We prove that the poset algebra of every scattered poset with finite width is embeddable in the poset algebra of a well ordered poset.

We show that the separative quotient of the poset 〈P,⊂〉 of isomorphic suborders of a countable scattered linear order L is σ-closed and atomless. So, under the CH, all these posets are forcing-equivalent /Fin)+).

We develop the theory of layered posets and use the notion of layering to prove a new iteration theorem is κ-cc, as long as direct limits are used sufficiently often. This iteration theorem simplifies and generalizes the various chain condition arguments for universal Kunen iterations in the literature on saturated ideals, especially in situations where finite support iterations are not possible. We also provide two applications:1 For any n≥1, a wide variety of <ωn−1-closed, ωn+1-cc posets of size ωn+1 can consistently (...) be absorbed by quotients of saturated ideals on ωn.2 For any n∈ω, the tree property at ωn+3 is consistent with Chang’s conjecture ↠. (shrink)

We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a \Delta^0_2 maximal filter, and there is a computable poset with no \Pi^0_1 or \Sigma^0_1 maximal filter. There is a computable poset on which every maximal filter is Turing complete. We obtain the reverse mathematics result (...) that the principle "every countable poset has a maximal filter" is equivalent to ACA₀ over RCA₀. An unbounded filter is a filter which achieves each of its lower bounds in the poset. We show that every computable poset has a \Sigma^0_1 unbounded filter, and there is a computable poset with no \Pi^0_1 unbounded filter. We show that there is a computable poset on which every unbounded filter is Turing complete, and the principle "every countable poset has an unbounded filter" is equivalent to ACA₀ over RCA₀. We obtain additional reverse mathematics results related to extending arbitrary filters to unbounded filters and forming the upward closures of subsets of computable posets. (shrink)

A notion of interpretation between arbitrary logics is introduced, and the poset$\mathsf {Log}$of all logics ordered under interpretability is studied. It is shown that in$\mathsf {Log}$infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between$\mathsf {Log}$and the lattice of interpretability types of varieties are investigated.

When we interpret modal ◊ as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S α , where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we (...) interpret ◊ as closure in a topological space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H α , where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H α in terms of α-representations. We prove that ${X \in {\bf H}_{1}}$ iff X is submaximal. For a positive integer n, we generalize the notion of a submaximal space to that of an n-submaximal space, and prove that ${X \in {\bf H}_{n}}$ iff X is n-submaximal. This provides topological completeness and definability results for extensions of Grz. We show that the two partitions are related to each other as follows. For a successor ordinal α = β + n, with β a limit ordinal and n a positive integer, we have ${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\beta+2n-1} \cup {\bf S}_{\beta+2n}}$ , and for a limit ordinal α, we have ${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\alpha}}$ . As a result, we obtain full and faithful translations of ordinal complete extensions of Grz into ordinal complete extensions of GL, thus generalizing the Kuznetsov–Goldblatt–Boolos theorem. (shrink)

A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of (...) meet-prime logics only. (shrink)

A logic in a finite language is said to be finitely presentable if it is axiomatized by finitely many finite rules. It is proved that binary non-indexed products of logics that are both finitely presentable and finitely equivalential are essentially finitely presentable. This result does not extend to binary non-indexed products of arbitrary finitely presentable logics, as shown by a counterexample. Finitely presentable logics are then exploited to introduce finitely presentable Leibniz classes, and to draw a parallel between the Leibniz (...) and the Maltsev hierarchies. (shrink)

We introduce a relational semantics based on poset products, and provide sufficient conditions guaranteeing its soundness and completeness for various substructural logics. We also demonstrate that our relational semantics unifies and generalizes two semantics already appearing in the literature: Aguzzoli, Bianchi, and Marra’s temporal flow semantics for Hájek’s basic logic, and Lewis-Smith, Oliva, and Robinson’s semantics for intuitionistic Łukasiewicz logic. As a consequence of our general theory, we recover the soundness and completeness results of these prior studies in a (...) uniform fashion, and extend them to infinitely-many other substructural logics. (shrink)

