What are iteration theories

“In the setting of algebraic theories enriched with an external fixed-point operation, the no-tion of an iteration theory seems to axiomatize the equational properties of all the computa-tionally interesting structures of this kind.” S. L. Bloom and Z. ´Esik (1996), see [3] We prove that iteration theories can be introduced as al-gebras for the monad on the category of signatures as-signing to every signature the rational- -tree signature. This supports the claim that iteration theories axiomatize precisely the equational properties of least fixed points in domain theory: is the monad of free rational theories and every rational theory has a continuous completion. 1