We prove quantifier elimination for the field ${\Bbb Q}_{p}((t^{{\Bbb Q}}))$ (the completion of the field of Puiseux series over ${\Bbb Q}_{p}$ ) in Macintyre's language together with symbols for functions in a class containing both t-adically and p-adically overconvergent functions. We also show that the theory of ${\Bbb Q}_{p}((t^{{\Bbb Q}}))$ is b-minimal in this language.

Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove (...) a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $\mathbb {Q}_p$, we show that $G^0 = G^{00}$, and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb {Q}_p$. (shrink)

We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of “admissible topologies” with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kn. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable (...) group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over Qp is definably isomorphic to a definable group. (shrink)

The aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$, with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$. We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$, and (...) the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context.In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$, so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$. This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case. (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)

Let G be a definable group in a p-adically closed field M. We show that G has finitely satisfiable generics ( fsg $\text{fsg}$ ) if and only if G is definably compact. The case M = Q p $M = \mathbb {Q}_p$ was previously proved by Onshuus and Pillay.

We study a reduct ${\mathcal{L}_*}$ of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the ${\mathcal{L}_*}$ -definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K, ${\mathcal{L}_*}$ ) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells (...) similar to p-adic semi-algebraic cells. From this we can derive quantifier-elimination, and give a characterization of definable functions. In particular, we conclude that multiplication can only be defined on bounded sets, and we consider the existence of definable Skolem functions. (shrink)

It is known Hrushovski and Pillay that a group G definable in the field \ of p-adic numbers is definably locally isomorphic to the group \\) of p-adic points of a algebraic group H over \. We observe here that if H is commutative then G is commutative-by-finite. This shows in particular that any one-dimensional group G definable in \ is commutative-by-finite. This result extends to groups definable in p-adically closed fields. We prove our results in the (...) more general context of geometric structures. (shrink)

In [16], Peterzil and Steinhorn proved that if a group G definable in an o-minimal structure is not definably compact, then G contains a definable torsion-free subgroup of dimension 1. We prove here a p-adic analogue of the Peterzil–Steinhorn theorem, in the special case of abelian groups. Let G be an abelian group definable in a p-adically closed field M. If G is not definably compact then there is a definable subgroup H of dimension 1 which is not (...) definably compact. In a future paper we will generalize this to non-abelian G. (shrink)

We generalize two of our previous results on abelian definable groups in p-adically closed fields [12, 13] to the non-abelian case. First, we show that if G is a definable group that is not definably compact, then G has a one-dimensional definable subgroup which is not definably compact. This is a p-adic analogue of the Peterzil–Steinhorn theorem for o-minimal theories [16]. Second, we show that if G is a group definable over the standard model $\mathbb {Q}_p$, then (...) $G^0 = G^{00}$. As an application, definably amenable groups over $\mathbb {Q}_p$ are open subgroups of algebraic groups, up to finite factors. We also prove that $G^0 = G^{00}$ when G is a definable subgroup of a linear algebraic group, over any model. (shrink)

Let p be prime number, K be a p‐adically closed field, a semi‐algebraic set defined over K and the lattice of semi‐algebraic subsets of X which are closed in X. We prove that the complete theory of eliminates quantifiers in a certain language, the ‐structure on being an extension by definition of the lattice structure. Moreover it is decidable, contrary to what happens over a real closed field for. We classify these ‐structures up to elementary equivalence, (...) and get in particular that the complete theory of only depends on m, not on K nor even on p. As an application we obtain a classification of semi‐algebraic sets over countable p‐adically closed fields up to so‐called “pre‐algebraic” homeomorphisms. (shrink)

We give an answer to the question as to whether quantifier elimination is possible in some infinite algebraic extensions of Qp (‘infinite p-adic fields’) using a natural language extension. The present paper deals with those infinite p-adic fields which admit only tamely ramified algebraic extensions (so-called tame fields). In the case of tame fields whose residue fields satisfy Kaplansky’s condition of having no extension of p-divisible degree quantifier elimination is possible when the language of valued (...) fields is extended by the power predicates Pn introduced by Macintyre and, for the residue field, further predicates and constants. For tame infinite p-adic fields with algebraically closed residue fields an extension by Pn predicates is sufficient. (shrink)

In [12], P. Scowcroft and L. van den Dries proved a cell decomposition theorem for p-adically closed fields. We work here with the notion of P-minimal fields defined by D. Haskell and D. Macpherson in [6]. We prove that a P-minimal field K admits cell decomposition if and only if K has definable selection. A preprint version in French of this result appeared as a prepublication [8].

