On the behaviour of coalgebras with side effects and algebras with effectful iteration

For every finitary monad T on sets and every endofunctor F on the category of T-algebras, we introduce the concept of an ffg-Elgot algebra for F, i.e. an algebra admitting coherent solutions for finite systems of recursive equations with effects represented by the monad T. The goal is to study the existence and construction of free ffg-Elgot algebras. To this end, we investigate the locally ffg fixed point phi(F), i.e. the colimit of all F-coalgebras with free finitely generated carrier, which is shown to be the initial ffg-Elgot algebra. This is the technical foundation for our main result: the category of ffg-Elgot algebras is monadic over the category of T-algebras.