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: | , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/9c4e0bdf31f54a5aa85b112a5c6a13a9 |
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| Sumario: | 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. |
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