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This paper is the second in the series celebrating the mathematical works of Professor Themba Dube. In this sequel, we give prominence to Dube’s pivotal contributions on point free convergence at the unstructured frame level, in the category of locales, and on his noteworthy conceptions on extensions and frame quotients. We distill and draw attention to particular studies of Dube on filters and his novel characterizations of certain conservative pointfree properties by filter and ultrafilter convergence, notably normality, almost realcompactness, and pseudocompactness. We also feature Dube’s joint work on convergence and clustering of filters in Loc and coconvergence and coclustering of ideals in the category Frm. [ABSTRACT FROM AUTHOR]

This paper is the second in the series celebrating the mathematical works of Professor Themba Dube. In this sequel, we give prominence to Dube's pivotal contributions on pointfree convergence at the unstructured frame level, in the category of locales, and on his noteworthy conceptions on extensions and frame quotients. We distill and draw attention to particular studies of Dube on filters and his novel characterizations of certain conservative pointfree properties by filter and ultrafilter convergence, notably normality, almost realcompactness, and pseudocompactness. We also feature Dube's joint work on convergence and clustering of filters in Loc and coconvergence and coclustering of ideals in the category Frm.

In their recent paper on posets with a pseudocomplementation denoted by $*$ the first and the third author introduced the concept of a $*$-ideal. This concept is in fact an extension of a similar concept introduced in distributive pseudocomplemented lattices and semilattices by several authors, see References. Now we apply this concept of a c-ideal (dually, c-filter) to complemented posets where the complementation need neither be antitone nor an involution, but still satisfies some weak conditions. We show when an ideal or filter in such a poset is a c-ideal or c-filter, and we prove basic properties of them. Finally, we prove the so-called separation theorems for c-ideals. The text is illustrated by several examples.

Let D be a domain and let Int (D) and In t R (D) be the ring of integer-valued polynomials and the ring of integer-valued rational functions, respectively. Skolem proved that if I is a finitely-generated ideal of Int (Z) with all the value ideals of I not being proper, then I = Int (Z). This is known as the Skolem property, which does not hold in Z [ x ]. One obstruction to Int (D) having the Skolem property is the existence of unit-valued polynomials. This is no longer an obstruction when we consider the Skolem property on In t R (D). We determine that the Skolem property on In t R (D) is equivalent to the maximal spectrum being contained in the ultrafilter closure of the set of maximal pointed ideals. We generalize the Skolem property using star operations and determine an analogous equivalence under this generalized notion. [ABSTRACT FROM AUTHOR]

In order to define the concept of nearness between L -fuzzy rough sets, we introduce the concept of Čech L -fuzzy rough proximity spaces. On this new nearness structure, we prove some basic topological results. To support the proposed approach, examples are given. Further, we introduce L -fuzzy rough grills, L -fuzzy rough filters and L -fuzzy rough clusters and on Čech L -fuzzy rough proximity spaces, and find the relationship between them. [ABSTRACT FROM AUTHOR]

In their recent paper on posets with a pseudocomplementation denoted by * the first and the third author introduced the concept of a *-ideal. This concept is in fact an extension of a similar concept introduced in distributive pseudocomplemented lattices and semilattices by several authors, see References. Now we apply this concept of a c-ideal (dually, c-filter) to complemented posets where the complementation need neither be antitone nor an involution, but still satisfies some weak conditions. We show when an ideal or filter in such a poset is a c-ideal or c-filter, and we prove basic properties of them. Finally, we prove the so-called separation theorems for c-ideals. The text is illustrated by several examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the concept of nearness between L-fuzzy rough sets and the concepts of L-fuzzy rough grills and L-fuzzy rough δℱ-clusters on a Čech L-fuzzy rough proximity space (X,δℱ). Also, we investigate the relationship among L-fuzzy rough grills, L-fuzzy rough δℱ-clan and L-fuzzy rough δℱ-clusters. Moreover, we introduce the L-fuzzy rough LO-proximity spaces and study some of its basic properties. An adequate number of examples support the study. [ABSTRACT FROM AUTHOR]

