Iterated effective embeddings of abelian p-groups

The paper contributes to the theory of recursively presented (see Higman [22]) infinitely generated abelian groups with solvable word problem. Mal’cev [35] and independently Rabin [39] initiated the study of such groups in the early 1960’s. In the paper we develop a technique that we call iterated effective embeddings. The significance of our new technique is that it extends the iteration technique from the realm of iterated 0′′ arguments to iterated 0′′′ ones. This is a new phenomenom in computable algebra. As an illustration, we use this technique to confirm and extend a 30 year old conjecture of Ash, Knight and Oates [2]. More specifically, Ash, Knight and Oates [2] conjectured that there exists a reduced abelian p-group of Ulm type ω such that its effective invariants, limitwise monotonic functions, are not uniform. We construct a computable reduced abelian p-group of Ulm type ω having its invariants, limitwise monotonic functions, not only non-uniform but at the maximal potentially possible level of non-uniformity. The result confirms the conjecture in a strong way, and it also 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 [4].