We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group ${\mathrm {UT}}_3({\mathbb {Z}})$, the continuous Heisenberg group ${\mathrm {UT}}_3({\mathbb {R}})$, and, more generally, groups of upper unitriangular and invertible upper triangular (...) matrices over unital rings. (shrink)

For G a group definable in some structure M, we define notions of “definable” compactification of G and “definable” action of G on a compact space X , where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as View the MathML source and the universal definable G-ambit as the type space SG. We also point out the existence and uniqueness of “universal minimal definable G-flows”, and discuss issues (...) of amenability and extreme amenability in this definable category, with a characterization of the latter. For the sake of completeness we also describe the universal compactification and universal G-ambit in model-theoretic terms, when G is a topological group. (shrink)

Let G be a separated (Hausdorff) topological group and let *G be an enlargement of G (see [8]). Thus, *G (i) possesses the same formal properties as G in the sense explained in [8], and (ii) every set of subsets {Aν} of G with the finite intersection property—i.e. such that every nonempty finite subset of {Aν} has a nonempty intersection—satisfies ∩*Aν ≠ ø, where the *Aν are the extensions of the Aν in *G, respectively.

In this paper we extend the usual notion of model (asa structure) to the more general notion of CauchySequence of Structures in a similar way as rationalsare extending to real numbers by means of Cauchysequences of rationals. We show that the structurespace St is dense in thecomplete space CSt of Cauchysequences of structures and that CSt is compact in the (topo)logicalsense.

In this paper we extend the usual notion of model (asa structure) to the more general notion of CauchySequence of Structures in a similar way as rationalsare extending to real numbers by means of Cauchysequences of rationals. We show that the structurespace Stτ is dense in thecomplete space CStτ of Cauchysequences of structures and that CStτ is compact in the (topo)logicalsense.

Given a locale L and any set-indexed family of continuous mappings , fi:L→Li with compact and completely regular co-domain, a compactification η:L→Lγ of L is constructed enjoying the following extension property: for every a unique continuous mapping exists such that . Considered in ordinary set theory, this compactification also enjoys certain convenient weight limitations.Stone–Čech compactification is obtained as a particular case of this construction in those settings in which the class of [0,1]-valued continuous mappings is a set (...) for all L. This will follow by the proof that–also in the point-free context–a compactification that allows for the extension of [0,1]-valued mappings suffices for deriving the full reflection.A constructive proof that the class is a set whenever L is locally compact and L′ is set-presented and regular is also obtained; together with the described compactification, this makes it possible to characterize the class of locales for which Stone–Čech compactification can be defined constructively. (shrink)

In this paper, we discuss some questions about compactness in MV-topological spaces. More precisely, we first present a Tychonoff theorem for such a class of fuzzy topological spaces and some consequence of this result, among which, for example, the existence of products in the category of Stone MV-spaces and, consequently, of coproducts in the one of limit cut complete MV-algebras. Then we show that our Tychonoff theorem is equivalent, in ZF, to the Axiom of Choice, classical Tychonoff theorem, and Lowen’s (...) analogous result for lattice-valued fuzzy topology. Last, we show an extension of the Stone–Čech compactification functor to the category of MV-topological spaces, and we discuss its relationship with previous works on compactification for fuzzy topological spaces. (shrink)

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing (...) with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces. (shrink)

Let ${\beta(\mathbb{N})}$ denote the Stone–Čech compactification of the set ${\mathbb{N}}$ of natural numbers (with the discrete topology), and let ${\mathbb{N}^\ast}$ denote the remainder ${\beta(\mathbb{N})-\mathbb{N}}$ . We show that, interpreting modal diamond as the closure in a topological space, the modal logic of ${\mathbb{N}^\ast}$ is S4 and that the modal logic of ${\beta(\mathbb{N})}$ is S4.1.2.

In this work we provide a new topological representation for implication algebras in such a way that its one-point compactification is the topological space given in [1]. Some applications are given thereof.

We introduce and study the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with the trivial Bohr compactification, that is, the class of minimally almost periodic groups.