In this paper, convergences of a filter and a net have been characterized through ideal on topological spaces. Furthermore, we characterized the local function in an ideal topological space in terms of convergence of filter. Using Zorn's Lemma, we have found a maximal element in the collection of all proper ideals on a nonempty set which is called maximal ideal. We provide a convenient characterization of maximal ideals. We also consider simple properties of the image of an ideal, a net and various local functions under a homeomorphism. [ABSTRACT FROM AUTHOR]

Given a topological space X, let UX and ηX: X → UX denote, respectively, the Salbany compactification of X and the compactification map called the Salbany map of X. For every continuous function f: X → Y, there is a continuous function Uf: UX → UY, called the Salbany lift of f, satisfying (Uf) ◦ ηX = ηY ◦ f. If a continuous function f: X → Y has a stably compact codomain Y, then there is a Salbany extension F: UX → Y of f, not necessarily unique, such that F ◦ ηX = f. In this paper, we give a condition on a space such that its Salbany map is open. In particular, we prove that in a class of Hausdorff spaces, the spaces with open Salbany maps are precisely those that are almost discrete. We also investigate openness of the Salbany lift and a Salbany extension of a continuous function. Related to open continuous functions are initial maps as well as nearly open maps. It turns out that the Salbany map of every space is both initial and nearly open. We repeat the procedure done for openness of Salbany maps, Salbany lifts and Salbany extensions to their initiality and nearly openness. [ABSTRACT FROM AUTHOR]

The problem of reachability in a topological space is studied under constraints of asymptotic nature arising from weakening the requirement that the image of a solution belong to a given set. The attraction set that arises in this case in the topological space is a regularization of certain kind for the image of the preimage of the mentioned set (the image and the preimage are defined for generally different mappings). When constructing natural compact extensions of the reachability problem with constraints of asymptotic nature generated by a family of neighborhoods of a fixed set, the case was studied earlier where the topological space in which the results of one or another choice of solution are realized satisfies the axiom subscript 푇 2 . In the present paper, for a number of statements related to compact extensions, it is possible to use for this purpose a subscript 푇 1 space, which seems to be quite important from a theoretical point of view, since it is possible to find out the exact role of the axiom subscript 푇 2 in questions related to correct extensions of reachability problems. We study extension models using ultrafilters of a broadly understood measurable space with detailing of the main elements in the case of a reachability problem in the space of functionals with the topology of a Tychonoff power of the real line with the usual -topology. The general constructions of extension models are illustrated by an example of a nonlinear control problem with state constraints. [ABSTRACT FROM AUTHOR]

We give a characterization of Ramsey ultrafilters on ω in terms of functions and their ultrafilter extensions. To do this, we prove that for any partition of there is a finite partition of such that any set that is homogeneous for is a finite union of sets that are canonical for . [ABSTRACT FROM AUTHOR]

Ultrafilters of broadly understood measurable spaces and their application as generalized elements in abstract reachability problems with constraints of asymptotic nature are considered. Constructions for the immersion of conventional solutions, which are points of a fixed set, into the space of ultrafilters and representations of "limit" ultrafilters realized with topologies of Wallman and Stone types are studied. The structure of the attraction set is established using constraints of asymptotic nature in the form of a nonempty family of sets in the space of ordinary solutions. The questions of implementation up to any preselected neighborhood of the attraction sets in the topologies of Wallman and Stone types are studied. Some analogs of the mentioned properties are considered for the space of maximal linked systems. [ABSTRACT FROM AUTHOR]

