Iterated effective embeddings of abelian p-groups

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, sets, and degrees on computable domains

We extend the notion of limitwise monotonic functions to include arbitrary computable domains. We then study which sets and degrees aresupport increasing (support strictly increasing)limitwise monotonic on various computable domains. As applications, we provide a characterization of the setsSwith computableincreasing η-representationsusing support increasing limitwise monotonic sets on ℚ and note relationships between the class oforder-computablesets and the class of support increasing (support strictly increasing) limitwise monotonic sets on certain domains.

Bounding prime models

A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model of T decidable in X. It is easy to see that X = 0′ is prime bounding. Denisov claimed that every X <T 0′ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets X ≤τ 0′ are exactly the sets which are not low2. Recall that X is low2 if X″ ≤τ 0″. To prove that a low2 set X is not prime bounding we use a 0′ -computable listing of the array of sets {Y : Y ≤τX } to build a CAD theory T which diagonalizes against all potential X-decidable prime models of T, To prove that any non-low2X is indeed prime bounding. we fix a function f ≤TX that is not dominated by a certain 0′-computable function that picks out generators of principal types. Given a CAD theory T. we use f to eventually find, for every formula φ(x̄) con sistent with T. a principal type which contains it. and hence to build an X-decidable prime model of T. We prove the prime bounding property equivalent to several other combinatorial properties, including some related to the limitwise monotonic functions which have been introduced elsewhere in computable model theory.