Recent zbMATH articles in MSC 03Ehttps://zbmath.org/atom/cc/03E2024-07-17T13:47:05.169476ZWerkzeugWays of destructionhttps://zbmath.org/1536.030212024-07-17T13:47:05.169476Z"Farkas, Barnabás"https://zbmath.org/authors/?q=ai:farkas.barnabas"Zdomskyy, Lyubomyr"https://zbmath.org/authors/?q=ai:zdomskyy.lyubomyrThe authors study forcing destructibility of Borel ideals. More precisely, they define notions of \(*\)-destructibility, \(+\)-destructibility and \(\infty\)-destructibility. In particular, a forcing notion \(\mathbb{P}\) is said to \(+\)-destroy an ideal \(\mathcal{I}\) if \(\mathbb{P}\) adds an \(\mathcal{I}\)-positive set \(X\) such that \(|X\cap A|<\infty\) for all ground model \(A\in \mathcal{I}\).
The papers discusses in detail many classical examples of Borel ideals, their forcing destructibility and associated cardinal invariants. Among others, the authors give a combinatorial characterization of real forcings, which can \(+\)-destroy a given Borel ideal and show that the generic real added by the Mathias-Prikry poset \(\mathbb{M}(\mathcal{I}^*)\) \(+\)-destroys \(\mathcal{I}\) iff \(\mathbb{M}(\mathcal{I}^*)\) \(+\)-destroys \(\mathcal{I}\) iff there is a poset which \(+\)-destroys \(\mathcal{I}\) iff \[\hbox{cov}^*(\mathcal{I},+)=\min\{|\mathcal{C}: \mathcal{C}\subseteq\mathcal{I}, \forall Y\in\mathcal{I}^+\exists C\in\mathcal{C}(|Y\cap C|=\infty)\}\] is uncountable. In addition, they characterize when the generic real added by the Laver-Prikry poset \(\mathbb{L}(\mathcal{I}^*)\) \(+\)-destroys \(\mathcal{I}\).
Reviewer: Vera Fischer (Wien)Sigma-Prikry forcing. II: Iteration schemehttps://zbmath.org/1536.030222024-07-17T13:47:05.169476Z"Poveda, Alejandro"https://zbmath.org/authors/?q=ai:poveda.alejandro"Rinot, Assaf"https://zbmath.org/authors/?q=ai:rinot.assaf"Sinapova, Dima"https://zbmath.org/authors/?q=ai:sinapova.dimaThis is the second part of a series of three papers. In the first part [Can. J. Math. 73, No. 5, 1205--1238 (2021; Zbl 1509.03149)], the authors introduced a class of notions of forcing which they called \(\Sigma\)-Prikry. Many of the known Prikry-like forcings which deal with singular cardinals of countable cardinality are \(\Sigma\)-Prikry. They showed that for a \(\Sigma\)-Prikry poset \(\mathbb{P}\) and a \(\mathbb{P}\)-name for a nonreflecting stationary set \(T\) there exists a \(\Sigma\)-Prikry poset that projects to \(\mathbb{P}\) and kills the stationarity of \(T\).
Here in the second part, the authors study the iteration of \(\Sigma\)-Prikry forcings. They develop a general schema for iterating \(\Sigma\)-Prikry posets. They investigate iterations of length \(\kappa^{++}\) (for \(\kappa\) singular).
As an application they obtain a streamlined proof of a result of \textit{A. Sharon} [Weak squares, scales, stationary reflection and the failure of SCH. Tel Aviv: Tel Aviv University (PhD Thesis) (2005)]: If \(\kappa\) is the limit of a countable increasing sequence of supercompact cardinals, then there exists a cofinality-preserving forcing extension in which \(\kappa\) remains a strong limit, every finite collection of stationary subsets of \(\kappa^+\)reflects simultaneously, and \(2^{\kappa} = \kappa^{++}\).
Reviewer: Martin Weese (Potsdam)Capturing sets of ordinals by normal ultrapowershttps://zbmath.org/1536.030232024-07-17T13:47:05.169476Z"Habič, Miha E."https://zbmath.org/authors/?q=ai:habic.miha-e"Honzík, Radek"https://zbmath.org/authors/?q=ai:honzik.radekFor infinite cardinals \(\kappa, \lambda\) let the local capturing property \(\mathrm{LCP}(\kappa, \lambda)\) be the statement that for every \(x \subseteq \lambda,\) there is a normal measure \(U\) on \(\kappa\) such that \(x\) is in \(\mathrm{Ult}(V, U)\). Let also the capturing property \(\mathrm{CP}(\kappa, \lambda)\) be the statement that there is a normal measure \(U\) on \(\kappa\) such that \(P(\lambda)\) is in \(\mathrm{Ult}(V, U)\). So clearly \(\mathrm{CP}(\kappa, \lambda)\) implies \(\mathrm{LCP}(\kappa, \lambda)\) and \(\mathrm{CP}(\kappa, 2^{\kappa})\) fails. Also \(\mathrm{LCP}(\kappa, (2^{\kappa})^+)\) fails.