For a T 1 -space X , we denote by P R (X) the Pixley-Roy space over X. For p ∈ ω ⁎ , let X p = { p } ∪ ω be the subspace of the Stone-Čech compactification βω of the discrete space ω. Motivated by Gul'ko's theorem (Theorem 1.1), we show: (1) P R (X p) and P R (X q) are homeomorphic if and only if X p and X q are homeomorphic (i.e., p and q are type-equivalent), (2) if q is selective and X q can be embedded into P R (X p) , then X p and X q are homeomorphic, (3) if p is selective, then P R (X p) contains copies of some X q n (n ∈ N) which are pairwise non-homeomorphic, and (4) P R (X p) , P R (X p ⊕ X p) and P R (X p ⁎ X p) are pairwise non-homeomorphic, where X p ⁎ X p is the quotient space obtained by identifying the limit points of the topological sum X p ⊕ X p. [ABSTRACT FROM AUTHOR]

New classes , , and of topological spaces generalizing the class of -spaces are introduced. It is proved that all homogeneous compact subspaces of spaces in these classes and of some of their products are finite. Results on the Rudin–Keisler comparability of ultrafilters along which distinct sequences converge to the same point in - and -spaces are obtained. [ABSTRACT FROM AUTHOR]

In this work, we survey some research directions in which the ultraproduct construction and methods based on ultrafilters play significant roles. Rather different areas of mathematics have been considered: topics we are reviewing here include some aspects of the model theory of first-order and second-order existential logics, finite Ramsey theory and general topology. Special emphasis has been made for producing a uniform treatment and for highlighting interconnections between these different subjects. [ABSTRACT FROM AUTHOR]

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d'une function analytique d'une variable sur un corps value complet, Publ. Math. l'IHES 14(1) (1942) 47–75]. [ABSTRACT FROM AUTHOR]

Let $\mathcal {I}$ be an ideal on $\omega $. For $f,\,g\in \omega ^{\omega }$ we write $f \leq _{\mathcal {I}} g$ if $f(n) \leq g(n)$ for all $n\in \omega \setminus A$ with some $A\in \mathcal {I}$. Moreover, we denote $\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$ (in particular, $\mathcal {D}_{\mathrm {Fin}}$ denotes the family of all finite-to-one functions). We examine cardinal numbers $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$ and $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}}\times \mathcal {D}_{\mathrm {Fin}}))$ describing the smallest sizes of unbounded from below with respect to the order $\leq _{\mathcal {I}}$ sets in $\mathcal {D}_{\mathrm {Fin}}$ and $\mathcal {D}_{\mathcal {I}}$ , respectively. For a maximal ideal $\mathcal {I}$ , these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers. We show that $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}} \times \mathcal {D}_{\mathrm {Fin}})) =\mathfrak {b}$ for all ideals $\mathcal {I}$ with the Baire property and that $\aleph _1 \leq \mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) \leq \mathfrak {b}$ for all coanalytic weak P-ideals (this class contains all $\bf {\Pi ^0_4}$ ideals). What is more, we give examples of Borel (even $\bf {\Sigma ^0_2}$) ideals $\mathcal {I}$ with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\mathfrak {b}$ as well as with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) =\aleph _1$. We also study cardinals $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {J}} \times \mathcal {D}_{\mathcal {K}}))$ describing the smallest sizes of sets in $\mathcal {D}_{\mathcal {K}}$ not bounded from below with respect to the preorder $\leq _{\mathcal {I}}$ by any member of $\mathcal {D}_{\mathcal {J}}\!$. Our research is partially motivated by the study of ideal-QN-spaces: those cardinals describe the smallest size of a space which is not ideal-QN. [ABSTRACT FROM AUTHOR]

Given a coarse space (X, E), we endow X with the discrete topology and denote X ♯ = {p ∈ βG : each member P ∈ p is unbounded }. For p, q ∈ X ♯ , p||q means that there exists an entourage E ∈ E such that E[P] ∈ q for each P ∈ p. We say that (X, E) is orbitally discrete if, for every p ∈ X ♯ , the orbit p = {q ∈ X ♯ : p||q} is discrete in βG. We prove that every orbitally discrete space is almost finitary and scattered.