This study explores the transhuman from an East Asian perspective. In terms of cognitive science, mathematics, and theology, we define the transhuman system as characterized by transcendence, extension by compactification, and samtaegeuk. Compactification is conceptualized here in mathematical terms, as adding one or more elements so that a system becomes more complete—as one might join both ends of a line, and thereby create a circle. We assert that the East Asian transhuman could be defined as a three-point (...) class='Hi'>compactification: as an extension of biophysical objects and events such as robots, cyborgs, and environments ; as an extension of culture, science, and art ; and as an extension of the interaction between the human and Cosmic Absolute such as in religions. Such a notion of the Transhuman might be associated with God, but any description of God, the Absolute Infinite, will apply to something less than God. (shrink)

We continue the research of an extension of the divisibility relation to the Stone‐Čech compactification. First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in and nonstandard extensions of are answered, providing a few more equivalent conditions for divisibility in. Results on uncountable chains in are proved and used in a construction of a well‐ordered chain of maximal cardinality. Probably the most interesting result is the existence (...) of a chain of type in. Finally, we consider ultrafilters without divisors in and among them find the greatest class. (shrink)

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $. In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $. This improves the results of G. Bezhanishvili and J. Harding in [4], (...) where the authors prove these theorems under stronger assumptions. Our proof is also somewhat simpler. (shrink)

I discuss a model inspired from the string/brane framework, in which our Universe is represented (after perhaps appropriate compactification) as a three brane, propagating in a bulk space time punctured by D0-brane (D-particle) defects. As the D3-brane world moves in the bulk, the D-particles cross it, and from an effective observer on D3 the situation looks like a “space-time foam” with the defects “flashing” on and off (“D-particle foam”). The open strings, with their ends attached on the brane, which (...) represent matter in this scenario, can interact with the D-particles on the D3-brane universe in a topologically non-trivial manner, involving splitting and capture of the strings by the D0-brane defects. Such processes are consistently described by logarithmic conformal field theories on the world-sheet of the strings. Physically, they result in effective decoherence of the string matter on the D3 brane, and as a result, of CPT Violation, but of a type that implies an ill-defined nature of the effective CPT operator. Due to electric charge conservation, only electrically neutral (string) matter can exhibit such interactions with the D-particle foam. This may have unique, experimentally detectable (in principle), consequences for electrically-neutral entangled quantum matter states on the brane world, in particular the modification of the pertinent Einstein-Podolsky-Rosen (EPR) Correlation in neutral mesons in an appropriate meson factory. For the simplest scenarios, the order of magnitude of such effects might lie within the sensitivity of upgraded φ-meson factories. (shrink)

A Clifford-algebraic interpretation is proposed of the charge, mass, spin relationship found recently by Cooperstock and Faraoini, which was based on the Kerr–Newman metric solutions of the Einstein–Maxwell equations. The components of the polymomentum associated with a Clifford polyparticle in four dimensions provide for such a charge, mass, spin relationship without the problems encountered in Kaluza–Klein compactifications which furnish an unphysically large value for the electron charge. A physical reasoning behind such charge, mass, spin relationship is provided, followed by a (...) discussion on the geometrical derivation of the fine structure constant by Wyler, Smith, Gonzalez-Martin and Smilga. To finalize, the renormalization of electric charge is discussed and some remarks are made pertaining the modifications of the charge–scale relationship, when the spin of the polyparticle changes with scale, that may cast some light into the alleged Astrophysical variations of the fine structure constant. (shrink)

The purpose of this paper is first to show that if X is any locally compact but not compact perfect Polish space and stands for the one-point compactification of X, while E X is the equivalence relation which is defined on the Polish group C(X,R +*) by where f, g are in C(X,R +*), then E X is induced by a turbulent Polish group action. Second we show that given any if we identify the n-dimensional unit sphere S n (...) with the one-point compactification of R n via the stereographic projection, while E n , r is the equivalence relation which is defined on the Polish group C r (R n ,R +*) by where f, g are in C r (R n ,R +*), then E n , r is also induced by a turbulent Polish group action. (shrink)

Scroggs's theorem on the extensions of S5 is an early landmark in the modern mathematical studies of modal logics. From it, we know that the lattice of normal extensions of S5 is isomorphic to the inverse order of the natural numbers with infinity and that all extensions of S5 are in fact normal. In this paper, we consider extending Scroggs's theorem to modal logics with propositional quantifiers governed by the axioms and rules analogous to the usual ones for ordinary quantifiers. (...) We call them Π-logics. Taking S5Π, the smallest normal Π-logic extending S5, as the natural counterpart to S5 in Scroggs's theorem, we show that all normal Π-logics extending S5Π are complete with respect to their complete simple S5 algebras, that they form a lattice that is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N, that they have arbitrarily high Turing-degrees, and that there are non-normal Π-logics extending S5Π. (shrink)

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $. In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $. This improves the results of G. Bezhanishvili and J. Harding in [4], (...) where the authors prove these theorems under stronger assumptions. Our proof is also somewhat simpler. (shrink)

The purpose of this paper is to compare the notion of a Grzegorczyk point introduced in [19] (and thoroughly investigated in [3, 14, 16, 18]) to the standard notions of a filter in Boolean algebras and round filter in Boolean contact algebras. In particular, we compare Grzegorczyk points to filters and ultrafilters of atomic and atomless algebras. We also prove how a certain extra axiom influences topological spaces for Grzegorczyk contact algebras. Last but not least, we do not refrain from (...) a philosophical interpretation of the results from the paper. (shrink)

Although classically every open subspace of a locally compact space is also locally compact, constructively this is not generally true. This paper provides a locally compact remetrization for an open set in a compact metric space and constructs a one-point compactification. MSC: 54D45, 03F60, 03F65.

