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This article introduces a line of investigation into connections between creature forcings and topological Ramsey spaces. Three examples of sets of pure candidates for creature forcings are shown to contain dense subsets which are actually topological Ramsey spaces. A new variant of the product tree Ramsey theorem is proved in order to obtain the pigeonhole principles for two of these examples. [ABSTRACT FROM AUTHOR]

We investigate the question of whether Q carries an ultrafilter generated by perfect sets (such ultrafilters were called gruff ultrafilters by van Douwen). We prove that one can (consistently) obtain an affirmative answer to this question in three different ways: by assuming a certain parametrized diamond principle, from the cardinal invariant equality d = c , and in the Random real model. [ABSTRACT FROM AUTHOR]

In this paper we discuss what kind of constrains combinatorial covering properties of Menger, Scheepers, and Hurewicz impose on remainders of topological groups. For instance, we show that such a remainder is Hurewicz if and only it is $$\sigma $$ -compact. Also, the existence of a Scheepers non- $$\sigma $$ -compact remainder of a topological group follows from CH and yields a P-point, and hence is independent of ZFC. We also make an attempt to prove a dichotomy for the Menger property of remainders of topological groups in the style of Arhangel'skii. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss those kinds of statistical convergence, in terms of filter F, or ideal I-convergence, which are equivalent to measure convergence defined by a single statistical measure. We prove a number of characterizations of a single statistical measure μ-convergence by using properties of its corresponding quotient Banach space l/ l( I). We also show that the usual sequential convergence is not equivalent to a single measure convergence. [ABSTRACT FROM AUTHOR]

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers on as the prototype structures, we construct a class of continuum many topological Ramsey spaces which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces , extending the Pudlák-Rödl theorem for barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings to the countable transfinite. The -closed partial order is forcing equivalent to , which forces a non-p-point ultrafilter . This work forms the basis for further work classifying the Rudin-Keisler and Tukey structures for the hierarchy of the generic ultrafilters . [ABSTRACT FROM AUTHOR]

We consider the problem of the reachability of states that are elements of a topological space under constraints of asymptotic nature on the choice of an argument of a given objective mapping. We study constructions that have the sense of extensions of the original space and are implemented with the use of methods that are natural for applied mathematics but employ elements of extensions used in general topology. The study is oriented towards the application in the problem on the construction and investigation of properties of reachability sets for control systems. Constructions involving an approximate observation of constraints in control problems, as well as various generalized regimes, were widely used by N.N. Krasovskii and his students. In particular, this approach was applied in the proof of N.N. Krasovskii and A.I. Subbotin's fundamental theorem of the alternative, which made it possible to establish the existence of a saddle point in a nonlinear differential game. In the investigation of impulse control problems, Krasovskii used techniques from the theory of generalized functions, which formed the basis for many studies in this direction. A number of A.B. Kurzhanski's papers are devoted to the solution of control problems related in one way or another to the construction of reachability sets. Control problems with incomplete information, duality issues for control and observation problems, and team control problems constitute a far from exhaustive list of research areas where Kurzhanskii obtained profound results. These studies are characterized by the use of a wide range of tools and methods from applied mathematics and various constructions as well as by the combination of theoretical investigations and procedures related to the possibility of computer modeling. The research direction developed in the present paper mainly concerns the problem of constraint observation (including 'asymptotic' constraints) and involves other issues. Nevertheless, the idea of constructing generalized elements of various nature (in particular, generalized controls) seems to be useful for the purpose of asymptotic analysis of control problems that do not possess stability as well as problems on the comparison of different tendencies in the choice of control in the form of dependences on a complex of factors inherent in the original real-life problem. The use of such tools as the Stone-Čech compactification and Wallman's extension is, of course, oriented toward the study of qualitative issues. In the authors' opinion, the combined application of the approaches to the construction of extensions used in control theory and in general topology holds promise from the point of view of both pure and applied mathematics. Apparently, the present paper can be considered as a certain step in this direction. [ABSTRACT FROM AUTHOR]