This article studies different topological properties of the space of maximal d -elements of an M -frame with a unit. We characterize when the space M a x (d L) is Hausdorff, answering the question posed in [2]. We also characterize other topological properties of M a x (d L) , namely zero-dimensional, discrete, and clopen π -base. The concept of weak-component elements is introduced here, as a generalized idea from the theory of rings, which is essential in the study of d -semiprime frames. [ABSTRACT FROM AUTHOR]

We study the definability of ultrafilter bases on ω in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in L we can construct Π 1 1 P-point and Q-point bases. We also show that the existence of a Δ n + 1 1 ultrafilter is equivalent to that of a Π n 1 ultrafilter base, for n ∈ ω . Moreover we introduce a Borel version of the classical ultrafilter number and make some observations. [ABSTRACT FROM AUTHOR]

We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations ∣ M and ∣ ~ . The set of its minimal elements proves to be very rich, and the ∣ ~ -hierarchy is used to get a better intuition of this richness. We find the place of the set of ∣ ~ -maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness of subsets of N , and compare it to other such notions, important for infinite combinatorics and topological dynamics. [ABSTRACT FROM AUTHOR]

Propounding a general categorical framework for the extension of dualities, we present a new proof of the de Vries Duality Theorem for the category KHaus of compact Hausdorff spaces and their continuous maps, as an extension of a restricted Stone duality. Then, applying a dualization of the categorical framework to the de Vries duality, we give an alternative proof of the extension of the de Vries duality to the category Tych of Tychonoff spaces that was provided by Bezhanishvili, Morandi and Olberding. In the process of doing so, we obtain new duality theorems for both categories, KHaus and Tych. [ABSTRACT FROM AUTHOR]

We prove that it is relatively consistent with \mathsf {ZFC} that every maximal ideal is Katětov above of all F_\sigma ideals. In particular, we prove that it is consistent that there is no Hausdorff ultrafilter. The main theorem answers questions from Mauro Di Nasso and Marco Forti [Proc. Amer. Math. Soc. 134 (2006), pp. 1809–1818], Jana Flašková [WDS'05 proceedings of contributed papers: part I - mathematics and computer sciences, 2005; Comment. Math. Univ. Carolin. 47 (2006), pp. 617–621; 10th Asian logic conference, World Sci. Publ., Hackensack, NJ, 2010], Osvaldo Guzmán and Michael Hrušák [Topology Appl. 259 (2019), pp. 242–250], and Mauro Di Nasso and Marco Forti [Logic and its applications, Contemp. Math., Amer. Math. Soc., Providence, RI, 2005], and gives a different model for a question from Michael Benedikt [J. Symb. Log. 63 (1998), pp. 638–662], which was originally solved by S. Shelah [Logic colloquium '95 (Haifa), lecture notes logic, Springer, Berlin, 1998]. [ABSTRACT FROM AUTHOR]

Background and Objective: Global market growth of reverse osmosis (RO) has led to an increase in annual disposal of membrane wastes. Therefore, evaluation of membrane waste management strategies is important to reduce their adverse environmental impacts. Due to the widespread domestic RO membrane market and their economic considerations, this study aims at investigation the direct recycling methods of RO membranes to extend their life cycles. Materials and Methods: Academic search engines and citation databases such Scopus and PubMed was used to retrieve relevant papers. Selected documents were analyzed and compared in three aspects of technical, economic and environmental issues. Results: Direct recycling of RO is performed with fouling removal and degradation of polyamide layer (PA) using oxidizing agents like KMnO4 and NaOCl. The degradation rate of the PA layer is controlled by optimizing the oxidant concentration during the oxidation process. Factors such as the type of membrane used, its storage conditions, the operating units’ conditions and the final expected product will determine the required concentration-time values. Strategies to reduce these values are very important from an economic and environmental point of view. Decreasing the concentration of oxidizing agent reduces the chlorinated and halogenated compounds emitted from the oxidizing unit which subsequently lessen their harmful environmental impacts and reduces the energy consumption required for treatment. Conclusion: The conversion of RO membranes to porous filters is technically possible by optimizing the conditions. In addition, the proper choice of RO membrane and final product type lead to economic and environmental productivity.

