Abstract
We continue our work on the ideal version of the Lévy–Steinitz theorem on conditionally convergent series of vectors. In particular, we prove that for each series \({\sum_{n\in\omega}v_n}\), \({(v_n)_{n\in\omega} \subset\mathbb{R}^2}\), such that its sum range is \({\mathbb{R}^2}\) and its set of Lévy vectors is of power at least 3, it is possible to find \({A\in\mathcal{I}}\) such that the sum range of \({\sum_{n\in A}v_n}\) is still \({\mathbb{R}^2}\), for some proper ideal \({\mathcal{I}\subset\mathcal{P}(\omega)}\).
We also work on the summability of certain known ideals as well as introduce the cardinal number \({\kappa_{M}}\) as the minimal number of summable ideals required to cover an ideal, and prove some basic properties of it.
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Klinga, P., Nowik, A. Extendability to summable ideals. Acta Math. Hungar. 152, 150–160 (2017). https://doi.org/10.1007/s10474-017-0704-8
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DOI: https://doi.org/10.1007/s10474-017-0704-8
Key words and phrases
- Lévy–Steinitz theorem
- Riemann’s theorem
- conditionally convergent series
- summable ideal
- cardinal characteristic
- van der Waerden ideal