This book offers the first comprehensive introduction to wave scattering in nonstationary materials. G. F. Roach's aim is to provide an accessible, self-contained resource for newcomers to this important field of research that has applications across a broad range of areas, including radar, sonar, diagnostics in engineering and manufacturing, geophysical prospecting, and ultrasonic medicine such as sonograms. New methods in recent years have been developed to assess the structure and properties of materials and surfaces. When light, sound, or some other (...) wave energy is directed at the material in question, "imperfections" in the resulting echo can reveal a tremendous amount of valuable diagnostic information. The mathematics behind such analysis is sophisticated and complex. However, while problems involving stationary materials are quite well understood, there is still much to learn about those in which the material is moving or changes over time. These so-called non-autonomous problems are the subject of this fascinating book. Roach develops practical strategies, techniques, and solutions for mathematicians and applied scientists working in or seeking entry into the field of modern scattering theory and its applications. Wave Scattering by Time-Dependent Perturbations is destined to become a classic in this rapidly evolving area of inquiry. (shrink)

Paraorthomodular posets are bounded partially ordered sets with an antitone involution induced by quantum structures arising from the logico-algebraic approach to quantum mechanics. The aim of the present work is starting a systematic inquiry into paraorthomodular posets theory both from algebraic and order-theoretic perspectives. On the one hand, we show that paraorthomodular posets are amenable of an algebraic treatment by means of a smooth representation in terms of bounded directoids with antitone involution. On the other, we investigate their order-theoretical features (...) in terms of forbidden configurations. Moreover, sufficient and necessary conditions characterizing bounded posets with an antitone involution whose Dedekind–MacNeille completion is paraorthomodular are provided. (shrink)

Can there be relational universals? If so, how can they be exemplified? A monadic universal is by definition capable of having a scattered spatiotemporal localization of its different exemplifications, but the problem of relational universals is that one single exemplification seems to have to be scattered in the many places where the relata are. The paper argues that it is possible to bite this bullet, and to accept a hitherto un-discussed kind of exemplification relation called ‘scattered exemplification’. (...) It has no immediate symbolic counterpart in any Indo-European natural language or in any so far constructed logical language. In order to remedy this, a notion called ‘many-place copula’ is introduced, too. (shrink)

This book deconstructs the whole lineage of political philosophy, showing the ways democracy abuts and regularly undermines the sovereignist tradition across a range of texts from the Iliad to contemporary philosophy. Politics is an object of perennial difficulty for philosophy—as recalcitrant to philosophical mastery as is philosophy’s traditional adversary, poetry. That difficulty makes it an attractive topic for any deconstructive approach to the tradition from which we inherit our language and our concepts. Scatter 2 pursues that deconstruction, often starting with, (...) and sometimes departing from, the work of Jacques Derrida by attending to the concepts of sovereignty on the one hand and democracy on the other. The book begins by following the fate of a line from Homer’s Iliad, where Odysseus asserts that “the rule of many is no good thing, let there be one ruler, one king.” The line, Bennington shows, is quoted, misquoted, and progressively Christianized by Aristotle, Philo Judaeus, Suetonius, the early Church Fathers, Aquinas, Dante, Ockham, Marsilius of Padua, Jean Bodin, Etienne de la Boétie, up to Carl Schmitt and Erik Peterson, and even one of the defendants at the Nuremberg trials, before being discussed by Derrida himself. In the book’s second half, Bennington begins again with Plato and Aristotle and tracks the concept of democracy as it regularly abuts and undermines that sovereignist tradition. In detailed readings of Hobbes and Rousseau, Bennington develops a notion of “proto-democracy” as a possible name for the scatter that underlies and drives the political as such and that will always prevent politics from achieving its aim of bringing itself to an end. (shrink)

Different constructions of BL-chains are compared. We establish when the ordinal sum and the poset product of the same family of BL-chains coincide. We also compare the poset product of MV-chains and product chains with saturated BL-chains.

Like bundles of homogeneous light or particles that are scattered in physics experiments, says Goris, the homogeneous field of metaphysics was dispersed in the 14th century. He connects that dispersion to the contemporary rejection of metaphysics. His topics are the analysis of concepts, the contested primacy, scattering the homogeneous field of me.

In this paper we give new criterions for left distributive posets to have neatest representations. We also illustrate a construction that would embed left distributive posets into representable semilattices.

We show that any coherent complete partial order is obtainable as the fixed-point poset of the strong Kleene jump of a suitably chosen first-order ground model. This is a strengthening of Visser’s result that any finite ccpo is obtainable in this way. The same is true for the van Fraassen supervaluation jump, but not for the weak Kleene jump.

