This paper contributes to the theory of recursively presented (see Higman [Subgroups of finitely presented groups, Proc. R. Soc. Ser. A 262 (1961) 455–475]) infinitely generated abelian groups with solvable word problem. Mal'cev [On recursive Abelian groups, Dokl. Akad. Nauk SSSR 146 (1962) 1009–1012] and independently Rabin [Computable algebra, general theory and theory of computable fields, Trans. Amer. Math. Soc. 95 (1960) 341–360] initiated the study of such groups in the early 1960's. In this paper, we develop a technique that we call iterated effective embeddings. The significance of our new technique is that it extends existing methods from the realm of iterated 0″ arguments to iterated 0‴ ones. This is a new phenomenon in computable algebra. We use this technique to confirm a 30 year-old conjecture of Ash, Knight and Oates [Recursive abelian p-groups of small length, https://dl.dropbox.com/u/4752353/Homepage/AKO.pdf ]. More specifically, Ash, Knight and Oates [Recursive abelian p-groups of small length. https://dl.dropbox.com/u/4752353/Homepage/AKO.pdf ] conjectured that there exists a computable reduced abelian p-group of Ulm type ω such that its effective invariants, defined using limitwise monotonic functions, cannot be found uniformly. We construct a computable reduced abelian p-group of Ulm type ω where its invariants are at the maximum possible level of non-uniformity. The result confirms the conjecture in a strong way, and it provides us with an explanation of why computable reduced p-groups of Ulm type ω seem hard to classify in general. We also use p-basic trees and their iterated embeddings to solve a problem posed in [W. Calvert, D. Cenzer, V. S. Harizanov and A. Morozov, Effective categoricity of abelian p-groups, Ann. Pure Appl. Logic 159(1–2) (2009) 187–197].
LIMITWISE MONOTONIC FUNCTIONS AND THEIR APPLICATIONS
