We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a (...) model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability. (shrink)

Journal of Mathematical Logic, Ahead of Print. We study classes of modules closed under direct sums, [math]-submodules and [math]-epimorphic images where [math] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [math]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [math]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. (...) We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability. (shrink)

We characterize rings over which every cotorsion module is pure injective in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to (...) coherence results previously known for the special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts. (shrink)

In the model theory of modules the Ziegler spectrum, the space of indecomposable pure-injective modules, has played a key role. We investigate the possibility of defining a similar space in the context of G-sets where G is a group.

We study Martsinkovsky–Russell torsion modules with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pure-injective and the Σ-pure-injective modules relative to the class of torsion modules assuming that the torsion submodule is a pure submodule. Our characterization of relative Σ-pure-injective modules extends the classical characterization of Gruson and Jenson as well as Zimmermann. We study the limit models of the class and determine when the class is (...) superstable assuming that the torsion submodule is a pure submodule. As a corollary, we show that the class of torsion abelian groups with pure embeddings is strictly stable (i.e., stable not superstable). (shrink)

We show that certain classes of modules have universal models with respect to pure embeddings: Let R be a ring, T a first‐order theory with an infinite model extending the theory of R‐modules and (where ⩽pp stands for “pure submodule”). Assume has the joint embedding and amalgamation properties. If or, then has a universal model of cardinality λ. As a special case, we get a recent result of Shelah [28, 1.2] concerning the existence of universal reduced torsion‐free abelian (...) groups with respect to pure embeddings.We begin the study of limit models for classes of R‐modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure‐injective and we characterize limit models with chains of countable cofinality. This can be used to answer [18, Question 4.25].As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both. (shrink)

We consider the decision problem for modules over a group ring ℤ[G], where G is a cyclic group of prime order. We show that it reduces to the same problem for a class of certain abelian structures, and we obtain some partial decidability results for this class.

We characterize the non-modular models of a unidimensional first-order theory of modules as the elementary submodels of its prime pure-injective model. We show that in case the maximal non-modular submodel of a given model splits off this is true for every such submodel, and we thus obtain a cancellation result for this situation. Although the theories in question always have models whose maximal non-modular submodel do split off, they may as well have others where they don't. We present (...) a corresponding example. (shrink)

We use the Drozd-Warfield structure theorem for finitely presented modules over a serial ring to investigate the model theory of modules over a serial ring, in particular, to give a simple description of pp-formulas and to classify the pure-injective indecomposable modules. We also study the question of whether every pure-injective indecomposable module over a valuation ring is the hull of a uniserial module.

We extend the analysis of the decision problem for modules over a group ring ℤ[G] to the case when G is a cyclic group of squarefree order. We show that separated ℤ[G]-modules have a decidable theory, and we discuss the model theoretic role of these modules within the class of all ℤ[G]-modules. The paper includes a short analysis of the decision problem for the theories of modules over ℤ[ζm], where m is a positive integer and ζm is a primitive mth (...) root of 1. (shrink)

Fisher [10] and Baur [6] showed independently in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In [25, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq _p)$ such that K is a class of modules and $\leq _p$ is the pure submodule relation. In this paper we give some instances where this (...) is true:Theorem.Assume R is an associative ring with unity. Let $(K, \leq _p)$ be an AEC such that $K \subseteq R\text {-Mod}$ and K is closed under finite direct sums, then: •If K is closed under pure-injective envelopes, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda \geq \operatorname {LS}(\mathbf {K})$ such that $\lambda ^{|R| + \aleph _0}= \lambda $.•If K is closed under pure submodules and pure epimorphic images, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $.•Assume R is Von Neumann regular. If $\mathbf {K}$ is closed under submodules and has arbitrarily large models, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $.As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, Dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy.Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of $\mathfrak {s}$ -torsion modules. (shrink)

Elementary duality between left and right modules over a ring, especially its interpretation in terms of the relevant functor categories, is discussed, as is the relationship between these categories of functors and sorts in theories of modules. A topology on the set of indecomposable pure-injective modules over a ring is introduced. This topology is dual to the Ziegler topology and may be seen as a generalisation of the Zariski topology.