In discussing the cultural history of the 19th century, Walter Benjamin diagnosed the emergence of the modern novel and its form of narration as the sign of a fracturing experience. The split in experience is related to the scattering of a homogeneous idea of space and time, constituted especially during the Enlightenment and in the German historicism. Benjamin's claim reflected the fracturing temporality of modern communities as well as the transformations in the understanding of the meaning of tradition. Here, I (...) begin by discussing Benjamin's conceptions of experience and memory in detail. Secondly, I consider his ideas on history in the framework of challenging the new forms of narration. In the end, I consider the loss of a unified community, especially by indicating ways in which the after-modern community reflects the relationship between aesthetics and politics in Jean-François Lyotard's thought. Key Words: Benjamin • community • experience • history • Lyotard • memory • narrative • remembrance • time. (shrink)

A version of scattering theory that was developed many years ago to treat nuclear scattering processes, has provided a powerful tool to study universality in scattering processes involving open quantum systems with underlying classically chaotic dynamics. Recently, it has been used to make random matrix theory predictions concerning the statistical properties of scattering resonances in mesoscopic electron waveguides and electromagnetic waveguides. We provide a simple derivation of this scattering theory and we compare its predictions to those obtained from an exactly (...) solvable scattering model; and we use it to study the scattering of a particle wave from a random potential. This method may prove useful in distinguishing the effects of chaos from the effects of disorder in real scattering processes. (shrink)

Orthomodular posets form an algebraic formalization of the logic of quantum mechanics. A central question is how to introduce implication in such a logic. We give a positive answer whenever the orthomodular poset in question is of finite height. The crucial advantage of our solution is that the corresponding algebra, called implication orthomodular poset, i.e. a poset equipped with a binary operator of implication, corresponds to the original orthomodular poset and that its implication operator is everywhere (...) defined. We present here a complete list of axioms for implication orthomodular posets. This enables us to derive an axiomatization in Gentzen style for the algebraizable logic of orthomodular posets of finite height. (shrink)

We show that for any infinite cardinal κ, every strongly $(\kappa + 1)-strategically$ closed poset is strongly $\kappa^+-strategically$ closed if and only if $AP_\kappa$ (the approachability property) holds, answering the question asked in [5]. We also give a complete classification of strengths of strategic closure properties and that of strong strategic closure properties respectively.

We here use our nonperturbative, cluster decomposable relativistic scattering formalism to calculate photon–spinor scattering, including the related particle–antiparticle annihilation amplitude. We start from a three-body system in which the unitary pair interactions contain the kinematic possibility of single quantum exchange and the symmetry properties needed to identify and substitute antiparticles for particles. We extract from it a unitary two-particle amplitude for quantum–particle scattering. We verify that we have done this correctly by showing that our calculated photon–spinor amplitude reduces in the (...) weak coupling limit to the usual lowest order, manifestly covariant (QED) result with the correct normalization. That we are able to successfully do this directly demonstrates that renormalizability need not be a fundamental requirement for all physically viable models. (shrink)

Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite $\Pi _1^0 $ chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result of Hirschfeldt and Shore to show that every infinite computable weakly stable poset contains either an infinite low chain or an infinite (...) computable antichain. Our hardest result is that there is an infinite computable weakly stable poset with no infinite $\Pi _1^0 $ chains or antichains. On the other hand, it is easily seen that every infinite computable stable poset contains an infinite computable chain or an infinite $\Pi _1^0 $ antichain. In Reverse Mathematics, we show that SCAC, the principle that every infinite stable poset contains an infinite chain or antichain, is equivalent over RCA₀ to WSCAC, the corresponding principle for weakly stable posets. (shrink)

Alzheimer’s disease is emerging into public view in unprecedented ways. Foremost among these is the embodied form of elderly men and women appearing in commercial imagery for patient advocacy groups or pharmaceutical advertisements, but scientific imagery also seeps into the visual media cultures that surround us. The recent reconfiguration of Alzheimer’s disease is due to expanding ageing populations, an aggressive biopharmaceutical industry becoming a fast-growing material-semiotic realm that is providing powerful images of both gendered and racialized embodiment. Such a visual, (...) and yet highly material, realm is in need of feminist interventions, engaging with the images and ideas that circulate around ageing, medicine, human and non-human embodiment. From a non-representationalist and posthumanist perspective identified as feminist visual studies of technoscience, the authors seek to further the discussion in the direction of understanding the ‘scattered ontologies’ of Alzheimer’s — in laboratory practice, the realm of medical media and in commercial appeals to coherent individuality and human cognition — as gendered domains of figural reality and performative matter. (shrink)

We introduce a class of proper posets which is preserved under countable support iterations, includes \(\omega ^\omega \) -bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be considered as a possible path towards solving variations of the famous Roitman problem.

