More remarks on the intersection ideal ${\mathcal {M}}\cap {\mathcal {N}}$

Tomasz Weiss

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Displaying similar documents to “More remarks on the intersection ideal $ℳ \cap 𝒩$ ”

A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case

Fateme Kouchakinejad, Alexandra Šipošová (2017)

Kybernetika

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For an aggregation function $A$ we know that it is bounded by $A^{*}$ and $A_{*}$ which are its super-additive and sub-additive transformations, respectively. Also, it is known that if $A^{*}$ is directionally convex, then $A = A^{*}$ and $A_{*}$ is linear; similarly, if $A_{*}$ is directionally concave, then $A = A_{*}$ and $A^{*}$ is linear. We generalize these results replacing the directional convexity and concavity conditions by the weaker assumptions of overrunning a super-additive function and underrunning a sub-additive function, respectively. ...

Addendum to “On Meager Additive and Null Additive Sets in the Cantor space $2^{ω}$ and in ℝ” (Bull. Polish Acad. Sci. Math. 57 (2009), 91-99)

Tomasz Weiss (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove in ZFC that there is a set $A \subseteq 2^{ω}$ and a surjective function H: A → ⟨0,1⟩ such that for every null additive set X ⊆ ⟨0,1), $H^{- 1} (X)$ is null additive in $2^{ω}$ . This settles in the affirmative a question of T. Bartoszyński.

Completeness of $L^{p}$ -spaces over finitely additive set functions

Euline Green (1971)

Colloquium Mathematicae

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A note on the multiplier ideals of monomial ideals

Cheng Gong, Zhongming Tang (2015)

Czechoslovak Mathematical Journal

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Let $𝔞 \subseteq ℂ [x_{1}, ..., x_{n}]$ be a monomial ideal and $𝒥 (𝔞^{c})$ the multiplier ideal of $𝔞$ with coefficient $c$ . Then $𝒥 (𝔞^{c})$ is also a monomial ideal of $ℂ [x_{1}, ..., x_{n}]$ , and the equality $𝒥 (𝔞^{c}) = 𝔞$ implies that $0 < c < n + 1$ . We mainly discuss the problem when $𝒥 (𝔞) = 𝔞$ or $𝒥 (𝔞^{n + 1 - ε}) = 𝔞$ for all $0 < ε < 1$ . It is proved that if $𝒥 (𝔞) = 𝔞$ then $𝔞$ is principal, and if $𝒥 (𝔞^{n + 1 - ε}) = 𝔞$ holds for all $0 < ε < 1$ then $𝔞 = (x_{1}, ..., x_{n})$ . One global result is also obtained. Let $\tilde{𝔞}$ be the ideal sheaf on $ℙ^{n - 1}$ associated with $𝔞$ . Then it is proved that the equality $𝒥 (\tilde{𝔞}) = \tilde{𝔞}$ implies that $\tilde{𝔞}$ is principal.

On Meager Additive and Null Additive Sets in the Cantor Space $2^{ω}$ and in ℝ

Tomasz Weiss (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let T be the standard Cantor-Lebesgue function that maps the Cantor space $2^{ω}$ onto the unit interval ⟨0,1⟩. We prove within ZFC that for every $X \subseteq 2^{ω}$ , X is meager additive in $2^{ω}$ iff T(X) is meager additive in ⟨0,1⟩. As a consequence, we deduce that the cartesian product of meager additive sets in ℝ remains meager additive in ℝ × ℝ. In this note, we also study the relationship between null additive sets in $2^{ω}$ and ℝ.

Semi $n$ -ideals of commutative rings

Ece Yetkin Çelikel, Hani A. Khashan (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with identity. A proper ideal $I$ is said to be an $n$ -ideal of $R$ if for $a, b \in R$ , $a b \in I$ and $a \notin \sqrt{0}$ imply $b \in I$ . We give a new generalization of the concept of $n$ -ideals by defining a proper ideal $I$ of $R$ to be a semi $n$ -ideal if whenever $a \in R$ is such that $a^{2} \in I$ , then $a \in \sqrt{0}$ or $a \in I$ . We give some examples of semi $n$ -ideal and investigate semi $n$ -ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new...

Order-theoretic properties of some sets of quasi-measures

Zbigniew Lipecki (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let $𝔐$ and $ℜ$ be algebras of subsets of a set $Ω$ with $𝔐 \subset ℜ$ , and denote by $E (μ)$ the set of all quasi-measure extensions of a given quasi-measure $μ$ on $𝔐$ to $ℜ$ . We show that $E (μ)$ is order bounded if and only if it is contained in a principal ideal in $b a (ℜ)$ if and only if it is weakly compact and $extr E (μ)$ is contained in a principal ideal in $b a (ℜ)$ . We also establish some criteria for the coincidence of the ideals, in $b a (ℜ)$ , generated by $E (μ)$ and $extr E (μ)$ .

