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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.

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 |