Algebras with parametrized iterativity

Iterative algebras, as studied by Nelson and Tiuryn, are generalized to algebras whose iterativity is parametrized in the sense that only some variables can be used for iteration. For example, in the case of one binary operation, the free iterative algebra is the algebra of all rational binary trees; if only the left-hand variable is allowed to be iterated, then the free iterative algebra is the algebra of all right-well-founded rational binary trees. In order to express such parametrized iterativity, we work with parametrized endofunctors of , i.e. finitary endofunctors , and introduce the concept of iterativity for algebras for the endofunctor X↦H(X,X). We then describe free iterative H-algebras.