Hyperations, Veblen progressions and transfinite iteration of ordinal functions

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∈On, called its hyperation, in such a way that f0=id, f1=f and fα+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 cohyperation coH[g]=〈gξ〉ξ∈On, which is a family of initial functions such that gξ is a left adjoint to fξ for all ξ.