The generic ultrafilter ${\cal G}_2 $ forced by ${\cal P}\left( {\omega \times \omega } \right)/\left( {{\rm{Fin}} \otimes {\rm{Fin}}} \right)$ was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters ([1]), but it was left open where exactly in the Tukey order it lies. We prove ${\cal G}_2 $ that is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each ${\cal G}_2 $, the collection of all nonprincipal ultrafilters Tukey reducible to the generic ultrafilter ${\cal G}_k $ forced by ${\cal P}\left( {\omega ^k } \right)/{\rm{Fin}}^{ \otimes k} $ forms a chain of length k. Essential to the proof is the extraction of a dense subset εk from (Fin⊗k)+ which we prove to be a topological Ramsey space. The spaces εk, k ≥ 2, form a hierarchy of high dimensional Ellentuck spaces. New Ramsey-classification theorems for equivalence relations on fronts on εk are proved, extending the Pudlák–Rödl Theorem for fronts on the Ellentuck space, which are applied to find the Tukey and Rudin–Keisler structures below ${\cal G}_k $. [ABSTRACT FROM PUBLISHER]

We study two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras. To highlight the difference between them, we develop new techniques to construct incomparable ultrafilters in this setting. Furthermore, we discuss the relation with Tukey reducibility and prove that, assuming the Continuum Hypothesis, there exist ultrafilters on the Cohen algebra which are RK-equivalent in the generalized sense but Tukey-incomparable, in stark contrast with the classical setting. [ABSTRACT FROM AUTHOR]

We study a variant of the reachability problem with constraints of asymptotic character on the choice of controls. More exactly, we consider a control problem in the class of impulses of given intensity and vanishingly small length. The situation is complicated by the presence of discontinuous dependences, which produce effects of the type of multiplying a discontinuous function by a generalized function. The constructed extensions in the special class of finitely additive measures make it possible to present the required solution, defined as an asymptotic analog of a reachable set, in terms of a continuous image of a compact, which is described with the use of the Stone space corresponding to the natural algebra of sets of the control interval. One of the authors had the honor of communicating with Nikolai Nikolaevich Krasovskii for many years and discussed with him problems that led to the statement considered in the paper. Krasovskii's support of this research direction provided possibilities for its fruitful development. His disciples and colleagues will always cherish the memory of Nikolai Nikolaevich in their hearts. [ABSTRACT FROM AUTHOR]

This is a survey describing Mary Ellen Rudin's theorem about the number of composants of the Stone–Čech remainder of a half-line and subsequent related work, especially the principle of near coherence of filters. [ABSTRACT FROM AUTHOR]

For any Boolean algebra A, the character of A is χ( A)= min{ κ: every ultrafilter on A can be generated by at most κ elements}. A pseudo-tree is a partial order ( T,≤) such that for every t in T, the set { s∈ T: s≤ t} is a linear order. The pseudo-tree algebra on T, denoted Treealg T, is the subalgebra of $\mathcal {P}(T)$ generated by the cones { s∈ T: s≥ t}, for t in T. Ultrafilters on a pseudo-tree algebra Treealg T are in one-to-one correspondence with initial chains of T whenever T has a least element. We give an explicit description of a generating set of the ultrafilter corresponding to a given initial chain, and we use this to describe the character of Treealg T in terms of the structure of initial chains of T. [ABSTRACT FROM AUTHOR]