Le but de cet article est de présenter un modèle formel des processus dichotomiques platoniciens. Cette méthode, déjà utilisée dans le Gorgias et décrite dans le Phèdre, reçoit une grande extension dans les dialogues ultérieurs. Elle s'efforce d'obtenir une définition à partir des divisions successives d'un ensemble de concepts. Nous montrons que les chaînes de dichotomies ne fonctionnent pas comme des classifications, mais comme des « filtres convergents » sur l'espace des Idées. Cela veut dire que cet espace est, formellement (...) parlant, un espace topologique compact, donc fini. Mais, dans d'autres textes, Platon semble indiquer que les Idées sont en nombre infini. Dans cette hypothèse, il resterait possible d'atteindre, au moins à l'infini, quelque chose comme une définition, en considérant l'Idée du Bien comme une sorte de principe de compactification de l'univers platonicien. The purpose of this article is to present a formal model of platonician dichotomic processes. This method, already used in Gorgias and described in Phaedrus, takes a great development in further dialogues. It tries to get a definition from successive divisions of a set of concepts. We show that the chains of dichotomies do not work as classifications but as "convergent filters "on the space of Ideas. It means that this space is, formally speaking, a compact and therefore, finite, topological space. But in some other texts, Platon seems to specify that there is an infinite number of Ideas. In this case, it will remain possible to reach, at least in the infinite, something like a definition, by regarding the Idea of Good as a sort of compactness principle of the platonician universe. (shrink)

The theory of unconstrained membranes of arbitrary dimension is presented. Their relativistic dynamics is described by an action which is a generalization of the Stueckelberg point-particle action. In the quantum version of the theory, the evolution of a membrane's state is governed by the relativistic Schrödinger equation. Particular stationary solutions correspond to the conventional, constrained membranes. Contrary to the usual practice, our spacetime is identified, not with the embedding space (which brings the problem of compactification), but with a membrane (...) of dimension 4 or higher. A 4-membrane is thus assumed to represent spacetime. The Einstein-Hilbert action emerges as an effective action after functionally integrating out the membrane's embedding functions. (shrink)

The mathematical and physical aspects of the conformal symmetry of space-time and of physical laws are analyzed. In particular, the group classification of conformally flat space-times, the conformal compactifications of space-time, and the problem of imbedding of the flat space-time in global four-dimensional curved spaces with non-trivial topological and geometrical structure are discussed in detail. The wave equations on the compactified space-times are analyzed also, and the set of their elementary solutions constructed. Finally, the implications of global compactified space-times for (...) cosmology are discussed. It is argued that the recent discovery of periodic structure of matter distribution on large distances strongly suggests that the global cosmological space-time should be close. Next we analyze the inflation scalar field in the inflationary model of universe evolution considered on the spatially compact Robertson-Walker space-time. It is shown that the energy distribution in this model is periodic and the periods and density decrease with increasing distance, in striking agreement with experimental data. Our model of the universe also provides a definite predictions for the energy distribution, polar and azimuthal, considered as a function of angles θ and φ. These predictions should be tested with the new astronomical data. (shrink)

Physical arguments stemming from the theory of black-hole thermodynamics are used to put constraints on the dynamics of closed-string tachyon condensation in Scherk–Schwarz compactifications. A geometrical interpretation of the tachyon condensation involves an effective capping of a noncontractible cycle, thus removing the very topology that supports the tachyons. A semiclassical regime is identified in which the matching between the tachyon condensation and the black-hole instability flow is possible. We formulate a generalized correspondence principle and illustrate it in several different circumstances: (...) an Euclidean interpretation of the transition from strings to black holes across the Hagedorn temperature and instabilities in the brane-antibrane system. (shrink)

Globalization is defined as reduction in world’s diversity due to cultural Gleichschaltung and compactification. A question is asked how the process is linkedwith the presence of universalist ideologies as great religions, utilitarianism, Socialism or ecologism. Some suggestions concerning further research are made.

