In this paper, we give a complete algebraic description of groups elementarily equivalent to the P. Hall completion of a given free nilpotent group of finite rank over an arbitrary binomial domain. In particular, we characterize all groups elementarily equivalent to a free nilpotent group of finite rank.

In this paper we prove that in modules, MS-measurability depends on being able to define a measure function on the p.p. definable subgroups. We give a classification of abelian groups in terms of measurability. Finally we discuss the relation with Q[t]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}[t]}$$\end{document} -valued measures.

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Läuchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and (...) definability theory. Here we give a unified presentation, including some new results, of how to use the Feferman–Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL . We then extend the technique to graph polynomials where the range of the summation of the monomials is definable in MSOL . Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well. (shrink)

We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all V-modules is decidable.

Tarski's model-theoretic truth definition of the 1950s differs from his 1930s truth definition by allowing the language to have a set of parameters that are interpreted by means of structures. The paper traces how the model-theoretic theorems that Tarski and others were proving in the period between these two truth definitions became increasingly difficult to fit into the framework of the earlier truth definition, making the later one more or less inevitable. The paper also maintains that neither recursiveness nor satisfaction (...) are essential features of the truth definition . The recursive form was a ‘practical’ step towards the explicit definition that was Tarski's target. There are model-theoretic languages for which satisfaction is provably too crude a notion for expressing those properties of formulas that are needed for determining the truth of sentences. (shrink)

Let G be a finite group, T denote the theory of Z[G]-lattices . It is shown that T is undecidable when there are a prime p and a p-subgroup S of G such that S is cyclic of order p4, or p is odd and S is non-cyclic of order p2, or p = 2 and S is a non-cyclic abelian group of order 8 . More precisely, first we prove that T is undecidable because it interprets the word problem (...) for finite groups; then we lift undecidability from T to T. (shrink)

We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable \ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are “computable d-\” sentence and a (...) computable \ sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank 1. These are exactly the additive subgroups of \. We show that for some of these groups, the computable \ Scott sentence is best possible, while for others, there is a computable d-\ Scott sentence. (shrink)

We study definable groups in dense/codense expansions of geometric theories with a new predicate P such as lovely pairs and expansions of fields by groups with the Mann property. We show that in such expansions, large definable subgroups of groups definable in the original language \ are also \-definable, and definably amenable \-definable groups remain amenable in the expansion. We also show that if the underlying geometric theory is NIP, and G is a group definable in a model of T, (...) then the connected component \ of G in the expansion agrees with the connected component \ in the original language. We prove similar preservation results for \^n\), the P-part of G in a lovely pair, and for subgroups of G in pairs where R is a real closed field and G is a subgroup of \ or the unit circle \\) with the Mann property. (shrink)

Let n be a positive integer and FAℓ be the free abelian lattice-ordered group on n generators. We prove that FAℓ and FAℓ do not satisfy the same first-order sentences in the language if m≠n. We also show that is decidable iff n{1,2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.