This article is a part of the theory developed by the author in which the following problem is solved under natural assumptions: to find necessary and sufficient conditions under which the union of at most countable family of algebras on a certain set X is equal to $\mathcal{P}(X)$. Here the following new result is proved. Let $\{\mathcal{A}_{\lambda }\}_{\lambda \in \Lambda }$ be a finite collection of algebras of sets given on a set X with $\# (\Lambda ) =n>0$, and for each λ there exist at least $\frac{10}{3}n+\sqrt{\frac{2n}{3}}$ pairwise disjoint sets belonging to $\mathcal{P}(X)\setminus\mathcal{A}_{\lambda }$. Then there exists a family $\{U^{1}_{\lambda }, U^{2}_{\lambda }\}_{\lambda \in \Lambda }$ of pairwise disjoint subsets of X ( $U^{i}_{\lambda }\cap U^{j}_{\lambda '}=\emptyset$ except the case $\lambda =\lambda '$, $i=j$); and for each λ the following holds: if $Q\in \mathcal{P}(X)$ and Q contains one of the two sets $U^{1}_{\lambda }$, $U^{2}_{\lambda }$, and its intersection with the other set is empty, then $Q\notin \mathcal{A}_{\lambda }$. [ABSTRACT FROM AUTHOR]

We consider properties of broadly understood measurable spaces that provide the preservation of maximality when ultrafilters are restricted to filters of the corresponding subspace. We study conditions that guarantee the convergence of images of ultrafilters consisting of open sets under continuous mappings. [ABSTRACT FROM AUTHOR]

M. M. Clementino and W. Tholen presented recently a lax-algebraic generalization of the well-known result on the coincidence of two classes of continuous maps between topological spaces: proper (maps, whose pullbacks are closed) and perfect (closed maps with compact fibres). Their achievement depends on a particular lax-algebraic extension of the concept of closed map, which relies on constructively completely distributive quantales. This paper proposes a definition of closed maps, which is not dependant on the underlying quantale of lax algebras, and proves the results of M. M. Clementino and W. Tholen in the new setting. We also show that our presented notion fits the axiomatic definition of closed maps in a category. [ABSTRACT FROM AUTHOR]

Let X be a nonempty set and F(X) be the set of nonempty finite subsets of X. The paper deals with the extended metrics τ : F(X) → R recently introduced by Peter Balk. Balk's metrics and their restriction to the family of sets A with ∣A∣ ⩽ n make possible to consider "distance functions" with n variables and related them quantities. In particular, we study such type generalized diameters diamτn and find conditions under which B → diamτn B is a Balk's metric. We prove the necessary and sufficient conditions under which the restriction τ to the set of A ϵ F(X) with ∣A∣ ⩽ 3 is a symmetric G-metric. An infinitesimal analog for extended by Balk metrics is constructed. [ABSTRACT FROM AUTHOR]

In this paper, we determine, for a given t-norm T , the algebraic structures of the set of decisive coalitions of a Fuzzy Aggregation Rule (FAR) defined from the set of profiles of fuzzy T -pre-orders to the set of fuzzy T -pre-orders. More precisely, we show that the set of decisive coalitions is a pre-filter on the finite set of voters when the FAR satisfies Pareto Condition (PC). When the FAR satisfies PC and Independence of Irrelevant Alternatives (IIA), we determine additional conditions on social preferences (range) under which the set of decisive coalitions is respectively a filter and an ultrafilter on the set of voters. From these two results, we outline two fuzzy Arrow-type results: the generator of the filter is an oligarchy (that is a decisive coalition where every member has a veto) for the FAR and the generator of the ultrafilter is a dictator for the FAR. We deduce that the usual FAR, defined in Dutta (1987) [7] and analyzed in Fono and Andjiga (2005) [8] , is an oligarchy by showing that its set of decisive coalitions is the filter having the set of voters as unique element. [ABSTRACT FROM AUTHOR]

The operation of a discrete group G naturally extends to the Stone-Čech compactification βG of G so that for each a ∈ G, the left translation βG B ∋ x → βG is continuous, and for each q ∈ βG, the right translation βG B ∋ x → xq ∈ βG is continuous. We show that for every Abelian group G with finitely many elements of order 2 such that |G| is not Ulam-measurable and for every p,q ∈ G* = βG \ G, the multiplication βG X βG ∋ (x,y) → xy ∈ βG is discontinuous at (p,q). We also show that it is consistent with ZFC, the system of usual axioms of set theory, that for every Abelian group G and for every p,q ∈ G*, the left translation G* ∋ x → px ∈ G* is discontinuous at [ABSTRACT FROM AUTHOR]