By a result of \textit{J. Cummings} [J. Symb. Log. 58, No. 1, 240--248 (1993; Zbl 0785.03036)], the consistency of \(\mathrm{CP}(\kappa, \kappa^+)\) is equiconsistent with the existence of a \((\kappa+2)\)-strong cardinal.
In this paper, the authors continue this study and prove some results. The first main result of the paper is that \(\mathrm{LCP}(\kappa, \kappa^+)\) is equiconsistent with the existence of a measurable cardinal \(\kappa\) with \(o(\kappa)=\kappa^{++}\). Indeed, the authors show that if \(V=L[\mathcal{U}]\), where \(\mathcal{U}\) is a coherent sequence of normal measures of length \(\kappa+1\) with \(o^{\mathcal{U}}(\kappa)=\kappa^{++}\), then \(\mathrm{LCP}(\kappa, \kappa^+)\) holds in \(V\). The other direction is proved by showing that if \(\mathrm{LCP}(\kappa, 2^\kappa)\) holds, then \(o(\kappa)=(2^\kappa)^+\).
The second main result of th paper is a sterengthening of Cummings result [loc. cit.]. Starting from a \((\kappa+2)\)-strong cardinal \(\kappa\), the authors find a generic extension in which \(\mathrm{CP}(\kappa, \kappa^+)\) holds and \(\kappa\) is the least measurable cardinal. The proof of this result is rather long and takes most part of the paper. The main difficulty is to force \(\mathrm{CP}(\kappa, \kappa^+)\), while making \(\kappa\) the least measurable cardinal. The way the authors ovecome the second problem is that they kill all measurable cardinals smaller than \(\kappa\), using ideas of \textit{A. W. Apter} and \textit{S. Shelah} [Trans. Am. Math. Soc. 349, No. 5, 2007--2034 (1997; Zbl 0876.03030); ibid. 349, No. 1, 103--128 (1997; Zbl 0864.03036)]. Some related results are obtained too, by the same method, for example if GCH holds and \(\kappa\) is \(\kappa^+\)-supercompact, then in a forcing extension \(\mathrm{CP}(\kappa, \kappa^+)\) holds and \(\kappa\) is \(\kappa^+\)-supercompact and the least measurable cardinal.
They also force, from large cardinals, a model in which \(\mathrm{CP}(\kappa, \kappa^+)\) holds, while \(\mathrm{LCP}(\kappa, \kappa^{++})\) fails. The paper also contains a list of open questions which are distributed in the paper.
Reviewer: Mohammad Golshani (Tehran)A benchmark similarity measures for Fermatean fuzzy setshttps://zbmath.org/1536.030242024-07-17T13:47:05.169476Z"Khan, Faiz Muhammad"https://zbmath.org/authors/?q=ai:khan.faiz-muhammad"Khan, Imran"https://zbmath.org/authors/?q=ai:khan.imran-a|khan.imran-parvez|khan.imran-shahzad|khan.imran-f"Ahmad, Waqas"https://zbmath.org/authors/?q=ai:ahmad.waqasSummary: In this paper, we utilized triangular conorms (S-norm). The essence of using S-norm is that the similarity order does not change using different norms. In fact, we are investigating for a new conception for calculating the similarity of two Fermatean fuzzy sets. For this purpose, utilizing an S-norm, we first present a formula for calculating the similarity of two Fermatean fuzzy values, so that they are truthful in similarity properties. Following that, we generalize a formula for calculating the similarity of the two Fermatean fuzzy sets which prove truthful in similarity conditions. Finally, various numerical examples have been presented to elaborate this method.On the number of algebraically closed subfields of \(\mathbb C\)https://zbmath.org/1536.120012024-07-17T13:47:05.169476Z"Katre, S. A."https://zbmath.org/authors/?q=ai:katre.shashikant-aAssuming ZFC, the author shows that \( \mathbb C\) contains \(2^{2^\aleph_0}\) algebraically closed fields, and also \(|\Aut(\mathbb(C))|=2^{2^\aleph_0}\) while \(|\Aut \mathbb(R))|=\{\mathrm{id}\}\). See, e.g., [\textit{H. Kestelman}, Proc. Lond. Math. Soc. (2) 53, 1--12 (1951; Zbl 0042.39304)] for earlier work on this subject.
