Set-theoretic Assumptions Research Articles

Enochs’ conjecture for cotorsion pairs and more

Abstract Enochs’ conjecture asserts that each covering class of modules (over any ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this paper, we prove the conjecture for the classes Filt ⁡ ( 𝒮 ) {\operatorname{Filt}(\mathcal{S})} , where 𝒮 {\mathcal{S}} consists of ℵ n {\aleph_{n}} -presented modules for some fixed n < ω {n<\omega} . In particular, this applies to the left-hand class of any cotorsion pair generated by a class of ℵ n {\aleph_{n}} -presented modules. Moreover, we also show that it is consistent with ZFC that Enochs’ conjecture holds for all classes of the form Filt ⁡ ( 𝒮 ) {\operatorname{Filt}(\mathcal{S})} , where 𝒮 {\mathcal{S}} is a set of modules. This leaves us with no explicit example of a covering class where we cannot prove that Enochs’ conjecture holds (possibly under some additional set-theoretic assumption).

Afektivna vezanost kao prediktor zadovoljstva poslom - medijatorska uloga socijalnog poređenja

The aim of this study was to examine the possibility of prediction of job satisfaction based on the style of attachment (avoiding/anxious), with the mediating role of social comparison. 325 employees participated in the study, of which 125 female and 200 male. The data from the study shows statistically significant correlations between all variables. As for prediction, the obtained results suggest that based on the avoiding style of affective attachment, it is possible to directly predict a person's job satisfaction level, and that social comparison has a significant, albeit a partial mediation role in that relation. The anxious style of affective attachment did not achieve a statistically significant direct effect on the predictor, but, in combination with social comparison, a total effect that is statistically significant was found, which confirms that social comparison has the role of a complete mediator in the relation. The results were discussed in terms of set theoretical assumptions, the limitations of the study were brought up, as well as guidelines for future research.

Enochs’ conjecture for small precovering classes of modules

Enochs’ conjecture asserts that each covering class of modules (over any fixed ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this short paper, we prove the validity of the conjecture for small precovering classes, i.e., the classes of the form Add(M) where M is any module, under a mild additional set-theoretic assumption which ensures that there are enough non-reflecting stationary sets. We even show that M has a perfect decomposition if Add(M) is a covering class. Finally, the additional set-theoretic assumption is shown to be redundant if there exists an n < ω such that M decomposes into a direct sum of אn-generated mo dules.

Almost continuous Sierpiński–Zygmund functions under different set-theoretical assumptions

A function f:mathbb {R}rightarrow mathbb {R} is: almost continuous in the sense of Stallings, fin textrm{AC}, if each open set Gsubset mathbb {R}^2 containing the graph of f contains also the graph of a continuous function g:mathbb {R}rightarrow mathbb {R}; Sierpiński–Zygmund, fin textrm{SZ} (or, more generally, fin textrm{SZ}(textrm{Bor})), provided its restriction frestriction M is discontinuous (not Borel, respectively) for any Msubset mathbb {R} of cardinality continuum. It is known that an example of a Sierpiński–Zygmund almost continuous function f:mathbb {R}rightarrow mathbb {R} (i.e., an fin textrm{SZ}cap textrm{AC}) cannot be constructed in ZFC; however, an fin textrm{SZ}cap textrm{AC} exists under the additional set-theoretical assumption {{,textrm{cov},}}(mathcal {M})=mathfrak {c}, that is, that mathbb {R} cannot be covered by less than mathfrak {c}-many meager sets. The primary purpose of this paper is to show that the existence of an fin textrm{SZ}cap textrm{AC} is also consistent with ZFC plus the negation of {{,textrm{cov},}}(mathcal {M})=mathfrak {c}. More precisely, we show that it is consistent with ZFC+{{,textrm{cov},}}(mathcal {M})<mathfrak {c} (follows from the assumption that {{,textrm{non},}}(mathcal {N})<{{,textrm{cov},}}(mathcal {N})=mathfrak {c}) that there is an fin textrm{SZ}(textrm{Bor})cap textrm{AC} and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either {{,textrm{cov},}}(mathcal {M})=mathfrak {c} or {{,textrm{non},}}(mathcal {N})<{{,textrm{cov},}}(mathcal {N})=mathfrak {c}, the lineability and the additivity coefficient of the class of all almost continuous Sierpiński–Zygmund functions. Several open problems are also stated.

