Recent zbMATH articles in MSC 03Ehttps://zbmath.org/atom/cc/03E2021-05-28T16:06:00+00:00WerkzeugConsistency of a strong uniformization principle.https://zbmath.org/1459.030832021-05-28T16:06:00+00:00"Larson, Paul"https://zbmath.org/authors/?q=ai:larson.paul-b"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharonSummary: We prove the consistency of a strong uniformization principle for subsets of the Baire space of cardinality \(\aleph_{1}\).The isomorphism problem for tree-automatic ordinals with addition.https://zbmath.org/1459.030572021-05-28T16:06:00+00:00"Jain, Sanjay"https://zbmath.org/authors/?q=ai:jain.sanjay"Khoussainov, Bakhadyr"https://zbmath.org/authors/?q=ai:khoussainov.bakhadyr-m"Schlicht, Philipp"https://zbmath.org/authors/?q=ai:schlicht.philipp"Stephan, Frank"https://zbmath.org/authors/?q=ai:stephan.frankSummary: This paper studies tree-automatic ordinals (or equivalently, well-founded linearly ordered sets) together with the ordinal addition operation +. Informally, these are ordinals such that their elements are coded by finite trees for which the linear order relation of the ordinal and the ordinal addition operation can be determined by tree automata. We describe an algorithm that, given two tree-automatic ordinals with the ordinal addition operation, decides if the ordinals are isomorphic.On the comparability of cardinals in the absence of the axiom of choice.https://zbmath.org/1459.030812021-05-28T16:06:00+00:00"Tachtsis, Eleftherios"https://zbmath.org/authors/?q=ai:tachtsis.eleftheriosSummary: It is a well-known result of Hartogs' that the statement ``for all sets \(x\) and \(y\), \(x\preceq y\) or \(y\preceq x\) (where `\(x\preceq y\)' means that there is a one-to-one map \(f:x\rightarrow y\))'' is equivalent to the Axiom of Choice (\(\mathsf{AC}\)) (the latter in the disguise of the well-ordering theorem, i.e., ``every set can be well-ordered''). A considerably stronger result by Tarski states that for any natural number \(n\geq 2\), the statement ``if \(x\) is a set consisting of \(n\) sets, then there exist distinct elements \(y,z\in x\) such that \(y\preceq z\) or \(z\preceq y\)'' is equivalent to \(\mathsf{AC}\). In this paper, we investigate the set-theoretic strength of the variant of Tarski's statement which concerns \textit{infinite} sets of sets, that is, ``if \(x\) is an infinite set of sets, then there exist distinct elements \(y,z\in x\) such that \(y\preceq z\) or \(z\preceq y\)''. We are mostly interested in denumerable (i.e., countably infinite) and continuum sized sets of sets. Among other results, we show that the above statement: (a) restricted to denumerable sets of sets, implies ``every Dedekind-finite set is finite'' and Ramsey's Theorem for pairs, and that the two implications are not reversible in \(\mathsf{ZF}\), (b) is equivalent to its restriction to denumerable sets of sets; {this settles the corresponding open problem} in Feldman and Orhon (2008), (c) restricted to continuum sized sets of sets, implies a certain version of the Kinna-Wagner selection principle, (d) restricted to sets of cardinality \(2^{\aleph_{\alpha}}\), where \(\aleph_{\alpha}\) is a regular aleph greater than \(\aleph_{0}\), is not implied by \(\mathsf{DC}_{\lambda}\) in \(\mathsf{ZF}\), for any infinite cardinal \(\lambda < \aleph_{\alpha}\), (e) restricted to sets of cardinality \(2^{\aleph_{0}}\), is not implied by any of the following: (1) \(\mathsf{AC}^{\mathsf{LO}}\) (\(\mathsf{AC}\) restricted to linearly orderable sets of non-empty sets; \(\mathsf{AC}^{\mathsf{LO}}\) is equivalent to \(\mathsf{AC}\) in \(\mathsf{ZF}\), but not equivalent to \(\mathsf{AC}\) in \(\mathsf{ZFA}\)) in \(\mathsf{ZFA}\), (2) \(\mathsf{LW}\) (``every linearly ordered set can be well-ordered''; \(\mathsf{LW}\) is equivalent to \(\mathsf{AC}\) in \(\mathsf{ZF}\), but not equivalent to \(\mathsf{AC}\) in \(\mathsf{ZFA}\)) in \(\mathsf{ZFA}\), (3) \(\mathsf{AC}^{\mathsf{WO}}\) (\(\mathsf{AC}\) restricted to well-orderable sets of non-empty sets) in \(\mathsf{ZF}\).On turbulent relations.https://zbmath.org/1459.030782021-05-28T16:06:00+00:00"López, Jesús A. Álvarez"https://zbmath.org/authors/?q=ai:alvarez-lopez.jesus-a"Candel, Alberto"https://zbmath.org/authors/?q=ai:candel.albertoSummary: This paper extends the theory of turbulence of Hjorth to certain classes of equivalence relations that cannot be induced by Polish actions. The results are applied to analyze the quasi-isometry relation and finite Gromov-Hausdorff distance relation in the space of isometry classes of pointed proper metric spaces, called the Gromov space.Structurable equivalence relations.https://zbmath.org/1459.030732021-05-28T16:06:00+00:00"Chen, Ruiyuan"https://zbmath.org/authors/?q=ai:chen.ruiyuan"Kechris, Alexander S."https://zbmath.org/authors/?q=ai:kechris.alexander-sSummary: For a class \(\mathcal K\) of countable relational structures, a countable Borel equivalence relation \(E\) is said to be \(\mathcal K\)-structurable if there is a Borel way to put a structure in \(\mathcal K\) on each \(E\)-equivalence class. We study in this paper the global structure of the classes of \(\mathcal K\)-structurable equivalence relations for various \(\mathcal K\). We show that \(\mathcal K\)-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of \(\mathcal K\)-structurable equivalence relations for various \(\mathcal K\), under inclusion, and show that it is a distributive lattice; this implies that the Borel reducibility preordering among countable Borel equivalence relations contains a large sublattice. Finally, we consider the effect on \(\mathcal K\)-structurability of various model-theoretic properties of \(\mathcal K\). In particular, we characterize the \(\mathcal K\) such that every \(\mathcal K\)-structurable equivalence relation is smooth, answering a question of \textit{A. S. Marks} [J. Math. Log. 17, No. 1, Article ID 1750003, 50 p. (2017; Zbl 1420.03121)].On generalized soft equality and soft lattice structure.https://zbmath.org/1459.030862021-05-28T16:06:00+00:00"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahid"Ali, Basit"https://zbmath.org/authors/?q=ai:ali.basit"Romaguera, Salvador"https://zbmath.org/authors/?q=ai:romaguera.salvadorSummary: \textit{D. Molodtsov} [Comput. Math. Appl. 37, No. 4--5, 19--31 (1999; Zbl 0936.03049)] introduced soft sets as a mathematical tool to handle uncertainty associated with real world data based problems. In this paper, we propose some new concepts which generalize existing comparable notions. We introduce the concept of generalized soft equality (denoted as $g$-soft equality) of two soft sets and prove that the so-called lower and upper soft equality of two soft sets imply $g$-soft equality but the converse does not hold. Moreover, we give tolerance or dependence relation on the collection of soft sets and soft lattice structures. Examples are provided to illustrate the concepts and results obtained herein.Monadic second order logic with measure and category quantifiers.https://zbmath.org/1459.030542021-05-28T16:06:00+00:00"Mio, Matteo"https://zbmath.org/authors/?q=ai:mio.matteo"Skrzypczak, Michał"https://zbmath.org/authors/?q=ai:skrzypczak.michal"Michalewski, Henryk"https://zbmath.org/authors/?q=ai:michalewski.henrykSummary: We investigate the extension of Monadic Second Order logic, interpreted over infinite words