For every integer µ ≥ 3, there exists a function fµ: N+ → N+ such that the following holds: (1) fµ(k) = 2k -- µ for k large enough; (2) if ... is a finite nonempty collection of subalgebras of P(X) such that ∩... B is not fµ (#(...))-saturated, for all nonempty ... ⊆ ..., then ∪ ≄ P(X). [ABSTRACT FROM AUTHOR]

We use filter representations of the $${\mathcal {LUC}}$$ -compactification of a topological group $$G$$ and the Stone-Čech compactification of $$G_d$$ to describe the continuous homomorphism $$\pi :\beta G_d \rightarrow G^{{\mathcal {LUC}}}$$ . We apply this method to study algebraic properties of $$G^{{\mathcal {LUC}}}$$ . [ABSTRACT FROM AUTHOR]

The problem of how to rationally aggregate probability measures occurs in particular (i) when a group of agents, each holding probabilistic beliefs, needs to rationalise a collective decision on the basis of a single 'aggregate belief system' and (ii) when an individual whose belief system is compatible with several (possibly infinitely many) probability measures wishes to evaluate her options on the basis of a single aggregate prior via classical expected utility theory (a psychologically plausible account of individual decisions). We investigate this problem by first recalling some negative results from preference and judgment aggregation theory which show that the aggregate of several probability measures should not be conceived as the probability measure induced by the aggregate of the corresponding expected utility preferences. We describe how McConway's (Journal of the American Statistical Association, 76(374):410-414, ) theory of probabilistic opinion pooling can be generalised to cover the case of the aggregation of infinite profiles of finitely additive probability measures, too; we prove the existence of aggregation functionals satisfying responsiveness axioms à la McConway plus additional desiderata even for infinite electorates. On the basis of the theory of propositional-attitude aggregation, we argue that this is the most natural aggregation theory for probability measures. Our aggregation functionals for the case of infinite electorates are neither oligarchic nor integral-based and satisfy (at least) a weak anonymity condition. The delicate set-theoretic status of integral-based aggregation functionals for infinite electorates is discussed. [ABSTRACT FROM AUTHOR]

After defining continuous extensions of binary relations on the set N of natural numbers to its Stone-Çech compactification βN, we establish some results about one of such extensions. This provides us with one possible divisibility relation on βN, ⊥, and we introduce a few more, defined in a natural way. For some of them we find equivalent conditions for divisibility. Finally, we mention a few facts about prime and irreducible elements of (βN, ∙). The motivation behind all this is to try to translate problems in elementary number theory into βN. [ABSTRACT FROM AUTHOR]

Let N be the discrete additive semigroup of natural numbers and let βN be the Stone-Čech compactification of N. The addition on N extends to an operation + on βN making it a right topological semigroup, and to an operation * making it a left topological semigroup. The semigroup (βN,*) is the opposite of the semigroup (βN, +): p * q = q + p: We list all 3-element idempotent semigroups that have algebraic copies in (βN, +). As a consequence we obtain that (βN, +) and (βN, *) are not algebraically isomorphic. [ABSTRACT FROM AUTHOR]

We study ultrafilters of broadly understood measurable spaces and possibilities of their application as generalized elements in the construction of attraction sets in abstract reachability problems with constraints of asymptotic nature. A class of measurable spaces is specified for which all ultrafilters including free ultrafilters (with empty intersection of all of their sets) are built constructively. [ABSTRACT FROM AUTHOR]

