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].

Infinite Abelian groups

Iterated effective embeddings of abelian p-groups

In this paper, a semigroup algebra consisting of polynomial expressions with coefficients in a field $F$ and exponents in an additive submonoid $M$ of $\mathbb{Q}_{\ge 0}$ is called a Puiseux algebra and denoted by $F[M]$. Here we study the atomic structure of Puiseux algebras. To begin with, we answer the Isomorphism Problem for the class of Puiseux algebras, that is, we show that for a field $F$ if two Puiseux algebras $F[M_1]$ and $F[M_2]$ are isomorphic, then the monoids $M_1$ and $M_2$ are also isomorphic. Then we construct three classes of Puiseux algebras satisfying the following well-known atomic properties: the ACCP property, the bounded factorization property, and the finite factorization property. Finally, we give a full description of the seminormal closure, root closure, and complete integral closure of a Puiseux algebra, and use such description to provide a class of antimatter Puiseux algebras (i.e., Puiseux algebras containing no irreducibles).

Atomic and antimatter semigroup algebras with rational exponents

The čech homology of compact connected abelian topological groups with applications to shape theory

It is well known that the category of finite groups has no non-trivial injective objects. In general, a group is said to be quasi-injective if for every subgroup H of G and homomorphism f:H → G there exists an endomorphism F:G → G such that F | H = G . In other words, a group is quasi-injective whenever each homomorphism from a subgroup into the group can be extended to the whole group.

/pdf/finite-quasi-injective-groups-44pppa3iaj.pdf

Finite quasi-injective groups

A characteristic result is that if 2 ℵ 0 < µ < µ + < λ = cf(λ) < µ ℵ 0 , then among the separable reduced p-groups of cardinality λ which are (< λ)-stable there is no universal one.

Infinite Abelian groups

Citations

Iterated effective embeddings of abelian p-groups

Atomic and antimatter semigroup algebras with rational exponents

The čech homology of compact connected abelian topological groups with applications to shape theory

Finite quasi-injective groups

UNIVERSAL IN (<λ)-STABLE ABELIAN GROUP

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