Real Analysis Exchange

Algebras with inner MB-representation.

Marek Balcerzak, Artur Bartoszewicz, and Krzysztof Ciesielski

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Abstract

We investigate algebras of sets, and pairs $(\mathcal{A , I})$ consisting of an algebra $\mathcal{A}$ and an ideal $\mathcal{I} \subset \mathcal{A}$, that possess an inner MB-representation. We compare inner MB-representability of $(\mathcal{A , I})$ with several properties of $(\mathcal{A , I})$ considered by Baldwin. We show that $\mathcal{A}$ is inner MB-representable if and only if $\mathcal{A} =S(\mathcal{A} \setminus\mathcal{H}(\mathcal{A}))$, where $S(\cdot)$ is a Marczewski operation defined below and $\mathcal H$ consists of sets that are hereditarily in $\mathcal{A}$. We study the question of uniqueness of the ideal in that representation..

Article information

Source
Real Anal. Exchange, Volume 29, Number 1 (2003), 265-273.

Dates
First available in Project Euclid: 9 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149860191

Mathematical Reviews number (MathSciNet)
MR2061310

Zentralblatt MATH identifier
1065.03033

Subjects
Primary: 06E25: Boolean algebras with additional operations (diagonalizable algebras, etc.) [See also 03G25, 03F45]
Secondary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 54E52: Baire category, Baire spaces

Keywords
algebra of sets ideal of sets Marczewski-Burstin representation

Citation

Balcerzak, Marek; Bartoszewicz, Artur; Ciesielski, Krzysztof. Algebras with inner MB-representation. Real Anal. Exchange 29 (2003), no. 1, 265--273. https://projecteuclid.org/euclid.rae/1149860191


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