Reviewer: Olaf Teschke (Berlin)Michael spaces and ultrafiltershttps://zbmath.org/1536.540112024-07-17T13:47:05.169476Z"Martínez-Celis, Arturo"https://zbmath.org/authors/?q=ai:martinez-celis.arturoThe subject of the paper is Michael spaces (Lindelöf spaces which have a non-Lindelöf product with the Baire space) and ultrafilters. The most important theorem of the paper relates to Michael ultrafilters -- an ultrafilter \(\mathscr{U}\) is Michael if for every compact \(K\subseteq \omega^{\omega}\), if \(\operatorname{cof}(K/\mathscr{U})>\omega\), then \(\operatorname{cof}(K/\mathscr{U})\geqslant \operatorname{cof}(\omega^{\omega}/\mathscr{U})\). Theorem 1.2 states what the relationship is between Michael spaces and Michael ultrafilters -- if there is a Michael ultrafilter, then there is a Michael space.
The main result of the paper is Theorem 2.7, which states that a selective ultrafilter \(\mathscr{U}\) is Michael if and only if \(\operatorname{cof}(\omega^{\omega}/\mathscr{U})\leqslant \operatorname{cin}(\omega^{\omega}/\mathscr{U})\).
The implication to the right runs as follows: \(\operatorname{cin}(\omega^{\omega}/\mathscr{U})=\operatorname{cof}(K_0/\mathscr{U})\geqslant \operatorname{cof}(\omega^{\omega}/\mathscr{U})\). In equality, the author takes advantage of the selectivity of the ultrafilter (exactly being a q-point) and then he applies Proposition 2.4. which says that \(\operatorname{cin}(\omega^{\omega}/\mathscr{U})=\operatorname{cof}(K_0/\mathscr{U})\). In the inequality, the author uses Michael's definition of ultrafilter, the selectivity of ultrafilter (exactly being a p-point) and Proposition 2.1, which gives us: \(\operatorname{cof}(K_0/\mathscr{U})>\omega\).
In the implication in the other direction, the author uses the concept of an internally unbounded set: Given a filter \(\mathscr{F}\) on \(\omega\), a compact set \[K=\{f\in \omega^{\omega}: \mathop\forall_{n\in \omega} f_{\restriction{n}}\in T \}\subseteq \omega^{\omega}\] (where \(T\subseteq \omega^{<\omega}\) is a tree) is internally unbounded in \(\mathscr{F}\) if for every \(f\in K\) and every \(s\in T\) there is \(g\in K\) extending \(s\) such that \(\{n\in \omega: g(n)\geqslant f(n)\}\in \mathscr{F}\). It also uses Lemma 1.4 and Proposition 2.6 related to this concept.
Lemma 1.4: Given an ultrafilter \(\mathscr{U}\) and a compact set \(K\subseteq \omega^{\omega}\) such that \(\operatorname{cof}(K/\mathscr{U})>\omega\), there is a compact \(K'\subseteq K\) internally unbounded in \(\mathscr{U}\) such that \(\operatorname{cof}(K/\mathscr{U})=\operatorname{cof}(K'/\mathscr{U})\).
Proposition 2.6: Assume \(\mathscr{U}\) is a selective ultrafilter, if \[K=\{f\in \omega^{\omega}: \mathop\forall_{n\in \omega} f_{\restriction{n}}\in T \}\subseteq \omega^{\omega}\] is a compact set internally unbounded on \(\mathscr{U}\) then \(\operatorname{cof}(K/\mathscr{U})\geqslant \operatorname{cof}(K_0/\mathscr{U})\).
From Lemma 1.4, we know that there exists a compact set \(K'\subseteq K\) such that \(\operatorname{cof}(K/\mathscr{U})=\operatorname{cof}(K'/\mathscr{U})\). Then, the author gets a sequence of inequalities: \(\operatorname{cof}(K'/\mathscr{U})\geqslant \operatorname{cof}(K_0/\mathscr{U})=\operatorname{cin}(\omega^{\omega}/\mathscr{U})\geqslant \operatorname{cof}(\omega^{\omega}/\mathscr{U})\). The first inequality is obtained from Proposition 2.6, and the equality is obtained from Proposition 2.4. Thus, \(\mathscr{U}\) is a Michael ultrafilter.
