Infinite Abelian groups

Classification theory of abelian groups, II: Local theory

Two lines of research are involved here. One is a combinatorial principle, proved in ZFC for many cardinals (e.g., any λ = λℵ 0) enabling us to prove things which have been proven using the diamond or for strong limit cardinals of uncountable cofinality. The other direction is building abelian groups with few endomorphisms and/or prescribed rings of endomorphisms. We prove that for a ringR, whose additive group is thep-adic completion of a freep-adic module,R is isomorphic to the endomorphism ring of some separable abelianp-groupG divided by the ideal of small endomorphisms, withG of power λ for any λ = λℵ 0≧|R|.

A combinatorial principle and endomorphism rings I: Onp-groups

We show that algebraic analogues of universal group covers, surjective group homomorphisms from a $\mathbb{Q}$-vector space to $F^{\times}$ with "standard kernel", are determined up to isomorphism of the algebraic structure by the characteristic and transcendence degree of $F$ and, in positive characteristic, the restriction of the cover to finite fields. This extends the main result of "Covers of the Multiplicative Group of an Algebraically Closed Field of Characteristic Zero" (B. Zilber, JLMS 2007), and our proof fills a hole in the proof given there.

Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic

In this paper we prove that the pseudovariety of Abelian groups is hyperdecidable and moreover that it is completely tame. This is a consequence of the fact that a system of group equations on a free Abelian group with certain rational constraints is solvable if and only if it is solvable in every finite quotient.

Tameness of the pseudovariety of abelian groups

We introduce a notion of separativeness for positively ordered monoids (P.O.M.'s), similar in definition to the notion of separativeness for commutative semigroups but which has a simple categorical equivalent, weaker that injectivity, the transfer property. We show that existence in a separative extension of the ground P.O.M. of a solution of a given linear system is equivalent to the satisfaction by the ground P.O.M. of a certain set of equations and inequations, the resolvent. We deduce in particular a characterization of the P.O.M.'s which are infective relatively to the class of embeddings of countable P.O.M.'s; those include in particular divisible weak-cardinal algebras. We also deduce that finitely additive positive non-standard measures invariant relatively to a given exponentially bounded group separate equidecomposability types modulo this group.

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Infinite Abelian groups

Citations

Classification theory of abelian groups, II: Local theory

A combinatorial principle and endomorphism rings I: Onp-groups

Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic

Tameness of the pseudovariety of abelian groups

Restricted injectivity, transfer property and decompositions of separative positively ordered monoids.

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