Finitely Supported Binary Relations between Infinite Atomic Sets

In the framework of finitely supported atomic sets, by using the notion of atomic cardinality and the <i>T</i>-finite support principle (a closure property for supports in some higher-order constructions), we present some finiteness properties of the finitely supported binary relations b...

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Autores principales: Andrei Alexandru, Gabriel Ciobanu
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spelling oai:doaj.org-article:9c4e0bdf31f54a5aa85b112a5c6a13a92021-11-25T19:06:09ZFinitely Supported Binary Relations between Infinite Atomic Sets10.3390/sym131120282073-8994https://doaj.org/article/9c4e0bdf31f54a5aa85b112a5c6a13a92021-10-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2028https://doaj.org/toc/2073-8994In the framework of finitely supported atomic sets, by using the notion of atomic cardinality and the <i>T</i>-finite support principle (a closure property for supports in some higher-order constructions), we present some finiteness properties of the finitely supported binary relations between infinite atomic sets. Of particular interest are finitely supported Dedekind-finite sets because they do not contain finitely supported, countably infinite subsets. We prove that the infinite sets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>℘</mo><mrow><mi>f</mi><mi>s</mi></mrow></msub><mrow><mo stretchy="false">(</mo><msup><mi>A</mi><mi>k</mi></msup><mo>×</mo><msup><mi>A</mi><mi>l</mi></msup><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>℘</mo><mrow><mi>f</mi><mi>s</mi></mrow></msub><mrow><mo stretchy="false">(</mo><msup><mi>A</mi><mi>k</mi></msup><mo>×</mo><msub><mo>℘</mo><mi>m</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>℘</mo><mrow><mi>f</mi><mi>s</mi></mrow></msub><mrow><mo stretchy="false">(</mo><msub><mo>℘</mo><mi>n</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>×</mo><msup><mi>A</mi><mi>k</mi></msup><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>℘</mo><mrow><mi>f</mi><mi>s</mi></mrow></msub><mrow><mo>(</mo><msub><mo>℘</mo><mi>n</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>×</mo><msub><mo>℘</mo><mi>m</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> do not contain uniformly supported infinite subsets. Moreover, the functions space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>Z</mi><msup><mi>A</mi><mi>m</mi></msup></msup></semantics></math></inline-formula> does not contain a uniformly supported infinite subset whenever <i>Z</i> does not contain a uniformly supported infinite subset. All these sets are Dedekind-finite in the framework of finitely supported structures.Andrei AlexandruGabriel CiobanuMDPI AGarticlefinitely supported structuresatomic setsrelationscardinalityfiniteness propertiesMathematicsQA1-939ENSymmetry, Vol 13, Iss 2028, p 2028 (2021)
institution DOAJ
collection DOAJ
language EN
topic finitely supported structures
atomic sets
relations
cardinality
finiteness properties
Mathematics
QA1-939
spellingShingle finitely supported structures
atomic sets
relations
cardinality
finiteness properties
Mathematics
QA1-939
Andrei Alexandru
Gabriel Ciobanu
Finitely Supported Binary Relations between Infinite Atomic Sets
description In the framework of finitely supported atomic sets, by using the notion of atomic cardinality and the <i>T</i>-finite support principle (a closure property for supports in some higher-order constructions), we present some finiteness properties of the finitely supported binary relations between infinite atomic sets. Of particular interest are finitely supported Dedekind-finite sets because they do not contain finitely supported, countably infinite subsets. We prove that the infinite sets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>℘</mo><mrow><mi>f</mi><mi>s</mi></mrow></msub><mrow><mo stretchy="false">(</mo><msup><mi>A</mi><mi>k</mi></msup><mo>×</mo><msup><mi>A</mi><mi>l</mi></msup><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>℘</mo><mrow><mi>f</mi><mi>s</mi></mrow></msub><mrow><mo stretchy="false">(</mo><msup><mi>A</mi><mi>k</mi></msup><mo>×</mo><msub><mo>℘</mo><mi>m</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>℘</mo><mrow><mi>f</mi><mi>s</mi></mrow></msub><mrow><mo stretchy="false">(</mo><msub><mo>℘</mo><mi>n</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>×</mo><msup><mi>A</mi><mi>k</mi></msup><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>℘</mo><mrow><mi>f</mi><mi>s</mi></mrow></msub><mrow><mo>(</mo><msub><mo>℘</mo><mi>n</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>×</mo><msub><mo>℘</mo><mi>m</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> do not contain uniformly supported infinite subsets. Moreover, the functions space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>Z</mi><msup><mi>A</mi><mi>m</mi></msup></msup></semantics></math></inline-formula> does not contain a uniformly supported infinite subset whenever <i>Z</i> does not contain a uniformly supported infinite subset. All these sets are Dedekind-finite in the framework of finitely supported structures.
format article
author Andrei Alexandru
Gabriel Ciobanu
author_facet Andrei Alexandru
Gabriel Ciobanu
author_sort Andrei Alexandru
title Finitely Supported Binary Relations between Infinite Atomic Sets
title_short Finitely Supported Binary Relations between Infinite Atomic Sets
title_full Finitely Supported Binary Relations between Infinite Atomic Sets
title_fullStr Finitely Supported Binary Relations between Infinite Atomic Sets
title_full_unstemmed Finitely Supported Binary Relations between Infinite Atomic Sets
title_sort finitely supported binary relations between infinite atomic sets
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/9c4e0bdf31f54a5aa85b112a5c6a13a9
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