Abstract

Ordinal functions may be iterated transfinitely in a natural way by taking pointwise limits at limit stages. However, this has disadvantages, especially when working in the class of normal functions, as pointwise limits do not preserve normality. To this end we present an alternative method to assign to each normal function f a family of normal functions Hyp[f]=〈fξ〉ξ∈OnHyp[f]=〈fξ〉ξ∈On, called its hyperation, in such a way that f0=idf0=id, f1=ff1=f and fα+β=fα∘fβfα+β=fα∘fβ for all α, β.Hyperations are a refinement of the Veblen hierarchy of f. Moreover, if f is normal and has a well-behaved left-inverse g called a left adjoint, then g can be assigned a cohyperationcoH[g]=〈gξ〉ξ∈OncoH[g]=〈gξ〉ξ∈On, which is a family of initial functions such that gξgξ is a left adjoint to fξfξ for all ξ