A Characterization of Strongly Dependent Ordered Abelian Groups

We characterize all ordered Abelian groups whose ﬁrst-order theory in the language {+, <, 0} is strongly dependent. The main result of this note was obtained independently by Halevi and Hasson [7] and Farré [5].

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PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALS

Journal of Symbolic Logic ◽

10.1017/jsl.2015.4 ◽

2015 ◽

Vol 80 (2) ◽

pp. 433-449 ◽

Cited By ~ 10

Author(s):

KEVIN WOODS

Keyword(s):

Generating Functions ◽

Counting Function ◽

Order Theory ◽

Characteristic Functions ◽

Presburger Arithmetic ◽

First Order ◽

First Order Theory ◽

Rational Generating Function ◽

Rational Generating Functions

AbstractPresburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, ifp= (p1, . . . ,pn) are a subset of the free variables in a Presburger formula, we can define a counting functiong(p) to be the number of solutions to the formula, for a givenp. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.

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LOGICAL CHARACTERIZATION OF RECOGNIZABLE SETS OF POLYNOMIALS OVER A FINITE FIELD

International Journal of Foundations of Computer Science ◽

10.1142/s0129054111008878 ◽

2011 ◽

Vol 22 (07) ◽

pp. 1549-1563 ◽

Cited By ~ 3

Author(s):

MICHEL RIGO ◽

LAURENT WAXWEILER

Keyword(s):

Finite Field ◽

Order Theory ◽

Ring Of Integers ◽

First Order ◽

Polynomial Coefficients ◽

Bounded Number ◽

First Order Theory ◽

Ring Of Polynomials

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination of powers of a non-constant polynomial P playing the role of the base of the numeration. Having in mind the theorem of Cobham from 1969 about recognizable sets of integers, it is natural to study P-recognizable sets of polynomials. Based on the results obtained in the Ph.D. thesis of the second author, we study the logical characterization of such sets and related properties like decidability of the corresponding first-order theory.

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NIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES

Journal of Symbolic Logic ◽

10.1017/jsl.2016.59 ◽

2017 ◽

Vol 82 (1) ◽

pp. 35-61 ◽

Cited By ~ 1

Author(s):

ALLEN GEHRET

Keyword(s):

Order Theory ◽

Exchange Property ◽

Independence Property ◽

First Order ◽

Valued Field ◽

First Order Theory ◽

Order Language ◽

Complete Survey ◽

Image Position

AbstractThe derivation on the differential-valued field Tlog of logarithmic transseries induces on its value group ${{\rm{\Gamma }}_{{\rm{log}}}}$ a certain map ψ. The structure ${\rm{\Gamma }} = \left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ is a divisible asymptotic couple. In [7] we began a study of the first-order theory of $\left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ where, among other things, we proved that the theory $T_{{\rm{log}}} = Th\left( {{\rm{\Gamma }}_{{\rm{log}}} ,\psi } \right)$ has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether Tlog has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: Tlog does have NIP. Our method of proof relies on a complete survey of the 1-types of Tlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions ${\rm{\Gamma }}\left\langle \alpha \right\rangle$ of ${\rm{\Gamma }}$. We also show that Tlog does not have the Steinitz exchange property and we weigh in on the relationship between models of Tlog and the so-called precontraction groups of [9].

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The elementary class of products of totally ordered abelian group

Journal of Symbolic Logic ◽

10.2307/2274920 ◽

1991 ◽

Vol 56 (1) ◽

pp. 295-299 ◽

Cited By ~ 1

Author(s):

Daniel Gluschankof

Keyword(s):

Abelian Groups ◽

Algebraic Theory ◽

Unary Operation ◽

Order Theory ◽

Direct Products ◽

First Order ◽

First Order Theory ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

General Goal

A basic goal in model-theoretic algebra is to obtain the classification of the complete extensions of a given (first-order) algebraic theory.Results of this type, for the theory of totally ordered abelian groups, were obtained first by A. Robinson and E. Zakon [5] in 1960, later extended by Yu. Gurevich [4] in 1964, and further clarified by P. Schmitt in [6].Within this circle of ideas, we give in this paper an axiomatization of the first-order theory of the class of all direct products of totally ordered abelian groups, construed as lattice-ordered groups (l-groups)—see the theorem below. We think of this result as constituing a first step—undoubtedly only a small one—towards the more general goal of classifying the first-order theory of abelian l-groups.We write groups for abelian l-groups construed as structures in the language 〈 ∨, ∧, +, −, 0〉 (“−” is an unary operation). For unproved statements and unexplicated definitions, the reader is referred to [1].

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