Minimality ranks in the style of Deissler are one way of measuring the structural complexity of minimal extensions of first-order structures. In particular, positive Deissler rank measures the complexity of the injective envelope of a module as an extension of that module. In this paper we solve a problem of the second author by showing that certain injective envelopes have the maximum possible positive Deissler rank complexity. The proof shows that this complexity naturally reflects the internal (...) structure of the injective extension in the form of the levels of the Matlis hierarchy. (shrink)

In this paper, by considering the notion of MV-module, which is the structure that naturally correspond to lu-modules over lu-rings, we study injective MV-modules and we investigate some conditions for constructing injective MV-modules. Then we define the notions of essential A-homomorphisms and essential extension of A-homomorphisms, where A is a product MV-algebra, and we get some of there properties. Finally, we prove that a maximal essential extension of any A-ideal of an injective MV-module is an (...) injective A-module, too. (shrink)

We find the model completion of the theory modules over ������, where ������ is a finitely generated commutative algebra over a field K. This is done in a context where the field K and the module are represented by sorts in the theory, so that constructible sets associated with a module can be interpreted in this language. The language is expanded by additional sorts for the Grassmanians of all powers of $K^n $ , which are necessary to achieve (...) quantifier elimination. The result turns out to be that the model completion is the theory of a certain class of "big" injective modules. In particular, it is shown that the class of injective modules is itself elementary. We also obtain an explicit description of the types in this theory. (shrink)

We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finite-type arithmetic. As a consequence, some uniform boundedness principles are interpreted while maintaining unmoved the -sentences of arithmetic. We explain why this interpretation is tailored to yield conservation results.

This paper studies a family of monotonic extensions of first-order logic which we call modulated logics, constructed by extending classical logic through generalized quantifiers called modulated quantifiers. This approach offers a new regard to what we call flexible reasoning. A uniform treatment of modulated logics is given here, obtaining some general results in model theory. Besides reviewing the “Logic of Ultrafilters”, which formalizes inductive assertions of the kind “almost all”, two new monotonic logical systems are proposed here, the “Logic of (...) Many” and the “Logic of Plausibility”, that characterize assertions of the kind “many”, and “for a good number of”. Although the notion of simple majority (“more than half”) can be captured by means of a modulated quantifier semantically interpreted by cardinal measure on evidence sets, it is proven that this system, although sound, cannot be complete if checked against the intended model. This justifies the interest on a purely qualitative approach to this kind of quantification, what is guaranteed by interpreting the modulated quantifiers as notions of families of principal filters and reduced topologies, respectively. We prove that both systems are conservative extensions of classical logic that preserve important properties, such as soundness and completeness. Some additional perspectives connecting our approach to flexible reasoning through modulated logics to epistemology and social choice theory are also discussed. (shrink)

We characterize strict Mittag-Leffler modules in terms of free realizations of positive primitive formulas, and rings over which projectives are trivial in terms of various notions of separability of strict Mittag-Leffler modules. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Quantum B‐algebras are partially ordered algebras characterizing the residuated structure of a quantale. Examples arise in algebraic logic, non‐commutative arithmetic, and quantum theory. A quantum B‐algebra with trivial partial order is equivalent to a group. The paper introduces a corresponding analogue of quantale modules. It is proved that every quantum B‐module admits an injective envelope which is a quantale module. The injective envelope is constructed explicitly as a completion, a multi‐poset version of the completion of Dedekind (...) and MacNeille. (shrink)