and trees, with generalized ``for almost all'' quantifiers interpreted using the notions of Baire category and Lebesgue measure.Polish topologies for graph products of cyclic groups.https://zbmath.org/1459.030792021-05-28T16:06:00+00:00"Paolini, Gianluca"https://zbmath.org/authors/?q=ai:paolini.gianluca"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharonSummary: We give a complete characterization of the graph products of cyclic groups admitting a Polish group topology, and show that they are all realizable as the group of automorphisms of a countable structure. In particular, we characterize the right-angled Coxeter groups (resp. Artin groups) admitting a Polish group topology. This generalizes results from [the second author, Bull. Lond. Math. Soc. 35, No. 1, 1--7 (2003; Zbl 1033.20002); Isr. J. Math. 181, 477--507 (2011; Zbl 1233.20001); the authors, Axioms 6, No. 2, Paper No. 13, 4 p. (2017; Zbl 1422.20001)].A hierarchy of Ramsey-like cardinals.https://zbmath.org/1459.030852021-05-28T16:06:00+00:00"Holy, Peter"https://zbmath.org/authors/?q=ai:holy.peter"Schlicht, Philipp"https://zbmath.org/authors/?q=ai:schlicht.philippSummary: We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by \textit{V. Gitman} [J. Symb. Log. 76, No. 2, 519--540 (2011; Zbl 1222.03054)], and is based on certain infinite \textit{filter games}, but also has a range of equivalent characterizations in terms of elementary embeddings. The aim of this paper is to locate the Ramsey-like cardinals studied by Gitman, and other well-known large cardinal notions, in this hierarchy.Almost disjoint refinements and mixing reals.https://zbmath.org/1459.030752021-05-28T16:06:00+00:00"Farkas, Barnabás"https://zbmath.org/authors/?q=ai:farkas.barnabas"Khomskii, Yurii"https://zbmath.org/authors/?q=ai:khomskii.yurii"Vidnyánszky, Zoltán"https://zbmath.org/authors/?q=ai:vidnyanszky.zoltanSummary: We investigate families of subsets of \(\omega \) with almost disjoint refinements in the classical case as well as with respect to given ideals on \(\omega \). We prove the following generalization of a result due to J. Brendle: If \(V\subseteq W\) are transitive models, \(\omega_1^W\subseteq V\), \(\mathcal {P}(\omega)\cap V\not =\mathcal {P}(\omega)\cap W\), and \(\mathcal {I}\) is an analytic or coanalytic ideal coded in \(V\), then there is an \(\mathcal {I}\)-almost disjoint refinement of \(\mathcal {I}^+\cap V\) in \(W\). We study the existence of perfect \(\mathcal {I}\)-almost disjoint families, and the existence of \(\mathcal {I}\)-almost disjoint refinements in which any two distinct sets have finite intersection. We introduce the notion of mixing real (motivated by the construction of an almost disjoint refinement of \([\omega]^\omega \cap V\) after adding a Cohen real to \(V\)) and discuss logical implications between the existence of mixing reals in forcing extensions and classical properties of forcing notions.Cofinalities of Marczewski-like ideals.https://zbmath.org/1459.030802021-05-28T16:06:00+00:00"Brendle, Jörg"https://zbmath.org/authors/?q=ai:brendle.jorg"Khomskii, Yurii"https://zbmath.org/authors/?q=ai:khomskii.yurii"Wohofsky, Wolfgang"https://zbmath.org/authors/?q=ai:wohofsky.wolfgangSummary: We show in ZFC that the cofinalities of both the Miller ideal \(m^0\) (the \(\sigma \)-ideal naturally related to Miller forcing \({\mathbb M}\)) and the Laver ideal \(\ell ^0\) (related to Laver forcing \({\mathbb L}\)) are larger than the size \({\mathfrak c}\) of the continuum.Decision-making analysis based on fuzzy graph structures.https://zbmath.org/1459.901172021-05-28T16:06:00+00:00"Koam, Ali