The paper (for abstract, see below) Aczel P., Adamek J., Velebil J.: Iteration Monads, preprint is now available from http://math.feld.cvut.cz/velebil/ Jiri Velebil ---------------------------------------------------------------- Abstract: It has already been noticed by C. Elgot and his collaborators that the algebra of (finite and infinite) 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 final coalgebra for H(_)+X exists. Then these final 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^+: 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 tensor (_)+I of B has a final 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].