Characterizing existence of certain ultrafilters

Following Baumgartner (1995) [4], for an ideal I on ω, we say that an ultrafilter U on ω is an I-ultrafilter if for every function f:ω→ω there is A∈U with f[A]∈I.If there is an I-ultrafilter which is not a J-ultrafilter, then I is not below J in the Katětov order ≤K (i.e. for every function f:ω→ω there is A∈I with f−1[A]∉J). On the other hand, in general I≰KJ does not imply that existence of an I-ultrafilter which is not a J-ultrafilter is consistent.We provide some sufficient conditions on ideals to obtain the equivalence: I≰KJ if and only if it is consistent that there exists an I-ultrafilter which is not a J-ultrafilter. In some cases when the Katětov order is not enough for the above equivalence, we provide other conditions for which a similar equivalence holds. We are mainly interested in the cases when the family of all I-ultrafilters or J-ultrafilters coincides with some known family of ultrafilters: P-points, Q-points or selective ultrafilters (a.k.a. Ramsey ultrafilters). In particular, our results provide a characterization of Borel ideals I which can be used to characterize P-points as I-ultrafilters.Moreover, we introduce a cardinal invariant which is used to obtain a sufficient condition for the existence of an I-ultrafilter which is not a J-ultrafilter. Finally, we prove some new results concerning existence of certain ultrafilters under various set-theoretic assumptions.

Remarks on set selectively star-CCC spaces

Kočinac and Singh [4] introduced and studied the class of set selectively k-star-ccc spaces. Let k∈N, a space X is said to be set selectively k-star-ccc if for each nonempty subset B of X and for every collection U of open sets in X such that B‾⊂∪U and for every sequence (An:n∈N) of maximal cellular open families in X, there is a sequence (An:n∈N) such that for every n∈N, An∈An and B⊆Stk(⋃n∈NAn,U). In this paper, we show that for k∈N, there exists a Tychonoff pseudocompact selectively 2-star-ccc space X which is not set selectively k-star-ccc, which gives an answer to the Problem 5.1 of [4], and construct an example of a Tychonoff set selectively star-ccc space having a regular closed Gδ-subspace which is not set selectively star-ccc, which gives an answer to the Problem 5.3 of [4]. We also prove that every open Fσ-subset of a set selectively star-ccc space is set selectively star-ccc, which gives a positive answer to the Problem 5.6 of [4]. Finally, we obtain a normal example of a selectively 2-star-ccc space X which is neither set strongly star-Lindelöf nor set selectively star-ccc under some set-theoretic assumption, which gives a consistent positive answer to Problem 5.4 of [4].

Constructions of Lindelöf scattered P-spaces

We construct locally Lindelöf scattered P-spaces (LLSP spaces, for short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width $\omega _1$ and height $\omega _2$ and that

NULL SETS AND COMBINATORIAL COVERING PROPERTIES

AbstractA subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $\unicode{x3b3} $ , a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $\unicode{x3b3} $ that is not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin–Miller and Bartoszyński–Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties.

A non-diagonalizable pure state

We construct a pure state on the C*-algebra [Formula: see text] of all bounded linear operators on [Formula: see text], which is not diagonalizable [i.e., it is not of the form [Formula: see text] for any orthonormal basis [Formula: see text] of [Formula: see text] and an ultrafilter u on N]. This constitutes a counterexample to Anderson's conjecture without additional hypothesis and improves results of C. Akemann, N. Weaver, I. Farah, and I. Smythe who constructed such states making additional set-theoretic assumptions. It follows from results of J. Anderson and the positive solution to the Kadison-Singer problem due to A. Marcus, D. Spielman, and N. Srivastava that the restriction of our pure state to any atomic masa [Formula: see text] of diagonal operators with respect to an orthonormal basis [Formula: see text] is not multiplicative on [Formula: see text].