N. A."https://zbmath.org/authors/?q=ai:koam.ali-n-a"Akram, Muhammad"https://zbmath.org/authors/?q=ai:akram.muhammad"Liu, Peide"https://zbmath.org/authors/?q=ai:liu.peide|liu.peide.1Summary: A graph structure is a useful framework to solve the combinatorial problems in various fields of computational intelligence systems and computer science. In this research article, the concept of fuzzy sets is applied to the graph structure to define certain notions of fuzzy graph structures. Fuzzy graph structures can be very useful in the study of various structures, including fuzzy graphs, signed graphs, and the graphs having labeled or colored edges. The notions of the fuzzy graph structure, lexicographic-max product, and degree and total degree of a vertex in the lexicographic-max product are introduced. Further, the proposed concepts are explained through several numerical examples. In particular, applications of the fuzzy graph structures in decision-making process, regarding detection of marine crimes and detection of the road crimes, are presented. Finally, the general procedure of these applications is described by an algorithm.Heterogeneous Ramsey algebras and classification of Ramsey vector spaces.https://zbmath.org/1459.030722021-05-28T16:06:00+00:00"Teoh, Zu Yao"https://zbmath.org/authors/?q=ai:teoh.zu-yao"Teh, Wen Chean"https://zbmath.org/authors/?q=ai:teh.wen-cheanSummary: Carlson introduced the notion of a Ramsey space as a generalization to the Ellentuck space. When a Ramsey space is induced by an algebra, Carlson suggested a study of its purely combinatorial version now called Ramsey algebra. Some basic results for homogeneous algebras have been obtained. In this paper, we introduce the notion of a Ramsey algebra for heterogeneous algebras and derive some basic results. Then, we study the Ramsey-algebraic properties of vector spaces.Feasible set functions have small circuits.https://zbmath.org/1459.030582021-05-28T16:06:00+00:00"Beckmann, Arnold"https://zbmath.org/authors/?q=ai:beckmann.arnold"Buss, Sam"https://zbmath.org/authors/?q=ai:buss.sam"Friedman, Sy-David"https://zbmath.org/authors/?q=ai:friedman.sy-david"Müller, Moritz"https://zbmath.org/authors/?q=ai:muller.moritz"Thapen, Neil"https://zbmath.org/authors/?q=ai:thapen.neilSummary: The Cobham Recursive Set Functions (CRSF) provide an analogue of polynomial time computation which applies to arbitrary sets. We give three new equivalent characterizations of CRSF. The first is algebraic, using subset-bounded recursion and a form of Mostowski collapse. The second is our main result: the CRSF functions are shown to be precisely the functions computed by a class of uniform, infinitary, Boolean circuits. The third is in terms of a simple extension of the rudimentary functions by transitive closure and subset-bounded recursion.Incompleteness theorems, large cardinals, and automata over finite words.https://zbmath.org/1459.030552021-05-28T16:06:00+00:00"Finkel, Olivier"https://zbmath.org/authors/?q=ai:finkel.olivierSummary: We prove that one can construct various kinds of automata over finite words for which some elementary properties are actually independent from strong set theories like \(T_n =:\mathbf{ZFC} +\) ``There exist (at least) \(n\) inaccessible cardinals'', for integers \(n\geq 0\). In particular, we prove independence results for languages of finite words generated by context-free grammars, or accepted by 2-tape or 1-counter automata. Moreover we get some independence results for weighted automata and for some related finitely generated subsemigroups of the set \(\mathbb {Z}^{3\times 3}\) of 3-3 matrices with integer entries. Some of these latter results are independence results from the Peano axiomatic system \textbf{PA}.
