For finitary set functors, free algebras and free completely iterative algebras share the same conservative completion as posets, with omega-chains of approximations in the free algebra solving recursive equations in the iterative one.
On free completely iterative algebras
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abstract
For every finitary set functor F we demonstrate that free algebras carry a canonical partial order. In case F is bicontinuous, we prove that the cpo obtained as the conservative completion of the free algebra is the free completely iterative algebra. Moreover, the algebra structure of the latter is the unique continuous extension of the algebra structure of the free algebra. For general finitary functors the free algebra and the free completely iterative algebra are proved to be posets sharing the same conservative completion. And for every recursive equation e in the free completely iterative algebra we present an omega-chain of approximate solutions in the free algebra whose join is the solution of e.
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On free completely iterative algebras
For finitary set functors, free algebras and free completely iterative algebras share the same conservative completion as posets, with omega-chains of approximations in the free algebra solving recursive equations in the iterative one.