权力群的半环和对合恒等式

Pub Date : 2022-06-17 DOI:10.1017/S1446788722000374
S. V. Gusev, Mikhail Volkov
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Gusev, Mikhail Volkov","doi":"10.1017/S1446788722000374","DOIUrl":null,"url":null,"abstract":"\n\t <jats:p>For every group <jats:italic>G</jats:italic>, the set <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline1.png\" />\n\t\t<jats:tex-math>\n$\\mathcal {P}(G)$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> of its subsets forms a semiring under set-theoretical union <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline2.png\" />\n\t\t<jats:tex-math>\n$\\cup $\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> and element-wise multiplication <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline3.png\" />\n\t\t<jats:tex-math>\n$\\cdot $\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>, and forms an involution semigroup under <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline4.png\" />\n\t\t<jats:tex-math>\n$\\cdot $\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> and element-wise inversion <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline5.png\" />\n\t\t<jats:tex-math>\n${}^{-1}$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>. We show that if the group <jats:italic>G</jats:italic> is finite, non-Dedekind, and solvable, neither the semiring <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline6.png\" />\n\t\t<jats:tex-math>\n$(\\mathcal {P}(G),\\cup ,\\cdot )$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> nor the involution semigroup <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788722000374_inline7.png\" />\n\t\t<jats:tex-math>\n$(\\mathcal {P}(G),\\cdot ,{}^{-1})$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.</jats:p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"SEMIRING AND INVOLUTION IDENTITIES OF POWER GROUPS\",\"authors\":\"S. V. 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We show that if the group <jats:italic>G</jats:italic> is finite, non-Dedekind, and solvable, neither the semiring <jats:inline-formula>\\n\\t <jats:alternatives>\\n\\t\\t<jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1446788722000374_inline6.png\\\" />\\n\\t\\t<jats:tex-math>\\n$(\\\\mathcal {P}(G),\\\\cup ,\\\\cdot )$\\n</jats:tex-math>\\n\\t </jats:alternatives>\\n\\t </jats:inline-formula> nor the involution semigroup <jats:inline-formula>\\n\\t <jats:alternatives>\\n\\t\\t<jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S1446788722000374_inline7.png\\\" />\\n\\t\\t<jats:tex-math>\\n$(\\\\mathcal {P}(G),\\\\cdot ,{}^{-1})$\\n</jats:tex-math>\\n\\t </jats:alternatives>\\n\\t </jats:inline-formula> admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.</jats:p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-06-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/S1446788722000374\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/S1446788722000374","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 4

摘要

对于每一个群G,其子集的集合$\mathcal {P}(G)$在集合论并$\cup $和元素智能乘法$\cdot $下形成一个半环,在$\cdot $和元素智能反转${}^{-1}$下形成一个对合半群。证明了如果群G是有限的、非dedekind的、可解的,则半环$(\mathcal {P}(G),\cup,\cdot)$和对合半群$(\mathcal {P}(G),\cdot,{}^{-1})$都不存在有限恒等基。我们还解决了任意有限集上Hall关系半环的有限基问题。
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SEMIRING AND INVOLUTION IDENTITIES OF POWER GROUPS
For every group G, the set $\mathcal {P}(G)$ of its subsets forms a semiring under set-theoretical union $\cup $ and element-wise multiplication $\cdot $ , and forms an involution semigroup under $\cdot $ and element-wise inversion ${}^{-1}$ . We show that if the group G is finite, non-Dedekind, and solvable, neither the semiring $(\mathcal {P}(G),\cup ,\cdot )$ nor the involution semigroup $(\mathcal {P}(G),\cdot ,{}^{-1})$ admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.
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