Our understanding of body ownership largely relies on the Rubber Hand Illusion (RHI) paradigm where synchronous stroking of the real and fake hands leads to an illusion of ownership of RH provided its physical, anatomical, and spatial plausibility. Self-attribution of a fake hand also occurs during visuomotor synchrony, when the visual feedback of self-initiated movements follows the trajectory of the instantiated motor command. In both cases, the experience of ownership is established through bottom-up integration and top-down prediction of multisensory (proximodistal) (...) cues within the peripersonal space. It seems, however, that depending on whether the signals are externally or self-generated, top-down expectation signals are qualitatively different. On the one hand, in RHI, the sensory correlations are modulated by empirically induced priors about the internal model of the body. On the other hand, in mRHI, body ownership is actively shaped by processes allowing for continuous comparison between the actual and predicted sensory consequences of the actions. Ample research demonstrates that the differential processing of the predicted and the actual reafferent information is addressed by the central nervous system via an internal Forward Model (FM). Indeed, results from mRHI suggest that in action-contexts mechanism underlying body ownership could be similar to the FM. Crucially FMs integrate across all sensory input including not only proximodistal (visuomotor) but also purely distal sensory signals (visuoauditory). Thus if body ownership results from a consistency of FM, it would be affected by the (in)congruency of purely distal signals provided that they inform about the action-consequences relevant to the task. Here we explicitly address this question. To test our hypothesis, we devise an embodied VR-based task where action outcomes were signaled by distinct auditory cues. By manipulating the cues with respect to their spatial, temporal and semantic congruency, we show that purely distal feedback violating predictions about action outcomes compromises both body ownership and performance. These results demonstrate that body ownership is influenced by cues which not only pertain to the body itself but also those arising outside of the body, and suggest that in goal-oriented tasks body ownership might result from the consistency of forward models. (shrink)

In this paper we develop the elementary theory of modules in the category Sh of sheaves over right-sided idempotent quantales. The main ingredient is the construction of a logic sound for Sh . As an application we prove that in Sh , a finitely generated projective module is free , a result that is relevant to the study of representation of non-commutative C ∗ -algebras.

Rings which, from the ring-theoretic point of view, are very different may well have categories of modules which are extremely similar. More generally, the category of modules over a ring may contain many other categories of modules. Ideas from model theory are of use in elucidating this state of affairs. In particular we investigate the model-theoretic effect of tilting functors between categories of modules.

Van den Dries, L. and J. Holly, Quantifier elimination for modules with scalar variables, Annals of Pure and Applied Logic 57 161–179. We consider modules as two-sorted structures with scalar variables ranging over the ring. We show that each formula in which all scalar variables are free is equivalent to a formula of a very simple form, uniformly and effectively for all torsion-free modules over gcd domains . For the case of Presburger arithmetic with scalar variables the result takes (...) a still simpler form, and we derive in this way the polynomial-time decidability of the sets defined by such formulas. (shrink)

The main theorem characterizes Mittag-Leffler modules as ‘positively atomic’ modules . This is applied to reduced products of Mittag-Leffler modules and pure-semisimple.

We prove that the full completeness theorem for MLL+Mix holds by the simple interpretation via formulas as objects and proofs as Z-invariant morphisms in the *-autonomous category of topologized vector spaces. We do this by generalizing the recent work of Blute and Scott 101–142) where they used the semantical framework of dinatural transformation introduced by Girard–Scedrov–Scott , Logic from Computer Science, vol. 21, Springer, Berlin, 1992, pp. 217–241). By omitting the use of dinatural transformation, our semantics evidently allows the interpretation (...) of the cut-rule, while the original Blute–Scott's does not. Moreover, our interpretation for proofs is preserved automatically under the cut elimination procedure. In our semantics proofs themselves are characterized by the concrete algebraic notion “Z-invariance”, and our denotational semantics provides the full completeness. Our semantics is naturally extended to the full completeness semantics for CyLL+Mix owing to an elegant method of Blute–Scott 1413–1436) 101–142))). (shrink)

Ordered sheaves on a small quantaloid have been defined in terms of -enriched categorical structures; they form a locally ordered category . The free-cocompletion KZ-doctrine on has , the quantaloid of -modules, as its category of Eilenberg–Moore algebras. In this paper we give an intrinsic description of the Kleisli algebras: we call them the locally principally generated -modules. We deduce that is biequivalent to the 2-category of locally principally generated -modules and left adjoint module morphisms. The example of locally (...) principally generated modules on a locale X is worked out in full detail: relating X-modules to objects of the slice category , we show that ordered sheaves on X correspond with skew local homeomorphisms into X. (shrink)

