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Squeezing multisets into real numbers
Cantone, Domenico
Policriti, Alberto
2021
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e-ISSN
2464-8728
Abstract
In this paper we study the encoding
$\mathbb{R}_A(x) = \sum_{y\in x} 2^{-\mathbb{R}_A(y)}$,
mapping hereditarily finite sets and hypersets - hereditarily finite
sets admitting circular chains of memberships - into real numbers.
The map $\mathbb{R}_A$ somewhat generalizes the well-known Ackermann's
encoding
$\mathbb{N}_A(x) = \sum_{y\in x} 2^{\mathbb{N}_A(y)}$,
whose co-domain is $\mathbb{N}$, to nonnegative real numbers.
In this work we define and study the further natural extension of the
map $\mathbb{R}_A$ to the so-called multisets. Such an extension is
simply obtained by multiplying by $k$ the code of each element having
multiplicity equal to $k$.
We prove that, under a rather natural injectivity assumption of
$\mathbb{R}_A$ on the universe of multisets, the map $\mathbb{R}_A$
sends almost all multisets into transcendental numbers.
Publisher
EUT Edizioni Università di Trieste
Source
Domenico Cantone, Alberto Policriti, "Squeezing multisets into real numbers" in: "Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.53 (2021)", EUT Edizioni Università di Trieste, Trieste, 2021. pp.
Languages
en
Rights
Attribution-NonCommercial-NoDerivatives 4.0 Internazionale
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