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On Levi’s duality between permutations and convergent series. (English) Zbl 0633.40004
F. W. Levi [Duke Math. J. 13, 579-585 (1946; Zbl 0060.154)] has introduced an interesting duality between subsets of $${\mathbb{C}}$$, denoting the set of all convergent real series, and subsets of $${\mathbb{P}}$$, denoting the set of permutations of $${\mathbb{N}}=\{1,2,3,...\}$$. Given a set $$A\subseteq {\mathbb{C}}$$, let $$A^{\times}=\{\pi \in {\mathbb{P}}:\sum a_ i=\sum a_{\pi (i)}$$ for all $$a\in A\}$$, and given a $$P\subseteq {\mathbb{P}}$$, let $$P^+=\{\sum a_ i\in {\mathbb{C}}:\sum a_ i=\sum a_{\pi (i)}$$ for all $$\pi\in P\}$$. Levi called $$A^{\times +}$$ the “closure of A” and $$P^{+\times}$$ the “closure of P”, and noted that $$\times$$ and $$+$$ are inverses of each other when considered as maps between closed sets of series and closed sets of permutations.
An unusual property of this duality is that $${\mathbb{C}}$$ is an unnormed linear space, while $${\mathbb{P}}$$ has a natural multiplicative group structure. Levi showed that $${\mathbb{P}}$$ is the only closed subgroup; conjectured that for any convergent series $$\sum a_ i$$ which is not absolutely convergent, $$(\sum a_ i)^{\times}$$ is not a semigroup; and asked if $${\mathbb{P}}$$ and $${\mathbb{C}}^{\times}$$ were the only closed semigroups. P. A. B. Pleasants [J. Lond. Math. Soc., II. Ser. 15, 134-142 (1977; Zbl 0344.40001)] and J. H. Smith [Proc. Am. Math. Soc. 47, 167-170 (1975; Zbl 0304.40003)] have also considered several problems relating convergent series and the group structure of $${\mathbb{P}}.$$
The paper under review examines some connections between Levi’s duality, the linearity of $${\mathbb{C}}$$, and the group structure of $${\mathbb{P}}$$. Levi’s question is answered in affirmative, and a study of the structure of the lattices of closed sets of $${\mathbb{P}}$$ and $${\mathbb{C}}$$ is initiated. The paper is divided into six sections, containing: basic definitions and previous results; the semigroup properties of permutations of the form $$(\sum a_ i)^{\times}$$; results on $$\pi^{+\times}$$; the characterization of the dual and second dual of the set of alternating series; and finally, some interesting open problems.
Reviewer: I.Sandor

##### MSC:
 40A05 Convergence and divergence of series and sequences
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