The interpretability of AI is just as important as its performance. In the LegalAI field, there have been efforts to enhance the interpretability of models, but a trade-off between interpretability and prediction accuracy remains inevitable. In this paper, we introduce a novel framework called LK-IB for compulsory measure prediction (CMP), one of the critical tasks in LegalAI. LK-IB leverages Legal Knowledge and combines an Interpretable model and a Black-box model to balance interpretability and prediction performance. Specifically, LK-IB involves three steps: (...) (1) inputting cases into the first module, where first-order logic (FOL) rules are used to make predictions and output them directly if possible; (2) sending cases to the second module if FOL rules are not applicable, where a case distributor categorizes them as either “simple” or “complex“; and (3) sending simple cases to an interpretable model with strong interpretability and complex cases to a black-box model with outstanding performance. Experimental results demonstrate that the LK-IB framework provides more interpretable and accurate predictions than other state-of-the-art models. Given that the majority of cases in LegalAI are simple, the idea of model combination has significant potential for practical applications. (shrink)

People categorize objects slowly when visual input is highly impoverished instead of optimal. While bottom-up models may explain a decision with optimal input, perceptual hypothesis testing (PHT) theories implicate top-down processes with impoverished input. Brain mechanisms and the time course of PHT are largely unknown. This event-related potential study used a neuroimaging paradigm that implicated prefrontal cortex in top-down modulation of occipitotemporal cortex. Subjects categorized more impoverished and less impoverished real and pseudo objects. PHT theories predict larger impoverishment effects for (...) real than pseudo objects because top-down processes modulate knowledge only for real objects, but different PHT variants predict different timing. Consistent with parietal-prefrontal PHT variants, around 250 ms, the earliest impoverished real object interaction started on an N3 complex, which reflects interactive cortical activity for object cognition. N3 impoverishment effects localized to both prefrontal and occipitotemporal cortex for real objects only. The N3 also showed knowledge effects by 230 ms that localized to occipitotemporal cortex. Later effects reflected (a) word meaning in temporal cortex during the N400, (b) internal evaluation of prior decision and memory processes and secondary higher-order memory involving anterotemporal parts of a default mode network during posterior positivity (P600), and (c) response related activity in posterior cingulate during an anterior slow wave (SW) after 700 ms. Finally, response activity in supplementary motor area during a posterior SW after 900 ms showed impoverishment effects that correlated with RTs. Convergent evidence from studies of vision, memory, and mental imagery which reflects purely top-down inputs, indicates that the N3 reflects the critical top-down processes of PHT. A hybrid multiple-state interactive, PHT and decision theory best explains the visual constancy of object cognition. (shrink)

Insulin is stored within the pancreas in an inactive Zn2+‐bound hexameric form prior to release. Similarly, clinical insulins contain Zn2+ and form multimeric complexes. Upon release from the pancreas or upon injection, insulin only becomes active once Zn2+ disengages from the complex. In plasma and other extracellular fluids, the majority of Zn2+ is bound to human serum albumin (HSA), which plays a vital role in controlling insulin pharmacodynamics by enabling removal of Zn2+. The Zn2+‐binding properties of HSA are attenuated by (...) non‐esterified fatty acids (NEFAs) also transported by HSA. Elevated NEFA concentrations are associated with obesity and type 2 diabetes. Here we present the hypothesis that higher NEFA levels in obese and/or diabetic individuals may contribute to insulin resistance and affect therapeutic insulin dose‐response profiles, through modulation of HSA/Zn2+ dynamics. We envisage this novel concept to have important implications for personalized treatments and management of diabetes‐related conditions in the future. (shrink)

We consider valued fields with a distinguished isometry or contractive derivation as valued modules over the Ore ring of difference operators. Under certain assumptions on the residue field, we prove quantifier elimination first in the pure module language, then in that language augmented with a chain of additive subgroups, and finally in a two-sorted language with a valuation map. We apply quantifier elimination to prove that these structures do not have the independence property.