C⁎-, C- and P-embedded subsets in products and the undecidability of a certain property on [formula omitted

Let X=∏α∈ω1Xα be a product of ω1 many separable metric spaces. It is proved that every C⁎-embedded subset in X is C-embedded in X under a certain set-theoretic assumption. Combining a result of E. Pol and R. Pol for N2ω, it is undecidable in ZFC that there is a (closed) C⁎-embedded but not C-embedded subset in Nω1. It is also proved in ZFC that every C-embedded subset in X is P-embedded in X. Next, we discuss when C⁎- or C-embedding implies P-embedding in products of generalized metrizable spaces, such as M-spaces, Σ-spaces and semi-stratifiable spaces.

On countably saturated linear orders and certain class of countably saturated graphs

The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality mathfrak {c}. We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality mathfrak {c}, under different set-theoretic assumptions. We give a new proof of the old theorem of Harzheim, that the class of countably saturated linear orders has a uniquely determined one-element basis. From our proof it follows that this minimal linear order is a Fraïssé limit of certain Fraïssé class. In particular, it is homogeneous with respect to countable subsets. Next we prove the existence and uniqueness of the uncountable version of the random graph. This graph is isomorphic to (H(omega _1),in cup ni ), where H(omega _1) is the set of hereditarily countable sets, and two sets are connected if one of them is an element of the other. In the last section, an example of a prime countably saturated Boolean algebra is presented.

Precompact groups and convergence

We investigate precompact topological groups with topologies determined by convergent sequences (sequential and Fréchet) and show that some natural constructions of such topologies always result in metrizable groups answering a question of D. Dikranjan et al. We show that it is consistent that all sequential precompact topologies on countable groups are Fréchet (or even metrizable). For some classes of groups (for example boolean) the extra set-theoretic assumptions may be omitted (although in this case such groups are not necessarily metrizable).We also build (using ⋄) an example of a countably compact Fréchet group that is not α3 and obtain a counterexample to a conjecture of D. Shakhmatov as a corollary.

Infinite monochromatic sumsets for colourings of the reals

N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of $\mathbb R$ so that no infinite sumset $X+X=\{x+y:x,y\in X\}$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any $c:\mathbb R\to r$ with $r$ finite there is an infinite $X\subseteq \mathbb R$ so that $c$ is constant on $X+X$.

UNIVERSAL CLASSES NEAR ${\aleph _1}$

AbstractShelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $-sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal:Theorem. Assume ${2^{{\aleph _0}}} &lt; {2^{{\aleph _1}}}$. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $-sentence.(1)If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left( {\psi ,\aleph _1 } \right) &lt; 2^{\aleph _1 } $, then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.(2)If ψ is categorical in ${\aleph _1}$, then ψ is categorical in all uncountable cardinals.The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $-definable tame abstract elementary classes with primes.

An infinite cardinal-valued Krull dimension for rings

We define and study two generalizations of the Krull dimension for rings, which can assume cardinal number values of arbitrary size. The first, which we call the cardinal Krull dimension, is the supremum of the cardinalities of chains of prime ideals in the ring. The second, which we call the strong cardinal Krull dimension, is a slight strengthening of the first. Our main objective is to address the following question: for which cardinal pairs (κ,λ) does there exist a ring of cardinality κ and (strong) cardinal Krull dimension λ? Relying on results from the literature, we answer this question completely in the case where κ≥λ. We also give several constructions, utilizing valuation rings, polynomial rings, and Leavitt path algebras, of rings having cardinality κ and (strong) cardinal Krull dimension λ>κ. The exact values of κ and λ that occur in this situation depend on set-theoretic assumptions.