Abstract.In this paper we show that the theory of fields together with an integrally closed subring, the theory of formally real fields with a real holomorphy ring and the theory of formally \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $p$\end{document}-adic fields with a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $p$\end{document}-adic holomorphy ring have no model companions in the language of fields augmented by a unary predicate for the corresponding (...) ring. (shrink)

We study function fields over p-adically closed fields in the first-order language of fields. Using ideas of Duret [D], we show that the field of constants is definable, and that the genus is an elementary property.

We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and p-adically closed valued fields. We give an effectiveness condition, related to Hensel’s lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally p-adic field which does not embed into any computable p-adic closure, but we give an effectiveness condition on the divisibility relation (...) in the value group which is sufficient to find such an embedding. By checking that algebraically closed valued fields and p-adically closed valued fields of infinite transcendence degree have the Mal’cev property, we show that they have computable dimension \. (shrink)

We study first-order properties of the quotient rings C(V)/P by a prime ideal P, where C(V) is the ring of p-adic valued continuous definable functions on some affine p-adic variety V. We show that they are integrally closed Henselian local rings, with a p-adically closed residue field and field of fractions, and they are not valuation rings in general but always satisfy ∀ x, y(x|y 2 ∨ y|x 2 ).

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively (...) saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA. (shrink)

In this article we present a p-adic valued probabilistic logic equation image which is a complete and decidable extension of classical propositional logic. The key feature of equation image lies in ability to formally express boundaries of probability values of classical formulas in the field equation image of p-adic numbers via classical connectives and modal-like operators of the form Kr, ρ. Namely, equation image is designed in such a way that the elementary probability sentences Kr, ρα actually do have their (...) intended meaning—the probability of propositional formula α is in the equation image-ball with the center r and the radius ρ. Due to modal nature of the operators Kr, ρ, it was natural to use the probability Kripke like models as equation image-structures, provided that probability functions range over equation image instead of equation image or equation image. (shrink)

Let and T be a set‐valued map from E to. We prove that if T is p‐adic semi‐algebraic, lower semi‐continuous and is closed for every, then T has a p‐adic semi‐algebraic continuous selection. In addition, we include three applications of this result. The first one is related to Fefferman's and Kollár's question on existence of p‐adic semi‐algebraic continuous solution of linear equations with polynomial coefficients. The second one is about the existence of p‐adic semi‐algebraic continuous extensions of continuous functions. (...) The other application is on the characterization of right invertible p‐adic semi‐algebraic continuous functions under the composition. (shrink)

We introduce a very weak language L M on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language L M are trivial functions. We also give a definitional expansion $L\begin{array}{*{20}{c}} ' \\ M \\ \end{array} $ of L M in which K has quantifier (...) elimination, and we obtain a cell decomposition result for L M -definable sets. Our language L M can serve as a p-adic analogue of the very weak language (<) on the real numbers, to define a notion of minimality on the field of p-adic numbers and on related valued fields. These fields are not necessarily Henselian and may have positive characteristic. (shrink)

Let us call an integer part of an ordered field any subring such that every element of the field lies at distance less than 1 from a unique element of the ring. We show that every real closed field has an integer part.

Let be the field of p‐adic numbers in the language of rings. In this paper we consider the theory of expanded by two predicates interpreted by multiplicative subgroups and where are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if α and β have positive p‐adic valuation. If either α or β has zero valuation we show that the theory of has the NIP (“negation of the independence property”) and therefore does not interpret Peano arithmetic. (...) In that case we also prove that the theory is decidable if and only if the theory of is decidable. (shrink)

