On Factoring Groups into Thin Subsets
A subset <i>X</i> of a group <i>G</i> is called thin if, for every finite subset <i>F</i> of <i>G</i>, there exists a finite subset <i>H</i> of <i>G</i> such that <inline-formula><math xmlns="http://www.w3.org/1998...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-05-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/10/2/89 |
| _version_ | 1827692565592276992 |
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| author | Igor Protasov |
| author_facet | Igor Protasov |
| author_sort | Igor Protasov |
| collection | DOAJ |
| description | A subset <i>X</i> of a group <i>G</i> is called thin if, for every finite subset <i>F</i> of <i>G</i>, there exists a finite subset <i>H</i> of <i>G</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>x</mi><mo>∩</mo><mi>F</mi><mi>y</mi><mo>=</mo><mo>∅</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mi>F</mi><mo>∩</mo><mi>y</mi><mi>F</mi><mo>=</mo><mo>∅</mo></mrow></semantics></math></inline-formula> for all distinct <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi><mo>\</mo><mi>H</mi></mrow></semantics></math></inline-formula>. We prove that every countable topologizable group <i>G</i> can be factorized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mi>A</mi><mi>B</mi></mrow></semantics></math></inline-formula> into thin subsets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T11:25:54Z |
| format | Article |
| id | doaj.art-774eaac89cea4abc96de683d66f3306b |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T11:25:54Z |
| publishDate | 2021-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-774eaac89cea4abc96de683d66f3306b2023-11-21T19:41:39ZengMDPI AGAxioms2075-16802021-05-011028910.3390/axioms10020089On Factoring Groups into Thin SubsetsIgor Protasov0Department of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Academic Glushkov pr. 4d, 03680 Kyiv, UkraineA subset <i>X</i> of a group <i>G</i> is called thin if, for every finite subset <i>F</i> of <i>G</i>, there exists a finite subset <i>H</i> of <i>G</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>x</mi><mo>∩</mo><mi>F</mi><mi>y</mi><mo>=</mo><mo>∅</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mi>F</mi><mo>∩</mo><mi>y</mi><mi>F</mi><mo>=</mo><mo>∅</mo></mrow></semantics></math></inline-formula> for all distinct <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi><mo>\</mo><mi>H</mi></mrow></semantics></math></inline-formula>. We prove that every countable topologizable group <i>G</i> can be factorized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mi>A</mi><mi>B</mi></mrow></semantics></math></inline-formula> into thin subsets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/10/2/89factorizations of a groupthin subset of a group |
| spellingShingle | Igor Protasov On Factoring Groups into Thin Subsets Axioms factorizations of a group thin subset of a group |
| title | On Factoring Groups into Thin Subsets |
| title_full | On Factoring Groups into Thin Subsets |
| title_fullStr | On Factoring Groups into Thin Subsets |
| title_full_unstemmed | On Factoring Groups into Thin Subsets |
| title_short | On Factoring Groups into Thin Subsets |
| title_sort | on factoring groups into thin subsets |
| topic | factorizations of a group thin subset of a group |
| url | https://www.mdpi.com/2075-1680/10/2/89 |
| work_keys_str_mv | AT igorprotasov onfactoringgroupsintothinsubsets |