Let${\mathbb K}$be an algebraically bounded structure, and letTbe its theory. IfTis model complete, then the theory of${\mathbb K}$endowed with a derivation, denoted by$T^{\delta }$, has a model completion. Additionally, we prove that if the theoryTis stable/NIP then the model completion of$T^{\delta }$is also stable/NIP. Similar results hold for the theory with several derivations, either commuting or non-commuting.

Pre-H-fields are ordered valued differential fields satisfying some basic axioms coming from transseries and Hardy fields. We study pre-H-fields that are differential-Hensel–Liouville closed, that is, differential-henselian, real closed, and closed under exponential integration, establishing an Ax–Kochen/Ershov theorem for such structures: the theory of a differential-Hensel–Liouville closed pre-H-field is determined by the theory of its ordered differential residue field; this result fails if the assumption of closure under exponential integration is dropped. In a two-sorted setting with one sort for a differential-Hensel–Liouville (...) closed pre-H-field and one sort for its ordered differential residue field, we eliminate quantifiers from the pre-H-field sort, from which we deduce that the ordered differential residue field is purely stably embedded and if it has NIP, then so does the two-sorted structure. Similarly, the one-sorted theory of differential-Hensel–Liouville closed pre-H-fields with closed ordered differential residue field has quantifier elimination, is the model completion of the theory of pre-H-fields with gap $0$, and is complete, distal, and locally o-minimal. (shrink)

We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

We consider the structure $({\Bbb Z}, +,0,|_{p_1 }, \ldots,|_{p_n } )$, where $x|_p y$ means $v_p \left( x \right) \leqslant v_p \left( y \right)$ and v p is the p-adic valuation. We prove that this structure has quantifier elimination in a natural expansion of the language of abelian groups, and that it has dp-rank n. In addition, we prove that a first order structure with universe ${\Bbb Z}$ which is an expansion of $({\Bbb Z}, +,0)$ and a reduct of $({\Bbb (...) Z}, +,0,|_p )$ must be interdefinable with one of them. We also give an alternative proof for Conant’s analogous result about $({\Bbb Z}, +,0, < )$. (shrink)

In model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example, linear independence in vector spaces yields such a relation.Some important classes in the stability hierarchy are stable, simple, and NSOP $_1$, each being contained in the next. For each of (...) these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example, simplicity is equivalent to admitting a certain independence relation, which must then be unique.All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic, and even a very general category-theoretic framework. In this thesis we continue this work.We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple, or NSOP $_1$ -like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing, and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness.Switching frameworks, we generalise Kim-dividing in NSOP $_1$ theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP $_1$ if and only if there is a nice enough independence relation, which then must be given by Kim-dividing.Abstract prepared by Mark Kamsma.E-mail:[email protected]:https://markkamsma.nl/phd-thesis. (shrink)

We introduce prime products as a generalization of ultraproducts for positive logic. Prime products are shown to satisfy a version of Łoś’s Theorem restricted to positive formulas, as well as the following variant of the Keisler Isomorphism Theorem: under the generalized continuum hypothesis, two models have the same positive theory if and only if they have isomorphic prime powers of ultrapowers.

We prove the existence of a model companion of the two-sorted theory of c-nilpotent Lie algebras over a field satisfying a given theory of fields. We describe a language in which it admits relative quantifier elimination up to the field sort. Using a new criterion which does not rely on a stationary independence relation, we prove that if the field is NSOP $_1$, then the model companion is NSOP $_4$. We also prove that if the field is algebraically closed, then (...) the model companion is c-NIP. (shrink)

We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written φ) if φ is true in some extension of that structure, and φ is necessary (written φ) if it is true in all extensions of the structure. A principal case for us will be the class Mod(T) (...) of all models of a given theory T—all graphs, all groups, all fields, or what have you—considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, k-colorability, finiteness, countability, size continuum, size ℵ1, ℵ2, ℵω, ℶω, first ℶ-fixed point, first ℶ-hyper-fixed point, and much more. A graph obeys the maximality principle φ(a)→φ(a) with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs. (shrink)

Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion implies (...) that the variety has equationally definable principal congruences. This result is then used to provide necessary and sufficient conditions for the existence of a model completion for theories of Hamiltonian varieties of pointed residuated lattices, a broad family of varieties that includes lattice-ordered abelian groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of pointed residuated lattices has a model completion, it must have equationally definable principal congruences. In particular, the theories of lattice-ordered abelian groups and MV-algebras do not have a model completion, as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is shown that certain varieties of pointed residuated lattices generated by their linearly ordered members, including lattice-ordered abelian groups and MV-algebras, can be extended with a binary operation to obtain theories that do have a model completion. (shrink)

We consider structures of the form (Z;+,C), where C is an additive cyclic order on (Z;+). We show that such structures are dp-minimal and in this way produce a continuum-size family of dp-minimal expansions of (Z;+) such that no two members of the family define the same subsets of Z.