Let {Ri}i2I be an infinite family of rings and R = Qi2I Ri their product. In this paper, we investigate the prime spectrum of R by F-limits. Special attention is paid to relationship between the elements of Spec(Ri) and the elements of Spec(Qi2I Ri) use F-lim, also we give a new condition so that Qi2I Ri is a zero dimensional ring. [ABSTRACT FROM AUTHOR]

The Ultrapower Axiom is an abstract combinatorial principle inspired by the fine structure of canonical inner models of large cardinal axioms. In this paper, it is established that the Ultrapower Axiom implies that the Generalized Continuum Hypothesis holds above the least supercompact cardinal. [ABSTRACT FROM AUTHOR]

In this paper, a new generalization of semi-linear uniform spaces is introduced for the first time. This generalization will dispense with the chain condition imposed on a semi-linear uniform space which is given in Tallafha and Khalil (Eur J Pure Appl Math 2:231–238, 2009). We will show that several facts and results have been verified without that strong condition. Finally, we use the concept of ultrafilter and Cauchy filter to study the concept of precompact in this generalization of semi-linear uniform spaces. [ABSTRACT FROM AUTHOR]

Criteria are obtained for a filter F of subsets of a set I to be an intersection of finitely many ultrafilters, respectively, finitely many κ-complete ultrafilters for a given uncountable cardinal κ. From these, general results are deduced concerning homomorphisms on infinite direct product groups, which yield quick proofs of some results in the literature: the Łoś–Eda theorem (characterizing homomorphisms from a not-necessarily-countable direct product of modules to a slender module), and some results of Nahlus and the author on homomorphisms on infinite direct products of not-necessarily-associative k-algebras. The same tools allow other results of Nahlus and the author to be nontrivially strengthened, and yield an analog to one of their results, with nonabelian groups taking the place of k-algebras. We briefly examine the question of how the common technique used in applying the general results of this note to k-algebras on the one hand, and to nonabelian groups on the other, might be extended to more general varieties of algebras in the sense of universal algebra. In a final section, the Erd˝os–Kaplansky theorem on dimensions of vector spaces DI (D a division ring) is extended to reduced products DI /F, and an application is noted.

Let ∑ (X) be the collection of subrings of C(X) containing C ∗ (X) , where X is a Tychonoff space. For any A (X) ∈ ∑ (X) there is associated a subset υ A (X) of β X which is an A-analogue of the Hewitt real compactification υ X of X. For any A (X) ∈ ∑ (X) , let [A(X)] be the class of all B (X) ∈ ∑ (X) such that υ A (X) = υ B (X) . We show that for first countable non compact real compact space X, [A(X)] contains at least 2 c many different subalgebras no two of which are isomorphic in Theorem 3.8. [ABSTRACT FROM AUTHOR]

There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them (Goranko in Filter and ultrafilter extensions of structures: universal-algebraic aspects, preprint, 2007) comes from modal logic and universal algebra, and in fact goes back to Jónsson and Tarski (Am J Math 73(4):891–939, 1951; 74(1):127–162, 1952). Another one (Saveliev in Lect Notes Comput Sci 6521:162–177, 2011; Saveliev in: Friedman, Koerwien, Müller (eds) The infinity project proceeding, Barcelona, 2012) comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups (Hindman and Strauss in Algebra in the Stone–Čech Compactification, W. de Gruyter, Berlin, 2012) 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 (Lect Notes Comput Sci 6521:162–177, 2011; in: Friedman, Koerwien, Müller (eds) 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 (in On two types of ultrafilter extensions of binary relations. arXiv:2001.02456). 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 (in: Kennedy, de Queiroz (eds) On two concepts of ultrafilter extensions of first-order models and their generalizations, Springer, Berlin, 2017). [ABSTRACT FROM AUTHOR]