Ideals in big Lipschitz algebras of analytic functions

Thomas Vils Pedersen (2004)

Studia Mathematica

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For 0 < γ ≤ 1, let $Λ ⁺_{γ}$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_{γ} (E, Q)$ be the closed ideal in $Λ ⁺_{γ}$ consisting of those functions $f \in Λ ⁺_{γ}$ for which (i) f = 0 on E, (ii) $| f (z) - f (w) | = o (| z - w |^{γ})$ as d(z,E),d(w,E) → 0, (iii) $f / Q \in Λ ⁺_{γ}$ . Also, for a closed ideal I in $Λ ⁺_{γ}$ , let $E_{I}$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_{I}$ be the greatest common divisor of the inner parts of non-zero functions...

The strong persistence property and symbolic strong persistence property

Mehrdad Nasernejad, Kazem Khashyarmanesh, Leslie G. Roberts, Jonathan Toledo (2022)

Czechoslovak Mathematical Journal

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Let $I$ be an ideal in a commutative Noetherian ring $R$ . Then the ideal $I$ has the strong persistence property if and only if $(I^{k + 1} :_{R} I) = I^{k}$ for all $k$ , and $I$ has the symbolic strong persistence property if and only if $(I^{(k + 1)} :_{R} I^{(1)}) = I^{(k)}$ for all $k$ , where $I^{(k)}$ denotes the $k$ th symbolic power of $I$ . We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial...

Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík (2016)

Mathematica Bohemica

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Let $\sum_{n = 1}^{\infty} a_{n}$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_{n})$ is non-increasing, then $lim_{n \to \infty} n a_{n} = 0$ . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $lim_{n \to \infty} n a_{n} = 0$ ; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $ℐ$ -convergence, that is a convergence according to an ideal $ℐ$ of subsets of $ℕ$ . Again, Olivier’s theorem is a consequence...

Duality of measures of non-𝒜-compactness

Juan Manuel Delgado, Cándido Piñeiro (2015)

Studia Mathematica

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Let be a Banach operator ideal. Based on the notion of -compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non–compactness of an operator. We consider a map $χ_{}$ (respectively, $n_{}$ ) acting on the operators of the surjective (respectively, injective) hull of such that $χ_{} (T) = 0$ (respectively, $n_{} (T) = 0$ ) if and only if the operator T is -compact (respectively, injectively -compact). Under certain conditions on the ideal , we prove an equivalence inequality involving...

On the regularity and defect sequence of monomial and binomial ideals

Keivan Borna, Abolfazl Mohajer (2019)

Czechoslovak Mathematical Journal

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When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$ , with homogeneous maximal ideal $𝔪$ , it is known that for an ideal $I$ of $S$ , the regularity of powers of $I$ becomes eventually a linear function, i.e., $reg (I^{m}) = d m + e$ for $m ≫ 0$ and some integers $d$ , $e$ . This motivates writing $reg (I^{m}) = d m + e_{m}$ for every $m \geq 0$ . The sequence $e_{m}$ , called the of the ideal $I$ , is the subject of much research and its nature is still widely unexplored. We know that $e_{m}$ is eventually constant. In this article, after proving...

When spectra of lattices of $z$ -ideals are Stone-Čech compactifications

Themba Dube (2017)

Mathematica Bohemica

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Let $X$ be a completely regular Hausdorff space and, as usual, let $C (X)$ denote the ring of real-valued continuous functions on $X$ . The lattice of $z$ -ideals of $C (X)$ has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) $β X$ precisely when $X$ is a $P$ -space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a $d$ -ideal if whenever two elements have the same annihilator...

On strong measure zero subsets of $^{κ} 2$

Aapo Halko, Saharon Shelah (2001)

Fundamenta Mathematicae

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We study the generalized Cantor space $^{κ} 2$ and the generalized Baire space $^{κ} κ$ as analogues of the classical Cantor and Baire spaces. We equip $^{κ} κ$ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of $^{κ} 2$ . We prove for successor $κ = κ^{< κ}$ that the ideal of strong measure zero sets of $^{κ} 2$ is $_{κ}$ -additive, where $_{κ}$ is the size of the smallest unbounded family in $^{κ} κ$ , and that the generalized Borel...

Ultrafilter extensions of asymptotic density

Jan Grebík (2019)

Commentationes Mathematicae Universitatis Carolinae

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We characterize for which ultrafilters on $ω$ is the ultrafilter extension of the asymptotic density on natural numbers $σ$ -additive on the quotient boolean algebra $𝒫 (ω) / d_{𝒰}$ or satisfies similar additive condition on $𝒫 (ω) / fin$ . These notions were defined in [Blass A., Frankiewicz R., Plebanek G., Ryll-Nardzewski C., A Note on extensions of asymptotic density, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3313–3320] under the name $A P$ (null) and $A P$ (*). We also present a characterization of a $P$ - and semiselective...

Displaying similar documents to “More remarks on the intersection ideal ℳ ∩ 𝒩 ”

Displaying similar documents to “More remarks on the intersection ideal $ℳ \cap 𝒩$ ”