2012/05/09 by David Fernández–Duque, Joost J. Joosten, Fernández-Duque, David +1 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Logic, programming, and type systems #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1205.2036
openalex publication_date 2012/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce hyperations and cohyperations, which are forms of\ntransfinite iteration of ordinal functions.\n Hyperations are iterations of normal functions. Unlike iteration by pointwise\nconvergence, hyperation preserves normality. The hyperation of a normal\nfunction f is a sequence of normal functions so that f0= id, f1 = f and for\nall ordinals \α, \β we have that f^(\α + \β) = f^\α f^\β.\nThese conditions do not determine f^\α uniquely; in addition, we require\nthat the functions be minimal in an appropriate sense. We study hyperations\nsystematically and show that they are a natural refinement of Veblen\nprogressions.\n Next, we define cohyperations, very similar to hyperations except that they\nare left-additive: given \α, \β, f^(\α + \β)= f^\β f^\α.\nCohyperations iterate initial functions which are functions that map initial\nsegments to initial segments. We systematically study cohyperations and see how\nthey can be employed to define left inverses to hyperations.\n Hyperations provide an alternative presentation of Veblen progressions and\ncan be useful where a more fine-grained analysis of such sequences is called\nfor. They are very amenable to algebraic manipulation and hence are convenient\nto work with. Cohyperations, meanwhile, give a novel way to describe slowly\nincreasing functions as often appear, for example, in proof theory.\n