ABSTRACT A bilattice is said to be regular provided its truth conjunction and disjunction are monotonic with respect to its knowledge ordering. The principal result of this paper is that the following properties of a bilattice B are equivalent: 1. B is regular; 2. the truth conjunction and disjunction of B are definable through the rest of the operations and constants of B; 3. B is isomorphic to a bilattice of the form L 1 · L 2 where L 1 (...) and L 2 are bounded lattices. We also derive from this metaequivalence a number of corollaries concerning distributive bilattices, degenerated bilattices and bilattices with negation. (shrink)

We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$, as $\kappa (...) $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$ ). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness. (shrink)

We study the theory T m,n of existentially closed incidence structures omitting the complete incidence structure K m,n, which can also be viewed as existentially closed K m,n -free bipartite graphs. In the case m = n = 2, this is the theory of existentially closed projective planes. We give an $\forall \exists$ -axiomatization of T m,n, show that T m,n does not have a countable saturated model when m, n ≥ 2, and show that the existence of a prime (...) model for T2,2 is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model theoretic notions of complexity for T m,n. We show that T m,n is NSOP1, but not simple when m, n ≥ 2, and we show that T m,n has weak elimination of imaginaries but not full elimination of imaginaries. These results rely on combinatorial characterizations of various notions of independence, including algebraic independence, Kim independence, and forking independence. (shrink)

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we also prove (...) a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators. (shrink)

This work presents a model-theoretic approach to the study of the amalgamation property for varieties of semilinear commutative residuated lattices. It is well-known that if a first-order theory T enjoys quantifier elimination in some language L, the class of models of the set of its universal consequences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm T_\forall}$$\end{document} has the amalgamation property. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Th}(\mathbb{K})}$$\end{document} be the theory of an elementary subclass (...) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{K}}$$\end{document} of the linearly ordered members of a variety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{V}}$$\end{document} of semilinear commutative residuated lattices. We show that whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Th}(\mathbb{K})}$$\end{document} has elimination of quantifiers, and every linearly ordered structure in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{V}}$$\end{document} is a model of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Th}_\forall(\mathbb{K})}$$\end{document}, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{V}}$$\end{document} has the amalgamation property. We exploit this fact to provide a purely model-theoretic proof of amalgamation for particular varieties of semilinear commutative residuated lattices. (shrink)

One measure of the complexity of a first-order theory, and similarly a type, is the complexity of the formulas required to axiomatize it. We say that a theory is bounded if there is an axiomatization involving only $\forall _n$ -formulas for some finite n, and unbounded otherwise. One might expect bounded theories to have only bounded types. In fact, an analogue holds in infinitary logic, where the complexity of a Scott sentence roughly agrees with the complexity of the most complicated (...) automorphism orbit. Our main result, however, shows that this is not the case in the first-order setting: Namely, there can be a bounded theory, in fact $\forall _1$ -axiomatizable, which has unbounded types. (shrink)

In this paper, we consider reducts of p-adically closed fields. We introduce a notion of shadows: sets Mf={∈K2∣|y|=|f|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_f = \{ \in K^2 \mid |y| = |f|\}}$$\end{document}, where f is a semi-algebraic function. Adding symbols for such sets to a reduct of the ring language, we obtain expansions of the semi-affine language where multiplication is nowhere definable, thus giving a negative answer to a question posed by Marker, Peterzil and Pillay. The second (...) main result of this paper is the fact that in p-adic fields, full multiplication becomes definable if we add a rational function to the semi-affine language. (shrink)

We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra script upper M $\mathcal {M}$ M is never model complete if its direct integral decomposition contains upper I upper I Subscript 1 $\mathrm {II}_1$ II1 factors script upper N $\mathcal {N}$ N such that upper M 2 left parenthesis script upper N right parenthesis $M_2(\mathcal {N})$ M2(N) embeds into an ultrapower of (...) script upper N $\mathcal {N}$ N. The proof in the case of upper I upper I Subscript 1 $\mathrm {II}_1$ II1 factors uses an explicit construction based on random matrices and quantum expanders. (shrink)

We investigate the existence of “generic derivations” in exponential fields. We show that exponential fields without additional compatibility conditions between derivation and exponentiation cannot support a generic derivation.