On properties related to star countability

We prove that a Hausdorff metaLindelöf weakly star countable space is feebly Lindelöf and a Hausdorff metacompact weakly star finite space is almost compact which partially answers a question of Alas and Wilson (2017) [2, Question 3.14]. We also obtain a normal example of a weakly star countable space which is neither almost star countable nor star Lindelöf without any set-theoretic assumptions, which answers a question implicitly asked by Song (2015) [13, Remark 2.8] and a question asked by Alas, Junqueira and Wilson (2011) [3, Question 4]. Under MA+¬CH, there even exists a normal weakly star countable Moore space which is not almost star countable. An example of a Tychonoff star compact and weakly star finite space which is not star countable is also given. Finally, we prove that every weakly star countable Hausdorff space with a rank 4-diagonal has cardinality at most 2ω.

ωω-dominated function spaces and ωω-bases in free objects of topological algebra

A topological space X is defined to have an ωω-base if at each point x∈X the space X has a neighborhood base (Uα[x])α∈ωω such that Uβ[x]⊂Uα[x] for all α≤β in ωω.For a Tychonoff space X consider the following conditions:(A)the free Abelian topological group A(X) of X has an ωω-base;(B)the free Boolean topological group B(X) of X has an ωω-base;(F)the free topological group F(X) of X has an ωω-base;(L)the free locally convex space L(X) of X has an ωω-base;(V)the free topological vector space V(X) of X has an ωω-base;(U)the universal uniformity UX of X has a base (Uα)α∈ωω such that Uβ⊂Uα for all α≤β in ωω;(C)the function space C(X) is ωω-dominated;(σ)X is σ-compact;(σ′)the set X′ of non-isolated points in X is σ-compact;(s)the space X is separable;(S)X is separable or cov♯(X)≤add(X);(D)X is discrete. Then (L)⇔(V)⇔(U∧C)⇔(U∧σ)⇔(U∧s)⇒(U∧S)⇒(F)⇒(A)⇔(B)⇔(U) and moreover (U∧S)⇔(F) under the set-theoretic assumption e♯=ω1 (which is weaker than b=d).If X is not a P-space, then (L)⇔(V)⇔(U∧C)⇔(U∧σ)⇔(U∧s)⇔(F)⇒(A)⇔(B)⇔(U).If the space X is metrizable, then (L)⇔(V)⇔(σ)⇒(D∨σ)⇔(F)⇒(A)⇔(B)⇔(σ′).

Uncountable equilateral sets in Banach spaces of the form C(K)

The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin’s axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form C(K) has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form C(K) which has no uncountable equilateral set (or equivalently no uncountable (1+ε)-separated set in the unit sphere for any ε > 0) making another consistent combinatorial assumption. The compact K is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than C(K) which has no uncountable equilateral set. It follows from the results of S. Mercourakis and G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples of nonseparable Banach spaces which have no uncountable equilateral sets in any equivalent renorming but it follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in which it has an uncountable equilateral set.

Extension operators and twisted sums of c0 and C(K) spaces

We investigate the following problem posed by Cabello Sanchéz, Castillo, Kalton, and Yost:Let K be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of c0 and C(K), i.e., does there exist a Banach space X containing a non-complemented copy Y of c0 such that the quotient space X/Y is isomorphic to C(K)?Using additional set-theoretic assumptions we give the first examples of compact spaces K providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either K is the Cantor cube 2ω1 or K is a separable scattered compact space of height 3 and weight ω1, then every twisted sum of c0 and C(K) is trivial.We also construct nontrivial twisted sums of c0 and C(K) for K belonging to several classes of compacta. Our main tool is an investigation of pairs of compact spaces K⊆L which do not admit an extension operator C(K)→C(L).

Exact saturation in simple and NIP theories

A theory [Formula: see text] is said to have exact saturation at a singular cardinal [Formula: see text] if it has a [Formula: see text]-saturated model which is not [Formula: see text]-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.