Given a coarse space (X, Ɛ), we endow X with the discrete topology and denote X# = {p 2 G: each member P 2 p is unbounded }. For p, q 2 X#, p||q means that there exists an entourage E 2 E such that E[P] 2 q for each P 2 p. We say that (X, Ɛ) is orbitally discrete if, for every p 2 X#, the orbit p = {q 2 X#: p||q} is discrete in G. We prove that every orbitally discrete space is almost finitary and scattered. [ABSTRACT FROM AUTHOR]

We say that an ultrafilter on an infinite group G is DTC if it determines the topological centre of the semigroup β G . If G has a subgroup of finite index in which conjugacy classes are all finite and uniformly bounded in size, then G does not admit a DTC ultrafilter. On the other hand, if G has no subgroup of finite index in which all conjugacy classes are finite, then G does admit a DTC ultrafilter. It follows that an infinite finitely generated group admits a DTC ultrafilter if and only if it has no abelian subgroup of finite index. [ABSTRACT FROM AUTHOR]

In this paper we investigate the structure of the set of maximal ideals of G (Ω) . The method of investigation passes through the use of the m - reduction and the ideas are analoguous to those in Gillman and Jerison (Rings of Continuous Functions, N.J. Van Nostrand, Princeton, 1960) for the investigation of maximal ideals of continuous functions on a Hausdorff space K. [ABSTRACT FROM AUTHOR]

The interplay between ultrafilters and unbounded subsets of ω ω with the order < ∗ of strict eventual domination is studied. Among the tools are special kinds of non-principal ("free") ultrafilters on ω . These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order ⊂ ∗ of almost inclusion. It is shown that the cofinality of such a base must be either b , the least cardinality of < ∗ -unbounded ("undominated") set, or d , the least cardinality of a < ∗ -cofinal ("dominating") set. The small uncountable cardinal π p is introduced. Consequences of b < π p and of r < d are explored; in particular, both imply b < d . Here r is the reaping number, and is also the least cardinality of a π -base for a free ultrafilter. Both of these inequalities are shown to occur if there exist simple P-points of different cofinalities (in other words, if b < d and there exist simple P b -points and P d -points), but this is a long-standing open problem. Six axioms on nonprincipal ultrafilters on ω and the relationships between them are discussed along with various models of set theory in which one or more are known to hold (or are known to fail). The strongest of these, Axiom 1, is that for every free ultrafilter U and for every < ∗ -unbounded < ∗ -chain C of increasing functions in ω ω , C is also unbounded in the ultraproduct ω ω , < U . The other axioms replace one or both quantifiers with "there exists." The negation of Axiom 3 in a model provides a family of normal sequentially compact spaces whose product is not countably compact. The question of whether such a family exists in ZFC, even with "normal" weakened to "regular", is a famous unsolved problem of set-theoretic topology, known as the Scarborough–Stone problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider natural generalizations of properties of the linkedness of families (of sets) and the supercompactness of topological spaces. In the first case, we analyze the "multiple" linkedness, assuming the nonemptiness of the intersection of sets from subfamilies, whose cardinality does not exceed some given positive integer . In the second case, we study the question of the existence of an (open) prebase such that any its covering has a subcovering, whose cardinality does not exceed . We consider maximal -linked (in the mentioned sense) subfamilies of a -system with "zero" and "unit" (a -system is a nonempty family closed with respect to finite intersections); these subfamilies are said to be maximal -linked systems or (for short) -MLS. We are interested in correlations between -MLS and ultrafilters (u/f) of a -system, including the "dynamics" in dependence of . Moreover, we consider bitopological spaces (BTS), whose elements are -MLS and u/f; in both cases, for constructing a BTS (a nonempty set with a pair of comparable topologies) we use topologies of Wallman and Stone types. The Wallman-type topology on the set of -MLS realizes an -supercompact (in the sense mentioned above) T1-space which represents an abstract analog of a superextension of a T1-space. We prove that the BTS of u/f of the initial -system is a subspace of the BTS whose points are -MLS; i. e., the corresponding "Wallman" and "Stone" topologies on the set of u/f are induced by the corresponding topologies on the set of -MLS. [ABSTRACT FROM AUTHOR]