This short note contains the proofs of two small but somewhat surprising results about ultrafilters on $$\mathbb {N}$$ : (1) strongly summable ultrafilters are rapid, (2) every rapid ultrafilter induces a closed left ideal of rapid ultrafilters. As a consequence, there will be rapid minimal idempotents in all models of set theory with rapid ultrafilters. The history of this result has been published as an experiment in mathematical writing on the author's website (Krautzberger, One Day in Colorado or Strongly Summable Ultrafilters are Rapid, ) and (Krautzberger, Rapid Idempotent Ultrafilters, ) where you can can also find additional remarks by Blass and Hindman, offering a form of peer-review. [ABSTRACT FROM AUTHOR]

A ballean is a set X endowed with some family F of its subsets, called the balls, in such a way that (X,F) can be considered as an asymp- totic counterpart of a uniform topological space. Given a cardinal κ, we define F using a natural order structure on κ. We characterize balleans up to coarse equivalence, give the criterions of metrizability and cellularity, calculate the basic cardinal invariant of these balleans. We conclude the paper with discussion of some special ultrafilters on cardinal balleans. [ABSTRACT FROM AUTHOR]

Let G be a countably infinite discrete group, let ßG be the Stone--Cech compactification of G, and let ... . The left (right) preordering on idempotents of ... is defined by ... . As any compact Hausdorff right topological semigroup, ... has right maximal idempotents. We show (in ZFC) that there are left maximal idempotents in ... . In the case G=ZG=Z, this is the answer to a long standing question. [ABSTRACT FROM AUTHOR]

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In memory of my father. LetXbe the prime spectrum of a ring. In Fontana and Loper [5] the authors define a topology onXby using ultrafilters and show that this topology is precisely the constructible topology. In this paper we generalize the construction given in Fontana and Loper [5] and, starting from a setXand a collection of subsets ℱ ofX, we define by using ultrafilters a topology onXin which ℱ is a collection of clopen sets. We use this construction for giving a new characterization of spectral spaces and several examples of spectral spaces. [ABSTRACT FROM PUBLISHER]

We study representations of ultrafilters of broadly understood measurable spaces that are realized by means of generalized Cartesian products. The structure of the arising ultrafilter space is established; in more traditional measurable spaces, this structure reduces to the realization of a Stone compact space in the form of a Tychonoff product. The developed methods can be applied to the construction of extensions of abstract reachability problems with constraints of asymptotic nature; in such extensions, ultrafilters can be used as generalized elements, which admits a conceptual analogy with the Stone-Čech compactification. The proposed implementation includes the possibility of using measurable spaces, for which the set of free ultrafilters can be described completely. This yields an exhaustive representation of the corresponding Stone compact space for measurable spaces with algebras of sets. The present issue is devoted to I.I. Eremin's jubilee; the author had many discussion with him on very different topics related to mathematical investigations, and the discussions inevitably led to a deeper understanding of their essence. The author appreciates the possibility of such communication and is grateful to Eremin, who contributed significantly to the development of the mathematical science and education in the Urals. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a countably infinite discrete group and let βG be the Stone–Čech compactification of G. Let denote the finest decomposition of into closed left ideals of βG with the property that the corresponding quotient space of is Hausdorff, and the finest decomposition of into closed left ideals of βG. We show that there is a dense subset of points such that . In particular, is finer than . [Copyright &y& Elsevier]

Abstract: Let X be an unbounded metric space, for all and . We endow X with the discrete topology and identify the Stone–Čech compactification βX of X with the set of all ultrafilters on X. Our aim is to reveal some features of algebra in βX similar to the algebra in the Stone–Čech compactification of a discrete semigroup [6]. We denote and, for , write if and only if there is such that for each , where . A subset is called invariant if and imply . We characterize the minimal closed invariant subsets of X, the closure of the set , and find the number of all minimal closed invariant subsets of . For a subset and , we denote and say that a subset is an ultracompanion of Y if for some . We characterize large, thick, prethick, small, thin and asymptotically scattered spaces in terms of their ultracompanions. [Copyright &y& Elsevier]