This thesis is concerned with the expansions of algebraic structures and their fit in Shelah’s classification landscape.The first part deals with the expansion of a theory by a random predicate for a substructure model of a reduct of the theory. Let T be a theory in a language $\mathcal {L}$. Let $T_0$ be a reduct of T. Let $\mathcal {L}_S = \mathcal {L}\cup \{S\}$, for S a new unary predicate symbol, and $T_S$ be the $\mathcal {L}_S$ -theory that axiomatises the (...) following structures: $$ consist of a model $\mathscr {M}$ of T and S is a predicate for a model $\mathscr {M}_0$ of $T_0$ which is a substructure of $\mathscr {M}$. We present a setting for the existence of a model-companion $TS$ of $T_S$. As a consequence, we obtain the existence of the model-companion of the following theories, for $p>0$ a prime number: • $\mathrm {ACF}_p$, $\mathrm {SCF}_{e,p}$, $\mathrm {Psf}_p$, $\mathrm {ACFA}_p$, $\mathrm {ACVF}_{p,p}$ in appropriate languages expanded by arbitrarily many predicates for additive subgroups;• $\mathrm {ACF}_p$, $\mathrm {ACF}_0$ in the language of rings expanded by a single predicate for a multiplicative subgroup;• $\mathrm {PAC}_p$ -fields, in an appropriate language expanded by arbitrarily many predicates for additive subgroups.From an independence relation in T, we define independence relations in $TS$ and identify which properties of are transferred to those new independence relations in $TS$, and under which conditions. This allows us to exhibit hypotheses under which the expansion from T to $TS$ preserves $\mathrm {NSOP}_{1}$, simplicity, or stability. In particular, under some technical hypothesis on T, we may draw the following picture : Configuration $T_0\subseteq T$ Generic expansion $TS$ $T_0 = T$ Preserves stability $T_0\subseteq T$ Preserves $\mathrm {NSOP}_{1}$ $T_0 = \emptyset $ Preserves simplicityIn particular, this construction produces new examples of $\mathrm {NSOP}_{1}$ not simple theories, and we study in depth a particular example: the expansion of an algebraically closed field of positive characteristic by a generic additive subgroup. We give a full description of imaginaries, forking, and Kim-forking in this example.The second part studies expansions of the group of integers by p-adic valuations. We prove quantifier elimination in a natural language and compute the dp-rank of these expansions: it equals the number of independent p-adic valuations considered. Thus, the expansion of the integers by one p-adic valuation is a new dp-minimal expansion of the group of integers. Finally, we prove that the latter expansion does not admit intermediate structures: any definable set in the expansion is either definable in the group structure or is able to “reconstruct” the valuation using only the group operation.prepared by Christian d’Elbée.E-mail: [email protected]: https://choum.net/~chris/page_perso. (shrink)

As a continuation of ideas initiated in [19], we study bi-colored (generic) expansions of geometric theories in the style of the Fraïssé–Hrushovski construction method. Here we examine that the properties $NTP_{2}$, strongness, $NSOP_{1}$, and simplicity can be transferred to the expansions. As a consequence, while the corresponding bi-colored expansion of a red non-principal ultraproduct of p-adic fields is $NTP_{2}$, the expansion of algebraically closed fields with generic automorphism is a simple theory. Furthermore, these theories are strong with (...) $\operatorname {\mathrm {bdn}}(\text {"}x=x\text {"})=(\aleph _0)_{-}$. (shrink)

Bounded PAC substructures of models of stable theory T are generalizations of bounded PAC fields and bounded PAC beautiful pairs generalize Poizat's beautiful pairs. Both notions were introduced in the authors Ph.D. thesis. In this paper, we prove that under the assumption that the PAC property is first order for T, the theory of any bounded PAC structure is simple. Moreover, if the PAC property is first order for T and T does not have the finite cover property, then (...) the theory of any bounded PAC beautiful pair is simple. We, also, give a characterization of dividing in both cases. (shrink)

This recent addition to the Studies in Logic series is a systematic treatise on the set-theoretic, or semantic, approach to mathematical logic and axiomatic method. The basic notions for the discussion are those of different kinds of languages, their realizations, and the models of a formula. The book begins with a preliminary "chapter 0," giving some general theorems about classes of functions defined by finite schemas. These results are directly applicable to the language of truth-functional propositional logic, and such application (...) is accomplished in detail in chapter 1, in which interpolation, finiteness and compactness theorems are proved for propositional calculus. Chapters 2 and 3 deal with the predicate calculus without and with identity, respectively. More advanced topics dealt with in subsequent chapters include elimination of quantifiers. Certain algebraic structures such as real closed fields and some Boolean rings are given in the form of axiomatic systems in which every formula is equivalent to a quantifier-free formula. This leads to completeness. Also covered are type theory and alternate methods for developing predicate logic, the theory of definability, and the theory of principal models and infinite formulas. Appendices deal with some additional matters of philosophic interest, including a detailed discussion of set-theoretic vs. combinatorial foundations of mathematics. The book lacks an index and a bibliography, but these are not serious defects. This is a highly rewarding work.—H. P. K. (shrink)

