On almost precipitous ideals

Transactions of the American Mathematical Society, 2005

We show that the reduced cofinality of the nonstationary ideal NS κ on a regular uncountable cardinal κ may be less than its cofinality, where the reduced cofinality of NS κ is the least cardinality of any family F of nonstationary subsets of κ such that every nonstationary subset of κ can be covered by less than κ many members of F . For this we investigate connections of the various cofinalities of NS κ with other cardinal characteristics of κ κ and we also give a property of forcing notions (called manageability) which is preserved in <κ-support iterations and which implies that the forcing notion preserves non-meagerness of subsets of κ κ (and does not collapse cardinals nor changes cofinalities).

arXiv (Cornell University), 2020

In this note we give equivalence of the existence of K-partitions with the existence of the precipitous ideal which is essentially topological. This way we strengthen the main result of Frankiewczi and Kunen (1987). * The author is partially supported by Wroc law Univercity of Science and Technology grant of K34W04D03 no. 8201003902.

2020

It is proved that a finite intersection of special preenveloping ideals in an exact category $({\mathcal A}; {\mathcal E})$ is a special preenveloping ideal. Dually, a finite intersection of special precovering ideals is a special precovering ideal. A counterexample of Happel and Unger shows that the analogous statement about special preenveloping subcategories does not hold in classical approximation theory. If the exact category has exact coproducts, resp., exact products, these results extend to intersections of infinite families of special peenveloping, resp., special precovering, ideals. These techniques yield the Bongartz-Eklof-Trlifaj Lemma: if $a \colon A \to B$ is a morphism in ${\mathcal A},$ then the ideal $a^{\perp}$ is special preenveloping. This is an ideal version of the Eklof-Trlifaj Lemma, but the proof is based on that of Bongartz&#39; Lemma. The main consequence is that the ideal cotorsion pair generated by a small ideal is complete.

Journal of Algebra, 1976

Annals of the Alexandru Ioan Cuza University - Mathematics, 2010

Yuksel et. al. introduced and studied pre-I-irresolute functions. The aim of this paper is to give a new class of functions called strongly pre-I-irresolute functions in ideal topological space which is stronger than the class of pre-I-irresolute functions. Some characterizations and several basic properties of this class of functions are obtained.

2017

We introduce the notion of $K$-ideals associated with Kuratowski partitions and we prove that each $\kappa$-complete ideal on a measurable cardinal $\kappa$ can be represented as a $K$-ideal. Moreover, we show some results concerning precipitous and Frechet ideals.

Journal of Pure and Applied Algebra, 1995

Mathematical Logic Quarterly, 2014

We use techniques due to Moti Gitik to construct models in which for an arbitrary ordinal ρ, ℵ ρ+1 is both the least measurable and least regular uncountable cardinal.

2003

We prove the following two theorems. Theorem 1. Let X be a strongly meager subset of 2 ω×ω . Then it is dual Ramsey null. We will say that a σ-ideal I of subsets of 2 satisfies the condition () iff for every X C 2, if ∀ f ∈ ω ↑ ω {g ∈ ω ω : ¬(f? g)} ∩ X ∈ I, then X ∈ I. Theorem 2. The σ-ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition ().

2020

If $\kappa$ is regular and $2^{&lt;\kappa}\leq\kappa^+$, then the existence of a weakly presaturated ideal on $\kappa^+$ implies $\square^*_\kappa$. This partially answers a question of Foreman and Magidor about the approachability ideal on $\omega_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $\omega_2$ such that $\mathcal{P}(\omega_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.

Journal of Pure and Applied Algebra, 2021

It is proved that a finite intersection of special preenveloping ideals in an exact category (A; E) is a special preenveloping ideal. Dually, a finite intersection of special precovering ideals is a special precovering ideal. A counterexample of Happel and Unger shows that the analogous statement about special preenveloping subcategories does not hold in classical approximation theory. If the exact category has exact coproducts, resp., exact products, these results extend to intersections of infinite families of special peenveloping, resp., special precovering, ideals. These techniques yield the Bongartz-Eklof-Trlifaj Lemma: if a : A → B is a morphism in A, then the ideal a ⊥ is special preenveloping. This is an ideal version of the Eklof-Trlifaj Lemma, but the proof is based on that of Bongartz' Lemma. The main consequence is that the ideal cotorsion pair generated by a small ideal is complete.

Notre Dame Journal of Formal Logic, 2019

Working under large cardinal assumptions such as supercompactness, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal κ. We show the consistency of E λ ++ ,λ ++ λ-club , the relation of equivalence modulo the non-stationary ideal restricted to S λ ++ λ in the space (λ ++) λ ++ , being continuously reducible to E 2,λ ++ λ +-club , the relation of equivalence modulo the non-stationary ideal restricted to S λ ++ λ + in the space 2 λ ++. Then we show that for κ ineffable E 2,κ reg , the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space 2 κ , is Σ 1 1-complete. We finish by showing, for Π 1 2-indescribable κ, that the isomorphism relation between dense linear orders of cardinality κ is Σ 1 1-complete.

Annals of Mathematical Logic, 1976

Proceedings of the American Mathematical Society, 1995

Let l 0 {l^0} and m 0 {m^0} be the ideals associated with Laver and Miller forcing, respectively. We show that a d d ( l 0 ) &gt; c o v ( l 0 ) {\mathbf {add}}({l^0}) &gt; {\mathbf {cov}}({l^0}) and a d d ( m 0 ) &gt; c o v ( m 0 ) {\mathbf {add}}({m^0}) &gt; {\mathbf {cov}}({m^0}) are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal ≤ h \leq \mathfrak {h} .

2013

In earlier work of the second and third author the equivalence of a finite square principle square^fin_{lambda,D} with various model theoretic properties of structures of size lambda and regular ultrafilters was established. In this paper we investigate the principle square^fin_{lambda,D}, and thereby the above model theoretic properties, at a regular cardinal. By Chang's Two-Cardinal Theorem, square^fin_{lambda,D} holds at regular cardinals for all regular filters D if we assume GCH. In this paper we prove in ZFC that for certain regular filters that we call "doubly^+ regular", square^fin_{lambda,D} holds at regular cardinals, with no assumption about GCH. Thus we get new positive answers in ZFC to Open Problems 18 and 19 in the book "Model Theory" by Chang and Keisler.