伪几何空间:组合自相似性组中的最小性到最大性

Pub Date : 2024-02-10 DOI:10.1515/agms-2023-0103
Viktoriia Bilet, Oleksiy Dovgoshey
{"title":"伪几何空间:组合自相似性组中的最小性到最大性","authors":"Viktoriia Bilet, Oleksiy Dovgoshey","doi":"10.1515/agms-2023-0103","DOIUrl":null,"url":null,"abstract":"The group of combinatorial self-similarities of a pseudometric space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the maximal subgroup of the symmetric group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Sym</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{\\rm{Sym}}\\left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula> whose elements preserve the four-point equality <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>d\\left(x,y)=d\\left(u,v)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let us denote by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℐP</m:mi> </m:math> <jats:tex-math>{\\mathcal{ {\\mathcal I} P}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> the class of all pseudometric spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> for which every combinatorial self-similarity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Φ</m:mi> <m:mo>:</m:mo> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>X</m:mi> </m:math> <jats:tex-math>\\Phi :X\\to X</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfies the equality <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_007.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">Φ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:math> <jats:tex-math>d\\left(x,\\Phi \\left(x))=0,</jats:tex-math> </jats:alternatives> </jats:inline-formula> but all permutations of metric reflection of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_008.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> are combinatorial self-similarities of this reflection. The structure of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_009.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℐP</m:mi> </m:math> <jats:tex-math>{\\mathcal{ {\\mathcal I} P}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-spaces is fully described.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Pseudometric spaces: From minimality to maximality in the groups of combinatorial self-similarities\",\"authors\":\"Viktoriia Bilet, Oleksiy Dovgoshey\",\"doi\":\"10.1515/agms-2023-0103\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The group of combinatorial self-similarities of a pseudometric space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0103_eq_001.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\\\left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the maximal subgroup of the symmetric group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0103_eq_002.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"normal\\\">Sym</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{\\\\rm{Sym}}\\\\left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula> whose elements preserve the four-point equality <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0103_eq_003.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>d\\\\left(x,y)=d\\\\left(u,v)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let us denote by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0103_eq_004.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"script\\\">ℐP</m:mi> </m:math> <jats:tex-math>{\\\\mathcal{ {\\\\mathcal I} P}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> the class of all pseudometric spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0103_eq_005.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\\\left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> for which every combinatorial self-similarity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0103_eq_006.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"normal\\\">Φ</m:mi> <m:mo>:</m:mo> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>X</m:mi> </m:math> <jats:tex-math>\\\\Phi :X\\\\to X</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfies the equality <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0103_eq_007.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\\\"normal\\\">Φ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:math> <jats:tex-math>d\\\\left(x,\\\\Phi \\\\left(x))=0,</jats:tex-math> </jats:alternatives> </jats:inline-formula> but all permutations of metric reflection of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_agms-2023-0103_eq_008.png\\\" /> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\\\left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> are combinatorial self-similarities of this reflection. 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引用次数: 0

摘要

伪几何空间 ( X , d ) \left(X,d)的组合自相似性群是对称群 Sym ( X ) {\rm{Sym}}\left(X) 的最大子群,其元素保持四点相等 d ( x , y ) = d ( u , v ) d\left(x,y)=d\left(u,v) 。让我们用 ℐP {\mathcal{ {\mathcal I} P}} 表示所有伪几何空间 ( X , d ) 的类 \left(X,d),其中每个组合自相似性 Φ : X → X \Phi :X\to X 满足等式 d ( x , Φ ( x ) ) = 0 , d\left(x,\Phi \left(x))=0,但是 ( X , d ) \left(X,d)的度量反射的所有排列都是这种反射的组合自相似性。对ℐP {\mathcal{ {\mathcal I} P}} 的结构进行了全面描述。 -空间的结构得到了充分描述。
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Pseudometric spaces: From minimality to maximality in the groups of combinatorial self-similarities
The group of combinatorial self-similarities of a pseudometric space ( X , d ) \left(X,d) is the maximal subgroup of the symmetric group Sym ( X ) {\rm{Sym}}\left(X) whose elements preserve the four-point equality d ( x , y ) = d ( u , v ) d\left(x,y)=d\left(u,v) . Let us denote by ℐP {\mathcal{ {\mathcal I} P}} the class of all pseudometric spaces ( X , d ) \left(X,d) for which every combinatorial self-similarity Φ : X X \Phi :X\to X satisfies the equality d ( x , Φ ( x ) ) = 0 , d\left(x,\Phi \left(x))=0, but all permutations of metric reflection of ( X , d ) \left(X,d) are combinatorial self-similarities of this reflection. The structure of ℐP {\mathcal{ {\mathcal I} P}} -spaces is fully described.
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