The Leuphana Semester at Leuphana University Lüneburg, together with the module “Science bears responsibility” demonstrate how innovative methods of teaching and learning can be combined with the topic of sustainable development and how new forms of university teaching can be introduced. With regard to module content, it has become apparent that, due to the complexity of the field of sustainability, a single discipline alone is unable to provide analyses and solutions. If teaching in higher education is to adequately (...) deal with this complexity, then it is necessary to develop inter- and transdisciplinary approaches that go beyond a purely specialist orientation. (shrink)

The primary function of smooth muscle cells is to contract and alter the stiffness or diameter of hollow organs such as blood vessels, the airways and the gastrointestinal and urogenital tracts. In addition to purely structural functions, smooth muscle cells may play important metabolic roles, particularly in various inflammatory responses. In cell culture, these cells have been shown to be metabolically dynamic, synthesizing and secreting extracellular matrix proteins, glycosaminoglycans and a wide variety of cell–cell signaling proteins, such as interleukins, chemokines (...) and peptide growth factors. Secreted cell signaling proteins participate in the inflammatory response of smooth muscle‐containing organs, and some can also stimulate smooth muscle migration, proliferation and contraction. The cellular signaling pathways controlling synthesis of these signaling proteins are similar to those used by cells mediating innate immunity and may contribute to pathogenesis of diverse diseases including atherosclerosis, asthma, inflammatory bowel diseases and preterm labor. Appreciating the role of smooth muscle cells in these diseases may lead to better understanding of the beneficial effects of anti‐inflammatory drugs as well as identification of new targets for anti‐inflammatory therapy. BioEssays 26:646–655, 2004. © 2004 Wiley Periodicals, Inc. (shrink)

Let R be an associative ring with identity. It is shown that every Σ-cotorsion left R-module satisfies the descending chain condition on divisibility formulae. If R is countable, the descending chain condition on M implies that it must be Σ-cotorsion. It follows that, for countable R, the class of Σ-cotorsion modules is closed under elementary equivalence and pure submodules. The modules M that satisfy this descending chain condition are the cotorsion analogues of totally transcendental modules; we characterize them (...) as the modules M for which , for every countably presented flat module F. (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.

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains whose localizations at maximal ideals have dense value groups. For Bézout domains, these conditions are also necessary.

On the basis of the Klingler–Levy classification of finitely generated modules over commutative noetherian rings we approach the old problem of classifying finite commutative rings R with a decidable theory of modules. We prove that if R is wild, then the theory of all R-modules is undecidable, and verify decidability of this theory for some classes of tame finite commutative rings.

Auditory feedback perturbation studies have indicated a link between feedback and feedforward mechanisms in speech production when participants compensate for applied shifts. In spectral perturbation studies, speakers with a higher perceptual auditory acuity typically compensate more than individuals with lower acuity. However, the reaction to feedback perturbation is unlikely to be merely a matter of perceptual acuity but also affected by the prediction and production of precise motor action. This interplay between prediction, perception, and motor execution seems to be crucial (...) for the timing of speech and non-speech motor actions. In this study, to examine the relationship between the responses to temporally perturbed auditory feedback and rhythmic abilities, we tested 45 adult speakers on the one hand with a temporal auditory feedback perturbation paradigm, and on the other hand with rhythm perception and production tasks. The perturbation tasks temporally stretched and compressed segments (onset + vowel or vowel + coda) in fluent speech in real-time. This technique sheds light on the temporal representation and the production flexibility of timing mechanisms in fluent speech with respect to the structure of the syllable. The perception tasks contained staircase paradigms capturing duration discrimination abilities and beat-alignment judgments. The rhythm production tasks consisted of finger tapping tasks taken from the BAASTA tapping battery and additional speech tapping tasks. We found that both auditory acuity and motor stability in finger tapping affected responses to temporal auditory feedback perturbation. In general, speakers with higher auditory acuity and higher motor variability compensated more. However, we observed a different weighting of auditory acuity and motor stability dependent on the prosodic structure of the perturbed sequence and the nature of the response as purely online or adaptive. These findings shed light on the interplay of phonological structure with feedback and feedforward integration for timing mechanisms in speech. (shrink)

This paper studies various equivalent characterizations of left semisimple rings from the standpoint of reverse mathematics. We first show that \ is equivalent to the statement that any left module over a left semisimple ring is semisimple over \. We then study characterizations of left semisimple rings in terms of projective modules as well as injective modules, and obtain the following results: \ is equivalent to the statement that any left module over a left semisimple ring is (...) projective over \; \ is equivalent to the statement that any left module over a left semisimple ring is injective over \; \ proves the statement that if every cyclic left R-module is projective, then R is a left semisimple ring; \ proves the statement that if every cyclic left R-module is injective, then R is a left semisimple ring. (shrink)