## Displaying similar documents to “On $\mathrm{K}$-Boolean Rings”

### The rings which are Boolean

Discussiones Mathematicae - General Algebra and Applications

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We study unitary rings of characteristic 2 satisfying identity ${x}^{p}=x$ for some natural number p. We characterize several infinite families of these rings which are Boolean, i.e., every element is idempotent. For example, it is in the case if $p={2}^{n}-2$ or $p={2}^{n}-5$ or $p={2}^{n}+1$ for a suitable natural number n. Some other (more general) cases are solved for p expressed in the form ${2}^{q}+2m+1$ or ${2}^{q}+2m$ where q is a natural number and $m\in 1,2,...,{2}^{q}-1$.

### Generalised irredundance in graphs: Nordhaus-Gaddum bounds

Discussiones Mathematicae Graph Theory

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For each vertex s of the vertex subset S of a simple graph G, we define Boolean variables p = p(s,S), q = q(s,S) and r = r(s,S) which measure existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p,q,r) may be considered as a compound existence property of S-pns. The subset S is called an f-set of G if f = 1 for all s ∈ S and the class of f-sets of G is denoted by ${\Omega }_{f}\left(G\right)$. Only 64 Boolean functions f can produce different classes ${\Omega }_{f}\left(G\right)$, special cases...

### Boolean algebras, splitting theorems, and ${\Delta }_{2}^{0}$ sets

Fundamenta Mathematicae

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### ${\forall }_{n}$-theories of Boolean algebras

Colloquium Mathematicae

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### FKN Theorem on the biased cube

Colloquium Mathematicae

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We consider Boolean functions defined on the discrete cube $-\gamma ,{\gamma }^{-1}ⁿ$ equipped with a product probability measure ${\mu }^{\otimes n}$, where $\mu =\beta {\delta }_{-\gamma }+\alpha {\delta }_{{\gamma }^{-1}}$ and γ = √(α/β). This normalization ensures that the coordinate functions ${\left({x}_{i}\right)}_{i=1,...,n}$ are orthonormal in $L₂\left(-\gamma ,{\gamma }^{-1}ⁿ,{\mu }^{\otimes n}\right)$. We prove that if the spectrum of a Boolean function is concentrated on the first two Fourier levels, then the function is close to a certain function of one variable. Our theorem strengthens the non-symmetric FKN Theorem due to Jendrej, Oleszkiewicz and Wojtaszczyk. Moreover,...

### Laslett’s transform for the Boolean model in ${ℝ}^{d}$

Kybernetika

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Consider a stationary Boolean model $X$ with convex grains in ${ℝ}^{d}$ and let any exposed lower tangent point of $X$ be shifted towards the hyperplane ${N}_{0}=\left\{x\in {ℝ}^{d}:{x}_{1}=0\right\}$ by the length of the part of the segment between the point and its projection onto the ${N}_{0}$ covered by $X$. The resulting point process in the halfspace (the Laslett’s transform of $X$) is known to be stationary Poisson and of the same intensity as the original Boolean model. This result was first formulated for the planar Boolean model (see N. Cressie...

### Coherent ultrafilters and nonhomogeneity

Commentationes Mathematicae Universitatis Carolinae

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We introduce the notion of a coherent $P$-ultrafilter on a complete ccc Boolean algebra, strengthening the notion of a $P$-point on $\omega$, and show that these ultrafilters exist generically under $𝔠=𝔡$. This improves the known existence result of Ketonen [On the existence of $P$-points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91–94]. Similarly, the existence theorem of Canjar [On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1,...

### A subclass of strongly clean rings

Communications in Mathematics

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In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let ${J}^{#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of ${J}^{#}$. A ring $R$ is said to be in case every element in $R$ is very ${J}^{#}$-clean. We prove that every very ${J}^{#}$-clean ring is strongly $\pi$-rad clean and has stable range one. It is shown that for a...

### Certain decompositions of matrices over Abelian rings

Czechoslovak Mathematical Journal

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A ring $R$ is (weakly) nil clean provided that every element in $R$ is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let $R$ be abelian, and let $n\in ℕ$. We prove that ${M}_{n}\left(R\right)$ is nil clean if and only if $R/J\left(R\right)$ is Boolean and ${M}_{n}\left(J\left(R\right)\right)$ is nil. Furthermore, we prove that $R$ is weakly nil clean if and only if $R$ is periodic; $R/J\left(R\right)$ is ${ℤ}_{3}$, $B$ or ${ℤ}_{3}\oplus B$ where $B$ is a Boolean ring, and that ${M}_{n}\left(R\right)$ is weakly nil clean if and only if ${M}_{n}\left(R\right)$ is nil clean for all $n\ge 2$.

### Cardinal sequences of length < ω₂ under GCH

Fundamenta Mathematicae

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Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put ${}_{\lambda }\left(\alpha \right)=s\in \left(\alpha \right):s\left(0\right)=\lambda =min\left[s\left(\beta \right):\beta <\alpha \right]$. We show that f ∈ (α) iff for some natural number n there are infinite cardinals $\lambda ₀i>\lambda ₁>...>{\lambda }_{n-1}$ and ordinals $\alpha ₀,...,{\alpha }_{n-1}$ such that $\alpha =\alpha ₀+\cdots +{\alpha }_{n-1}$ and $f=f₀⏜f₁⏜...⏜{f}_{n-1}$ where each ${f}_{i}{\in }_{{\lambda }_{i}}\left({\alpha }_{i}\right)$. Under GCH we prove that if α < ω₂ then (i) ${}_{\omega }\left(\alpha \right)=s{\in }^{\alpha }\omega ,\omega ₁:s\left(0\right)=\omega$; (ii) if λ > cf(λ) = ω, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,{s}^{-1}\lambda is\omega ₁-closedin\alpha$; (iii) if cf(λ) = ω₁, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,{s}^{-1}\lambda is\omega -closedandsuccessor-closedin\alpha$; (iv) if cf(λ) > ω₁, ${}_{\lambda }\left(\alpha \right){=}^{\alpha }\lambda$. This yields a complete characterization of the classes (α) for all...

### Semicommutativity of the rings relative to prime radical

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$-semicommutative. We prove that a ring $R$ is $P$-semicommutative if and only if $R\left[x\right]$ is $P$-semicommutative if and only if $R\left[x,{x}^{-1}\right]$ is $P$-semicommutative. Also, if $R\left[\left[x\right]\right]$ is $P$-semicommutative, then $R$ is $P$-semicommutative. The converse holds provided that $P\left(R\right)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$, $R$ is $P$-semicommutative...

### Bounds for quotients in rings of formal power series with growth constraints

Studia Mathematica

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In rings ${\Gamma }_{M}$ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $M={\left({M}_{l}\right)}_{l\ge 0}$ (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in ${\Gamma }_{M}$ such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to ${\Gamma }_{M}$, provided ${\Gamma }_{M}$ is stable under derivation. By a further development of the method, we obtain the main result of the paper,...

### Generalized E-algebras via λ-calculus I

Fundamenta Mathematicae

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An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra $En{d}_{R}A$ of the R-module ${}_{R}A$, taking any a ∈ A to the right multiplication ${a}_{r}\in En{d}_{R}A$ by a, is an isomorphism of algebras. In this case ${}_{R}A$ is called an E(R)-module. There is a proper class of examples constructed in . E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3, 5, 7, 8, 10, 13, 14, 15, 18,...