There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them comes from modal logic and universal algebra, and in fact goes back to Jónsson and Tarski :891–939, 1951; 74:127–162, 1952). Another one The infinity project proceeding, Barcelona, 2012) comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is (...) its largest compactification. The main result of Saveliev The infinity project proceeding, Barcelona, 2012), which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in Saveliev. Results of such kind are referred to as extension theorems. After a brief introduction, we offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We provide two specific operations which turn ultrafilter models into ordinary models, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and obtain a topological characterization of ultrafilter models. We generalize a restricted version of the extension theorem to ultrafilter models. To formulate the full version, we propose a wider concept of ultrafilter models with their semantics based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a particular case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn ultrafilter models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which ultrafilter models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we establish three full versions of the extension theorem for ultrafilter models in the wide sense. The results of the first three sections of this paper were partially announced in Poliakov and Saveliev On two concepts of ultrafilter extensions of first-order models and their generalizations, Springer, Berlin, 2017). (shrink)

The impact of the KK-modes in d-brane models of gravity with large compactification radii and TeV-scale quantum gravity on the hadronic potential at small impact parameters is examined. The effects of the gravitational hadron form factors obtained from the hadron generalized parton distributions (GPDs) on the behavior of the gravitational potential and the possible spin correlation effects are also analysed.

Uniformities describing the distinguishability of states and of observables are discussed in the context of general statistical theories and are shown to be related to distinguished subspaces of continuous observables and states, respectively. The usual formalism of quantum mechanics contains no such physical uniformity for states. Using recently developed tools of quantum harmonic analysis, a natural one-to-one correspondence between continuous subspaces of nonrelativistic quantum and classical mechanics is established, thus exhibiting a close interrelation between physical uniformities for quantum states and (...) compactifications of phase space. General properties of the completions of the quantum state space with respect to these uniformities are discussed. (shrink)

Ramsey type theorems are theorems of the form: if certain sets are partitioned at least one of the parts has some particular property. In its finite form, Ramsey's theory will ask how big the partitioned set should be to assure this fact. Proofs of such theorems usually require a process of multiple choice, so that this apparently pure combinatoric field is rich in proofs that use ideal guides in making the choices. Typically they may be ultrafilters or points in the (...) compactification of the given set. It is, therefore, not surprising that nonstandard elements are much more natural guides in some of the proofs and in the general abstract treatment.In Section 1 we start off with some very natural examples of Ramsey type exercises that illustrate our idea. In Section 2 we give a nonstandard proof of the infinite Ramsey theorem. Section 3 tries to do the same for Hindman's theorem, and points out, where nonstandard analysis must use some hard standard facts to make the proof go through. (shrink)

If there is a homeomorphic embedding of one set into another, the sets are said to be topologically comparable. Friedman and Hirst have shown that the topological comparability of countable closed subsets of the reals is equivalent to the subsystem of second order arithmetic denoted byATR 0. Here, this result is extended to countable closed locally compact subsets of arbitrary complete separable metric spaces. The extension uses an analogue of the one point compactification of ℝ.

Free bosonic fields are investigated at a classical level by imposing their characteristic de Broglie periodicities as constraints. In analogy with finite temperature field theory and with extra-dimensional field theories, this compactification naturally leads to a quantized energy spectrum. As a consequence of the relation between periodicity and energy arising from the de Broglie relation, the compactification must be regarded as dynamical and local. The theory, whose foundamental set-up is presented in this paper, turns out to be consistent (...) with special relativity and in particular respects causality. The non trivial classical dynamics of these periodic fields show remarkable overlaps with ordinary quantum field theory. This can be interpreted as a generalization of the AdS/CFT correspondence. (shrink)

Let G be a group, and let X be an infinite transitive G-space. A free ultrafilter U on X is called G-selective if, for any G-invariant partition P of X, either one cell of P is a member of U, or there is a member of U which meets each cell of P in at most one point. We show that in ZFC with no additional set-theoretical assumptions there exists a G-selective ultrafilter on X. We describe all G-spaces X such (...) that each free ultrafilter on X is G-selective, and we prove that a free ultrafilter U on ω is selective if and only if U is G-selective with respect to the action of any countable group G of permutations of ω. A free ultrafilter U on X is called G-Ramsey if, for any G-invariant coloring χ:[X]2→{0,1}, there is U∈U such that [U]2 is χ-monochromatic. We show that each G-Ramsey ultrafilter on X is G-selective. Additional theorems give a lot of examples of ultrafilters on Z that are Z-selective but not Z-Ramsey. (shrink)