For the entire collection see [Zbl 1360.68012].Infinite powers and Cohen reals.https://zbmath.org/1459.540082021-05-28T16:06:00+00:00"Medini, Andrea"https://zbmath.org/authors/?q=ai:medini.andrea"van Mill, Jan"https://zbmath.org/authors/?q=ai:van-mill.jan"Zdomskyy, Lyubomyr"https://zbmath.org/authors/?q=ai:zdomskyy.lyubomyrSummary: We give a consistent example of a zero-dimensional separable metrizable space \(Z\) such that every homeomorphism of \(Z^\omega\) acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of \textit{A. Dow} and \textit{E. Pearl} [Proc. Am. Math. Soc. 125, No. 8, 2503--2510 (1997; Zbl 0963.54002)] is sharp, and gives some insight into an open problem of \textit{T. Terada} [Yokohama Math. J. 40, No. 2, 87--93 (1993; Zbl 0819.54006)]. Our example \(Z\) is simply the set of \(\omega_1\) Cohen reals, viewed as a subspace of \(2^\omega\).Incompleteness theorems, large cardinals, and automata over finite words.https://zbmath.org/1459.030562021-05-28T16:06:00+00:00"Finkel, Olivier"https://zbmath.org/authors/?q=ai:finkel.olivierMore on uniform ultrafilters over a singular cardinal.https://zbmath.org/1459.030712021-05-28T16:06:00+00:00"Gitik, Moti"https://zbmath.org/authors/?q=ai:gitik.motiThe character \(\mathrm{ch}(D)\) of an ultrafilter \(D\) is the smallest cardinality of its basis (\(W\subseteq D\) such that for every \(A\in D\) there is \(B\in W\) such that \(B\subseteq^* A\)). The ultrafilter number of a cardinal \(\kappa\) is \(u(\kappa)=\min\{\mathrm{ch}(D):D\mbox{ is a uniform ultrafilter over }\kappa\}\).
The main goal of this paper is to find lower bounds for the character of certain types of ultrafilters over a singular cardinal \(\kappa\). Let \(\overrightarrow{\tau}=\langle\tau_\alpha:\alpha<\eta\rangle\) be a fixed increasing sequence of cardinals of length \(\eta=\mathrm{cf}(\kappa)\) cofinal in \(\kappa\).
If \(F\) is an ultrafilter over \(\eta\), an ultrafilter \(D\) on \(\kappa\) is called \((\overrightarrow{\tau},F)\)-uniform if, for every \(A\in D\), \(\{\alpha<\eta:|A\cap\tau_\alpha|=\tau_\alpha\}\in F\). \(u^{\mathrm{str}}(\kappa)\) is the smallest character of a \((\overrightarrow{\delta},F)\)-uniform ultrafilter on \(\kappa\) for some \(\overrightarrow{\delta}\) and \(F\). A typical result is the following.
Proposition 3.3. If:
(i) \(\delta\) is a regular cardinal such that \(\kappa<\delta\leq 2^\kappa\),
(ii) \(\overrightarrow{\delta}=\langle\delta_\alpha:\alpha<\eta\rangle\) is a sequence of regular cardinals such that \(\tau_\alpha<\delta_\alpha\leq\tau_{\alpha+1}\) for every \(\alpha<\eta\) and
(iii) \(\mathrm{tcf}(\prod_{\alpha<\eta}\delta_\alpha,<_F)=\delta\) for some ultrafilter \(F\) on \(\eta\) which extends the filter of co-bounded subsets of \(\eta\),
then \(\mathrm{ch}(D)\geq\delta\) for every \((\overrightarrow{\delta},F)\)-uniform ultrafilter \(D\) over \(\kappa\).
Using results of this type, in Section 4 the author finds lower bounds for the character of ultrafilters of type \(U=F-\lim_{\alpha<\eta}U_\alpha\) (defined by: \(A\in D\) if and only if \(\{\alpha<\eta:A\in U_\alpha\}\in F\)) obtained from uniform ultrafilters \(U_\alpha\) satisfying certain regularity properties, over cardinals \(\mu_\alpha\). In Section 5, these properties are replaced by \(\square(\mu_\alpha)\).
The cardinal \(r(\kappa)\) is defined as the smallest cardinality of a reaping (non-splittable) family of subsets of \(\kappa\). It is easy to see that \(r(\kappa)\leq u(\kappa)\). In Section 6, an upper bound for \(r(\kappa)\) is obtained, using conditions similar to those from previous results.