We construct a model with a uniform ultrafilter U over a singular strong limit cardinal κ such that κ < cof (Ch (U)) < Ch (U) < 2 κ . [ABSTRACT FROM AUTHOR]

Background and Objective: Global market growth of reverse osmosis (RO) has led to an increase in annual disposal of membrane wastes. Therefore, evaluation of membrane waste management strategies is important to reduce their adverse environmental impacts. Due to the widespread domestic RO membrane market and their economic considerations, this study aims at investigation the direct recycling methods of RO membranes to extend their life cycles. Materials and Methods: Academic search engines and citation databases such Scopus and PubMed was used to retrieve relevant papers. Selected documents were analyzed and compared in three aspects of technical, economic and environmental issues. Results: Direct recycling of RO is performed with fouling removal and degradation of polyamide layer (PA) using oxidizing agents like KMnO4 and NaOCl. The degradation rate of the PA layer is controlled by optimizing the oxidant concentration during the oxidation process. Factors such as the type of membrane used, its storage conditions, the operating units’ conditions and the final expected product will determine the required concentration-time values. Strategies to reduce these values are very important from an economic and environmental point of view. Decreasing the concentration of oxidizing agent reduces the chlorinated and halogenated compounds emitted from the oxidizing unit which subsequently lessen their harmful environmental impacts and reduces the energy consumption required for treatment. Conclusion: The conversion of RO membranes to porous filters is technically possible by optimizing the conditions .In addition,the proper choice of RO membrane and final product type lead to economic and environmental productivity. [ABSTRACT FROM AUTHOR]

This paper examines collective decision-making with an infinite-time horizon setting. First, we establish a result on the collection of decisive sets: if there are at least four alternatives and Arrow's axioms are satisfied on the selfish domain, then the collection of decisive sets forms an ultrafilter. Second, we impose generalized versions of stationarity axiom for social preferences, which are substantially weaker than the standard version. We show that if any of our generalized versions are satisfied in addition to Arrow's axioms, then some generation is dictatorial. Moreover, we specify a very weak stationarity axiom that guarantees a possibility result. [ABSTRACT FROM AUTHOR]

We prove a characterization theorem for filters, proper filters and ultrafilters which is a kind of converse of Los's theorem. It is more natural than the usual intuition of these terms as large sets of coordinates, which is actually unconvincing in the case of ultrafilters. As a bonus, we get a very simple proof of Los's theorem. [ABSTRACT FROM AUTHOR]

We study a partial order on countably complete ultrafilters introduced by Ketonen [2] as a generalization of the Mitchell order. The following are our main results: the order is wellfounded; its linearity is equivalent to the Ultrapower Axiom, a principle introduced in the author's dissertation [1]; finally, assuming the Ultrapower Axiom, the Ketonen order coincides with Lipschitz reducibility in the sense of generalized descriptive set theory. [ABSTRACT FROM AUTHOR]

We continue the study of multivalued mappings with the BAD (bounded angular distortion) property which was initiated in the author's works in 2018. Using the technique of ultrafilters, we obtain a full description for the mutual adherence domains of the images of points under mappings of BAD class. We exhibit the example that illustrates all possible cases in the general formula for adherence domains. [ABSTRACT FROM AUTHOR]

A Michael space is a Lindelöf space which has a non-Lindelöf product with the Baire space. In this work, we present the notion of Michael ultrafilter and we use it to construct a Michael space under the existence of a selective ultrafilter and max ⁡ { b , g } = d. [ABSTRACT FROM AUTHOR]