The generalization properties of algebraically closed fields $ACF_p$ of characteristic $p > 0$ and $ACF_0$ of characteristic 0 are investigated in the sequent calculus with blocks of quantifiers. It is shown that $ACF_p$ admits finite term bases, and $ACF_0$ admits term bases with primality constraints. From these results the analogs of Kreisel's Conjecture for these theories follow: If for some $k$ , $A(1 + \cdots + 1)$ ( $n$ 1's) is provable in $k$ steps, then $(\forall x)A(x)$ is (...) provable. (shrink)

They have watched, as insiders, the first fumbling attempts to transplant kidneys, then hearts, then live‐donated lobes of liver and lung. Now the two sociologists most closely identified with organ transplantation have concluded that they must leave the field.

Using the Lascar inequalities, we show that any finite rank δ-closed subset of a quasiprojective variety is definably isomorphic to an affine δ-closed set. Moreover, we show that if X is a finite rank subset of the projective space P n and a is a generic point of P n , then the projection from a is injective on X. Finally we prove that if RM = RC in DCF 0 , then RM = RU.

Tom Andersen’s reflecting team process, which allowed families to witness and respond to the talk of professionals during therapy sessions, has been described as revolutionary in the field of family therapy. Reflecting teams are prominent in a number of family therapy approaches, more recently in narrative and dialogical therapies. This way of working is considered more a philosophy than a technique, and has been received positively by both therapists and service users. This paper describes how dialogical therapists conceptualise the reflective (...) process, how they work to engage families in reflective dialogues and how this supports change. We conducted semi-structured, reflective interviews with 12 dialogical therapists with between 2 and 20 years of experience. Interpretative Phenomenological analysis of transcribed interviews identified varying conceptualisations of the reflecting process and descriptions of therapist actions that support reflective talk among network members. We adopted a dialogical approach to interpretation of this data. In this sense, we did not aim to condense accounts into consensus but instead to describe variations and new ways of understanding dialogical reflecting team practices. Four themes were identified: Lived experience as expertise; Listening to the self and hearing others; Relational responsiveness and fostering connection; and Opening space for something new. We applied these themes to psychotherapy process literature both within family therapy literature and more broadly to understand more about how reflecting teams promote helpful and healing conversations in practice. (shrink)

This paper explores the gray area of questionable research practices (QRPs) between responsible conduct of research and severe research misconduct in the form of fabrication, falsification, and plagiarism (Steneck in SEE 12(1): 53–57, 2006). Up until now, we have had very little knowledge of disciplinary similarities and differences in QRPs. The paper is the first systematic account of variances and similarities. It reports on the findings of a comprehensive study comprising 22 focus groups on practices and perceptions of QRPs across (...) main areas of research. The paper supports the relevance of the idea of epistemic cultures (Knorr Cetina in: Epistemic cultures: how the sciences make knowledge, Harvard University Press, Cambridge, 1999), also when it comes to QRPs. It shows which QRPs researchers from different areas of research (humanities, social sciences, medical sciences, natural sciences, and technical sciences) report as the most severe and prevalent within their fields. Furthermore, it shows where in the research process these self-reported QRPs can be found. This is done by using a five-phase analytical model of the research process (idea generation, research design, data collection, data analysis, scientific publication and reporting). The paper shows that QRPs are closely connected to the distinct research practices within the different areas of research. Many QRPs can therefore only be found within one area of research, and QRPs that cut across main areas often cover relatively different practices. In a few cases, QRPs in one area are considered good research practice in another. (shrink)

This work, which first appeared in 1936, offers in addition to an historical treatment displaying Cassirer's characteristic insight, an analysis of quantum mechanics largely unaffected by subsequent development in the field. The author argues, on the basis of epistemological considerations, that quantum mechanics necessitates no major revisions in our basic understanding of causality. The new laws simply refer to "definite collectives" rather than things or events and are no less determinate than the old. In the final part the author stresses (...) the independence of causality and continuity in nature and closes by sensibly warning the reader against attempting to establish ethical freedom within the gaps of physical law. Henry Margenau has expanded the bibliography and added a helpful preface which, in part, reports Cassirer's thoughts up to 1945.--R. P. (shrink)