Finally, in Section 7, forcing is used to construct a model in which \(\aleph_\omega\) is a strong limit, \(2^{\aleph_\omega}=\aleph_{\omega+2}\) and \(u^{\mathrm{str}}(\aleph_\omega)=\aleph_{\omega+2}\).
Reviewer: Boris Šobot (Novi Sad)A generalization of a theorem of Hurewicz for quasi-Polish spaces.https://zbmath.org/1459.030742021-05-28T16:06:00+00:00"de Brecht, Matthew"https://zbmath.org/authors/?q=ai:de-brecht.matthewSummary: We identify four countable topological spaces \(S_2\), \(S_1\), \(S_D\), and \(S_0\) which serve as canonical examples of topological spaces which fail to be quasi-Polish. These four spaces respectively correspond to the \(T_2\), \(T_1\), \(T_D\), and \(T_0\)-separation axioms. \(S_2\) is the space of rationals, \(S_1\) is the natural numbers with the cofinite topology, \(S_D\) is an infinite chain without a top element, and \(S_0\) is the set of finite sequences of natural numbers with the lower topology induced by the prefix ordering. Our main result is a generalization of Hurewicz's theorem showing that a co-analytic subset of a quasi-Polish space is either quasi-Polish or else contains a countable \(\boldsymbol{\Pi}^0_2\)-subset homeomorphic to one of these four spaces.Normal measures and strongly compact cardinals.https://zbmath.org/1459.030842021-05-28T16:06:00+00:00"Apter, Arthur W."https://zbmath.org/authors/?q=ai:apter.arthur-wSummary: We prove four theorems concerning the number of normal measures a non-\((\kappa + 2)\)-strong strongly compact cardinal \(\kappa \) can carry.On sets which can be moved away from sets of a certain family.https://zbmath.org/1459.030772021-05-28T16:06:00+00:00"Horbaczewska, Grażyna"https://zbmath.org/authors/?q=ai:horbaczewska.grazyna"Lindner, Sebastian"https://zbmath.org/authors/?q=ai:lindner.sebastian.1|lindner.sebastianSummary: An operation which assigns to an arbitrary family of sets the class of sets which can be translated away from every set from the fixed family is considered in abelian groups. Assuming CH it is proven that on the real line meager sets can be defined as sets ``shiftable'' from the family of strong measure zero sets \((\mathcal{K} = \mathcal{SMZ}^\ast)\). A similar result is shown for Lebesgue null sets and strongly meager sets \((\mathcal{N}= \mathcal{SM}^\ast)\). Additionally a certain characterization of the family of meager-additive sets is given.A \(\kappa\)-rough morass under \(2^{{<}\kappa}=\kappa\) and various applications.https://zbmath.org/1459.030402021-05-28T16:06:00+00:00"Villegas Silva, Luis Miguel"https://zbmath.org/authors/?q=ai:villegas-silva.luis-miguelAssuming that \(2^{<\kappa}=\kappa\) where \(\kappa\) is an uncountable regular cardinal, the author derives the existence of a \(\kappa\)-\textit{rough morass}. The author's stated aim is to study rough morasses.
From the author's introduction: ``[A] \(\kappa\)-rough morass allows us to derive some important combinatorial principles such as \(\square^{\ast}_{\kappa}\) and \(\lozenge^{\#}_{\kappa}\) for suitable \(\kappa\) in a constructible-like universe, and we can also obtain proofs for some cases of the gap-1 cardinal transfer theorem.''
Under the stated assumptions on \(\kappa\), the author produces a subset \(W\subseteq\kappa^{+}\) such that by working inside \(L_{{\kappa}^{+}}[W]\), he establishes the existence of the object that he calls a \(\kappa\)-rough morass. This is the article's main theorem. Following his existence result, the author presents several applications, namely the ones cited in the above quote.
Reviewer: J. M. Plotkin (East Lansing)Fixed point results for \((\phi ,\psi )\)-weak contraction in fuzzy metric spaces.https://zbmath.org/1459.540352021-05-28T16:06:00+00:00"Tiwari, Vandana"https://zbmath.org/authors/?q=ai:tiwari.vandana"Som, Tanmoy"https://zbmath.org/authors/?q=ai:som.tanmoySummary: In the present work, a fixed point result for generalized weakly contractive mapping in fuzzy metric space has been established. An example is cited to illustrate the obtained result.
