Counting Ω-ideals

Abstract

Let \({\mathbb{A}}\) be a universal algebra of signature Ω, and let \({\mathcal{I}}\) be an ideal in the Boolean algebra \({\mathcal{P}_{\mathbb{A}}}\) of all subsets of \({\mathbb{A}}\) . We say that \({\mathcal{I}}\) is an Ω-ideal if \({\mathcal{I}}\) contains all finite subsets of \({\mathbb{A}}\) and \({f(A^{n}) \in \mathcal{I}}\) for every n-ary operation \({f \in \Omega}\) and every \({A \in \mathcal{I}}\) . We prove that there are \({2^{2^{\aleph_0}}}\) Ω-ideals in \({\mathcal{P}_{\mathbb{A}}}\) provided that \({\mathbb{A}}\) is countably infinite and Ω is countable.

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Correspondence to Igor V. Protasov.

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Presented by S. Koppelberg.

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Protasov, I.V. Counting Ω-ideals. Algebra Univers. 62, 339–343 (2009). https://doi.org/10.1007/s00012-010-0032-0

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2000 Mathematics Subject Classification

  • 08A65
  • 06E25

Key words and phrases

  • Boolean algebra
  • Ω-ideal
  • ballean