Under the assumption \(\operatorname{cov}(\mathcal{M})=\mathfrak{c}\) the author shows using transfinite induction that any filter with a character smaller than \(\mathfrak{c}\) can be extended to a Michael ultrafilter (Theorem 1.5).
At the end of the paper, the author shows that there is a model without Michael ultrafilters.
Reading the paper, you can find 6 interesting open-ended questions.
Reviewer: Krzysztof Kowitz (Gdańsk)A q-rung orthopair fuzzy decision-making model with new score function and best-worst method for manufacturer selectionhttps://zbmath.org/1536.901012024-07-17T13:47:05.169476Z"Xiao, Liming"https://zbmath.org/authors/?q=ai:xiao.liming"Huang, Guangquan"https://zbmath.org/authors/?q=ai:huang.guangquan"Pedrycz, Witold"https://zbmath.org/authors/?q=ai:pedrycz.witold"Pamucar, Dragan"https://zbmath.org/authors/?q=ai:pamucar.dragan-s"Martínez, Luis"https://zbmath.org/authors/?q=ai:martinez.luis-manuel"Zhang, Genbao"https://zbmath.org/authors/?q=ai:zhang.genbaoSummary: Selecting the appropriate manufacturer is important in customized product development because a poor selection may delay the delivery schedule, increase costs, and even affect product quality. This selection task inherently involves ambiguous and imprecise information from decision-makers. The q-rung orthopair fuzzy (q-ROF) sets theory has been proven as a valuable instrument to model human uncertain expressions. However, the existing q-ROF score functions, an essential tool to rank q-ROF values, have some deficiencies, such as generating counterintuitive solutions and antilogarithm or division by zero problems. Furthermore, little research presented the modified best-worst methods (BWM) under q-ROF environments. Nevertheless, these models only generate crisp weights rather than q-ROF weights, which is against the original intention of the q-ROF BWM. To address these problems, this paper first introduces a novel q-ROF score function for measuring q-ROF values. A new q-ROF-BWM is then presented based on the q-ROF preference relations to determine the fuzzy criteria weights. Subsequently, an improved weighted aggregated sum product assessment (WASPAS) with q-ROF settings is presented. Based on them, an integrated q-ROF-BWM-WASPAS model is introduced to rank manufacturers. Additionally, a case study, sensitivity analysis, and several comparisons are conducted to illustrate and validate the usefulness of the developed model.On generating fuzzy Pareto solutions in fully fuzzy multiobjective linear programming \textit{via} a compromise methodhttps://zbmath.org/1536.902522024-07-17T13:47:05.169476Z"Arana-Jiménez, Manuel"https://zbmath.org/authors/?q=ai:arana-jimenez.manuelSummary: In the present paper, it is unified and extended recent contributions on fully fuzzy multiobjective linear programming, and it is proposed a new method for obtaining fuzzy Pareto solutions of a fully fuzzy multiobjective linear programming problem. For its formulation, triangular fuzzy numbers and variables are combined with fuzzy partial orders and fuzzy arithmetic, and no ranking functions are required. By means of solving related crisp multiobjective linear problems, it is provided algorithms to generate fuzzy Pareto solutions; in particular, to generate compromise fuzzy Pareto solutions, what is a novelty in this field.Optimization problems subject to addition-Łukasiewicz-product fuzzy relational inequalities with applications in urban sewage treatment systemshttps://zbmath.org/1536.902542024-07-17T13:47:05.169476Z"Qiu, Jianjun"https://zbmath.org/authors/?q=ai:qiu.jianjun"Yang, Xiaopeng"https://zbmath.org/authors/?q=ai:yang.xiaopengSummary: In this paper, fuzzy relational inequalities with addition-Łukasiewicz-product compositions are first introduced for describing the urban sewage treatment systems. It's well known that a classical fuzzy relational system usually consists of two composed operations. The most commonly used composed operations consists of max-min and max-product. But in our introduced fuzzy relational system, there exist three composed operations, i.e., addition-Łukasiewicz-product. Due to the specific composed operations, the construction for the corresponding solution set is much different. For a consistent fuzzy relation inequality system, composed by addition-Łukasiewicz-product, there exist some maximal solutions and a unique minimum solution. Especially, the number of the maximal solutions might be infinite. In fact, the solution set can be constructed by means of these maximal solutions and the sole minimum solution. Properties of the solutions to an addition-Łukasiewicz-product system is investigated. Following the presented properties, we further study the corresponding single-level and bilevel optimization problems. Two effective resolution approaches are developed and demonstrated by some numerical experiments.