For the entire collection see [Zbl 1411.65006].Unambiguous tree languages are topologically harder than deterministic ones.https://zbmath.org/1459.681022021-05-28T16:06:00+00:00"Hummel, Szczepan"https://zbmath.org/authors/?q=ai:hummel.szczepanSummary: The paper gives an example of a tree language \(G\) that is recognised by an unambiguous parity automaton and is \(\Sigma^1_1\)-complete as a set in Cantor space. This already shows that the unambiguous languages are topologically more complex than the deterministic ones, that are all \(\Pi^1_1\).
Using set \(G\) as a building block we construct an unambiguous language that is topologically harder than any countable Boolean combination of \(\Sigma^1_1\) and \(\Pi^1_1\) sets. In particular the language is harder than any set in difference hierarchy of analytic sets considered by O. Finkel and P. Simonnet in the context of nondeterministic automata.
For the entire collection see [Zbl 1392.68016].Fréchet Borel ideals with Borel orthogonal.https://zbmath.org/1459.030762021-05-28T16:06:00+00:00"Guevara, Francisco"https://zbmath.org/authors/?q=ai:guevara.francisco"Uzcátegui, Carlos"https://zbmath.org/authors/?q=ai:uzcategui.carlos-enriqueSummary: We study Borel ideals \(I\) on \(\mathbb {N}\) with the Fréchet property such that the orthogonal \(I^\perp \) is also Borel (where \(A\in I^\perp \) iff \(A\cap B\) is finite for all \(B\in I\), and \(I\) is Fréchet if \(I=I^{\perp \perp}\)). Let \(\mathcal {B}\) be the smallest collection of ideals on \(\mathbb {N}\) containing the ideal of finite sets and closed under countable direct sums and orthogonal. All ideals in \(\mathcal {B}\) are Fréchet, Borel and have Borel orthogonal. We show that \(\mathcal {B}\) has exactly \(\aleph_1\) non-isomorphic members. The family \(\mathcal {B}\) can be characterized as the collection of all Borel ideals which are isomorphic to an ideal of the form \(I_{\mathrm{wf}}{\upharpoonright}A\), where \(I_{\mathrm{wf}}\) is the ideal on \(\mathbb {N}^{ < \omega}\) generated by the well founded trees. Also, we show that \(A\subseteq \mathbb {Q}\) is scattered iff \(\mathrm{WO}(\mathbb {Q}){\upharpoonright} A\) is isomorphic to an ideal in \(\mathcal {B}\), where \(\mathrm{WO}(\mathbb {Q})\) is the ideal of well founded subsets of \(\mathbb Q\). We use the ideals in \(\mathcal {B}\) to construct \(\aleph_1\) pairwise non-homeomorphic countable sequential spaces whose topology is analytic.The notion of infinity within the Zermelo system and its relation to the axiom of countable choice.https://zbmath.org/1459.030822021-05-28T16:06:00+00:00"Chailos, George"https://zbmath.org/authors/?q=ai:chailos.georgeSummary: In this article we consider alternative definitions-descriptions of a set being Infinite within the primitive Axiomatic system of Zermelo, \(Z\). We prove that in this system the definitions of sets being Dedekind infinite, Cantor infinite and Cardinal infinite are equivalent each other. Additionally, we show that assuming the Axiom of countable choice, \(AC_{\aleph_0}\), these definitions are also equivalent to the definition of a set being Standard infinite, that is, of not being finite. Furthermore, we consider the relation of \(AC_{\aleph_0}\) (and some of its special cases) with the statement \(SD\) ``A set is standard infinite if and only if it is Dedekind infinite''. Among other results we show that the system \(Z+SD\) is `strictly weaker' than \(Z+AC_{\aleph_0}\).Localic completion of uniform spaces.https://zbmath.org/1459.030942021-05-28T16:06:00+00:00"Kawai, Tatsuji"https://zbmath.org/authors/?q=ai:kawai.tatsujiSummary: We extend the notion of localic completion of generalised metric spaces by Steven Vickers to the setting of generalised uniform spaces. A generalised uniform space (gus) is a set \(X\) equipped with a family of generalised metrics on \(X\), where a generalised metric on \(X\) is a map from the product of \(X\) to the upper reals satisfying zero self-distance law and triangle inequality.