As the class PCSL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {PCSL}$$\end{document} of pseudocomplemented semilattices is a universal Horn class generated by a single finite structure it has a ℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _0$$\end{document}-categorical model companion PCSL∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {PCSL}^*$$\end{document}. As PCSL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {PCSL}$$\end{document} is inductive the models of PCSL∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} (...) \begin{document}$$\mathcal {PCSL}^*$$\end{document} are exactly the existentially closed models of PCSL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {PCSL}$$\end{document}. We will construct the unique existentially closed countable model of PCSL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {PCSL}$$\end{document} as a direct limit of algebraically closed pseudocomplemented semilattices. (shrink)

We work in the language of rings augmented by a 1-ary predicate symbol Fin(x) with intended interpretation in a ring as “x is a finite union of atoms” in the Boolean algebra of idempotents of the ring. We exhibit a set of axioms in this language, and prove that any commutative unital ring R satisfying these axioms is elementarily equivalent to a restricted product, over the set of atoms e of R, of connected rings Re. Each connected ring Re is (...) the localization of R at the set of powers of e. This proves a Feferman–Vaught theorem for rings and a converse to the Feferman–Vaught theorem for restricted products of rings. The most important application is to axioms for rings closely resembling adele rings over number fields. Our axioms are inspired by the axioms of D’Aquino and Macintyre for products, and our results are an exact analogue of their results on products which have intriguing applications to nonstandard models of first-order Peano arithmetic. (shrink)

A homogenizable structure M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}$$\end{document} is a structure where we may add a finite number of new relational symbols to represent some ∅-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\emptyset-$$\end{document}definable relations in order to make the structure homogeneous. In this article we will divide the homogenizable structures into different classes which categorize many known examples and show what makes each class important. We will show that model completeness is vital (...) for the relation between a structure and the amalgamation bases of its age and give a necessary and sufficient condition for an ω-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega-$$\end{document}categorical model-complete structure to be homogenizable. (shrink)

We present a relatively simple description of binary, definable subsets of models of weakly quasi-o-minimal theories. In particular, we closely describe definable linear orders and prove a weak version of the monotonicity theorem. We also prove that weak quasi-o-minimality of a theory with respect to one definable linear order implies weak quasi-o-minimality with respect to any other such order.

We develop a notion of cell decomposition suitable for studying weak p‐adic structures (reducts of p‐adic fields where addition and multiplication are not (everywhere) definable). As an example, we consider a structure with restricted addition.

We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks, we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of “generic” used by Akin, Glasner, and Weiss is distinct from the notion of “generic” encountered in Fraïssé (...) theory. (shrink)

We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} of a computably enumerable, model complete theory, the entire elementary diagram E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} must be decidable. We prove that indeed a c.e. theory T is model complete if and only if there is a (...) uniform procedure that succeeds in deciding E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} from the atomic diagram Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta $$\end{document} for all countable models A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} of T. Moreover, if every presentation of a single isomorphism type A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} has this property of relative decidability, then there must be a procedure with succeeds uniformly for all presentations of an expansion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} by finitely many new constants. (shrink)

Generalizing Cooper’s method of quantifier elimination for Presburger arithmetic, we give a new proof that all parametric Presburger families \ [as defined by Woods ] are definable by formulas with polynomially bounded quantifiers in an expanded language with predicates for divisibility by f for every polynomial \. In fact, this quantifier bounding method works more generally in expansions of Presburger arithmetic by multiplication by scalars \: \alpha \in R, t \in X\}\) where R is any ring of functions from X (...) into \. (shrink)

We introduce new first‐order languages for the elementary n‐dimensional geometry and elementary n‐dimensional affine geometry (n ≥ 2), based on extending and, respectively, with new function symbols. Here, β stands for the betweenness relation and ≡ for the congruence relation. We show that the associated theories admit effective quantifier elimination.

We identify the locally finite graphs that are quantifier-eliminable and their first order theories in the signature of distance predicates. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Let R be a real closed field, and X⊆Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\subseteq R^m}$$\end{document} semi-algebraic and 1-dimensional. We consider complete first-order theories of modules over the ring of continuous semi-algebraic functions X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X\to R}$$\end{document} definable with parameters in R. As a tool we introduce -piecewise vector bundles on X and show that the category of piecewise vector bundles on X is equivalent to the category of syzygies of (...) finitely generated submodules of free modules. We give an explicit method to determine the Baur–Monk invariants of free modules in terms of pre-piecewise vector bundles. When R is a recursive real closed field this yields the decidability of the theory of free modules. Where it makes sense, we address the same questions for continuous definable functions in o-minimal expansions of a real closed field. From the free module case we are able to deduce generalisations of some results to arbitrary modules over the ring. We present a geometrically motivated quantifier elimination result down to the level of positive primitive formulae with a certain block decomposition of the matrix of coefficients. (shrink)