With the analysis of the scattering ashes in a Norwegian context as its point of departure, the article sets out to explore ash scattering and how it relates to the governance of deathscape and religion/worldview in the public space. Referring to ethnographic study, the focus is on the bereaved and the deceased in the governance process for ash scattering and on critically rethinking the governance of ash scattering from the private actors’ experiences. I argue that ash scattering is in the (...) process of establishing a spatial ritual institution, deregulated vis-à-vis organized religion/worldview, which, on the one hand, opens up the possibility for the privatization of death and provides ideals of individuality, privacy and discretion, and, on the other hand, this takes place paradoxically not in private but rather within public space. (shrink)

The scattering number and isolated scattering number of a graph have been introduced in relation to Hamiltonian properties and network vulnerability, and the isolated scattering number plays an important role in characterizing graphs with a fractional 1-factor. Here we investigate the computational complexity of one variant, namely, the weighted isolated scattering number. We give a polynomial time algorithm to compute this parameter of interval graphs, an important subclass of perfect graphs.

Background Recent publications reveal shortcomings in evidence review and summarization methods for patient decision aids. In the large-scale "Share to Care (S2C)" Shared Decision Making (SDM) project at the University Hospital Kiel, Germany, one of 4 SDM interventions was to develop up to 80 decision aids for patients. Best available evidence on the treatments' impact on patient-relevant outcomes was systematically appraised to feed this information into the decision aids. Aims of this paper were to (1) describe how PtDAs are developed (...) and how S2C evidence reviews for each PtDA are conducted, (2) appraise the quality of the best available evidence identified and (3) identify challenges associated with identified evidence. Methods The quality of the identified evidence was assessed based on GRADE quality criteria and categorized into high-, moderate-, low-, very low-quality evidence. Evidence appraisal was conducted across all outcomes assessed in an evidence review and for specific groups of outcomes, namely mortality, morbidity, quality of life, and treatment harms. Challenges in evidence interpretation and summarization resulting from the characteristics of decision aids and the type and quality of evidence are identified and discussed. Conclusions Evidence reviews in this project were carefully conducted and summarized. However, the evidence identified for our decision aids was indeed a "scattered landscape" and often poor quality. Facing a high prevalence of low-quality, non-directly comparative evidence for treatment alternatives doesn't mean it is not necessary to choose an evidence-based approach to inform patients. While there is an urgent need for high quality comparative trials, best available evidence nevertheless has to be appraised and transparently communicated to patients. (shrink)

In this paper, we aim at highlighting the significance of the A- and B-properties introduced by Finch (Bull Aust Math Soc 2:57–62, 1970b). These conditions turn out to capture interesting structural features of lattices of closed subspaces of complete inner vector spaces. Moreover, we generalise them to the context of effect algebras, establishing a novel connection between quantum structures (orthomodular posets, orthoalgebras, effect algebras) arising from the logico-algebraic approach to quantum mechanics.

The class of stable posets is defined and investigated. We give a forcing construction of a universe of set theory which satisfies a weak form of Martin's Axiom and $2^{\aleph_0} > \aleph_1$ and yet some propositions which follow from CH hold in this universe.

In the paper Randomizations of Scattered Sentences, Keisler showed that if Martin’s axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is “yes”. It follows that the absolute Vaught conjecture holds if and only if every \-sentence with few separable randomizations has countably many countable models.

Climate change is a temporally fragmented phenomenon: the causes and effects at work are dispersed over a remarkably long time period. Climate change exceeds human ability to forecast and quantify its effects in time. This creates serious epistemic, moral, and psychological difficulties and poses challenges to generating adequate ethical responses. Augustine’s understanding of time as a measure of imagination emphasizes the way in which human beings actively shape their sense of time. He sees “looking forward” in time as a matter (...) of spiritual vocation that collects the self out of dispersion and connects to a transgenerational collective. A notable example of how this “looking forward” may be practiced is singing the Psalms. The Augustinian “temporal imagination” links the imaginative, affective, moral, and vocational dimensions of measuring time. This offers some preliminary avenues for reimagining a sense of time responsive to climate change’s temporal fragmentation. (shrink)