For a symmetric generalised uniform space, the localic completion lifts its generalised uniform structure to a point-free generalised uniform structure. This point-free structure induces a complete generalised uniform structure on the set of formal points of the localic completion that gives the standard completion of the original gus with Cauchy filters.
We extend the localic completion to a full and faithful functor from the category of locally compact uniform spaces into that of overt locally compact completely regular formal topologies. Moreover, we give an elementary characterisation of the cover of the localic completion of a locally compact uniform space that simplifies the existing characterisation for metric spaces. These results generalise the corresponding results for metric spaces by Erik Palmgren.
Furthermore, we show that the localic completion of a symmetric gus is equivalent to the point-free completion of the uniform formal topology associated with the gus.
We work in Aczel's constructive set theory CZF with the Regular Extension Axiom. Some of our results also require Countable Choice.Multiple criteria decision-making based on vector similarity measures under the framework of dual hesitant fuzzy sets.https://zbmath.org/1459.910422021-05-28T16:06:00+00:00"García Guirao, Juan Luis"https://zbmath.org/authors/?q=ai:garcia-guirao.juan-luis"Sarwar Sindhu, M."https://zbmath.org/authors/?q=ai:sindhu.m-sarwar"Rashid, Tabasam"https://zbmath.org/authors/?q=ai:rashid.tabasam"Kashif, Agha"https://zbmath.org/authors/?q=ai:kashif.aghaSummary: Similarity measures have a great importance in the decision-making process. In order to identify the similarity between the options, many experts have established several types of similarity measures on the basis of vectors and distances. The Cosine, Dice, and Jaccard are the vector similarity measures. The present work enclosed the modified Jaccard and Dice similarity measures. Founded on the Dice and Jaccard similarity measures, we offered a multiple criteria decision-making (MCDM) model under the dual hesitant fuzzy sets (DHFSs) situation, in which the appraised values of the alternatives with respect to criteria are articulated by dual hesitant fuzzy elements (DHFEs). Since the weights of the criteria have a much influence in making the decisions, therefore decision makers (DMs) allocate the weights to each criteria according to their knowledge. In the present work, we get rid of the doubt to allocate the weights to the criteria by taking an objective function under some constraints and then extended the linear programming (LP) technique to evaluate the weights of the criteria. The Dice and Jaccard weighted similarity measures are practiced amongst the ideal and each alternative to grade all the alternatives to get the best one. Eventually, two practical examples, about investment companies and selection of smart phone accessories are assumed to elaborate the efficiency of the proposed methodology.On some results in intuitionistic fuzzy ideal convergence double sequence spaces.https://zbmath.org/1459.400052021-05-28T16:06:00+00:00"Khan, Vakeel A."https://zbmath.org/authors/?q=ai:khan.vakeel-a"Ahmad, Mobeen"https://zbmath.org/authors/?q=ai:ahmad.mobeen"Fatima, Hira"https://zbmath.org/authors/?q=ai:fatima.hira"Khan, Mohd Faisal"https://zbmath.org/authors/?q=ai:khan.mohd-faisalSummary: Recently, spaces of ideal convergent sequences of bounded linear operators were studied by \textit{V. A. Khan} et al. [Numer. Funct. Anal. Optim. 39, No. 12, 1278--1290 (2018; \url{doi:10.1080/01630563.2018.1477797})]. This has motivated us to propose the intuitionistic fuzzy \(I\)-convergent double sequence spaces determined by the bounded linear operator. In this paper, we investigate the algebraic and topological properties. We also study the concept of the ideal Cauchy and ideal convergence on the said spaces.