完整的表示和整齐的嵌入

Q2 Arts and Humanities

Bulletin of the Section of Logic Pub Date : 2022-09-09 DOI:10.18778/0138-0680.2022.17

T. Sayed Ahmed

{"title":"完整的表示和整齐的嵌入","authors":"T. Sayed Ahmed","doi":"10.18778/0138-0680.2022.17","DOIUrl":null,"url":null,"abstract":"Let \\(2<n<\\omega\\). Then \\({\\sf CA}_n\\) denotes the class of cylindric algebras of dimension \\(n\\), \\({\\sf RCA}_n\\) denotes the class of representable \\(\\sf CA_n\\)s, \\({\\sf CRCA}_n\\) denotes the class of completely representable \\({\\sf CA}_n\\)s, and \\({\\sf Nr}_n{\\sf CA}_{\\omega}(\\subseteq {\\sf CA}_n\\)) denotes the class of \\(n\\)-neat reducts of \\({\\sf CA}_{\\omega}\\)s. The elementary closure of the class \\({\\sf CRCA}_n\\)s (\\(\\mathbf{K_n}\\)) and the non-elementary class \\({\\sf At}({\\sf Nr}_n{\\sf CA}_{\\omega})\\) are characterized using two-player zero-sum games, where \\({\\sf At}\\) is the operator of forming atom structures. It is shown that \\(\\mathbf{K_n}\\) is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of \\(\\mathbf{S_c}{\\sf Nr}_n{\\sf CA}_{\\omega}\\) where \\(\\mathbf{S_c}\\) is the operation of forming complete subalgebras. For any class \\(\\mathbf{L}\\) such that \\({\\sf At}{\\sf Nr}_n{\\sf CA}_{\\omega}\\subseteq \\mathbf{L}\\subseteq {\\sf At}\\mathbf{K_n}\\), it is proved that \\({\\bf SP}\\mathfrak{Cm}\\mathbf{L}={\\sf RCA}_n\\), where \\({\\sf Cm}\\) is the dual operator to \\(\\sf At\\); that of forming complex algebras. It is also shown that any class \\(\\mathbf{K}\\) between \\({\\sf CRCA}_n\\cap \\mathbf{S_d}{\\sf Nr}_n{\\sf CA}_{\\omega}\\) and \\(\\mathbf{S_c}{\\sf Nr}_n{\\sf CA}_{n+3}\\) is not first order definable, where \\(\\mathbf{S_d}\\) is the operation of forming dense subalgebras, and that for any \\(2<n<m\\), any \\(l\\geq n+3\\) any any class \\(\\mathbf{K}\\) (such that \\({\\sf At}({\\sf Nr}_n{\\sf CA}_{m})\\cap {\\sf CRCA}_n\\subseteq \\mathbf{K}\\subseteq {\\sf At}\\mathbf{S_c}{\\sf Nr}_n{\\sf CA}_{l}\\), \\(\\mathbf{K}\\) is not not first order definable either.","PeriodicalId":38667,"journal":{"name":"Bulletin of the Section of Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Complete Representations and Neat Embeddings\",\"authors\":\"T. Sayed Ahmed\",\"doi\":\"10.18778/0138-0680.2022.17\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let \\\\(2<n<\\\\omega\\\\). Then \\\\({\\\\sf CA}_n\\\\) denotes the class of cylindric algebras of dimension \\\\(n\\\\), \\\\({\\\\sf RCA}_n\\\\) denotes the class of representable \\\\(\\\\sf CA_n\\\\)s, \\\\({\\\\sf CRCA}_n\\\\) denotes the class of completely representable \\\\({\\\\sf CA}_n\\\\)s, and \\\\({\\\\sf Nr}_n{\\\\sf CA}_{\\\\omega}(\\\\subseteq {\\\\sf CA}_n\\\\)) denotes the class of \\\\(n\\\\)-neat reducts of \\\\({\\\\sf CA}_{\\\\omega}\\\\)s. The elementary closure of the class \\\\({\\\\sf CRCA}_n\\\\)s (\\\\(\\\\mathbf{K_n}\\\\)) and the non-elementary class \\\\({\\\\sf At}({\\\\sf Nr}_n{\\\\sf CA}_{\\\\omega})\\\\) are characterized using two-player zero-sum games, where \\\\({\\\\sf At}\\\\) is the operator of forming atom structures. It is shown that \\\\(\\\\mathbf{K_n}\\\\) is not finitely axiomatizable and that it coincides with the class of atomic algebras in the elementary closure of \\\\(\\\\mathbf{S_c}{\\\\sf Nr}_n{\\\\sf CA}_{\\\\omega}\\\\) where \\\\(\\\\mathbf{S_c}\\\\) is the operation of forming complete subalgebras. For any class \\\\(\\\\mathbf{L}\\\\) such that \\\\({\\\\sf At}{\\\\sf Nr}_n{\\\\sf CA}_{\\\\omega}\\\\subseteq \\\\mathbf{L}\\\\subseteq {\\\\sf At}\\\\mathbf{K_n}\\\\), it is proved that \\\\({\\\\bf SP}\\\\mathfrak{Cm}\\\\mathbf{L}={\\\\sf RCA}_n\\\\), where \\\\({\\\\sf Cm}\\\\) is the dual operator to \\\\(\\\\sf At\\\\); that of forming complex algebras. It is also shown that any class \\\\(\\\\mathbf{K}\\\\) between \\\\({\\\\sf CRCA}_n\\\\cap \\\\mathbf{S_d}{\\\\sf Nr}_n{\\\\sf CA}_{\\\\omega}\\\\) and \\\\(\\\\mathbf{S_c}{\\\\sf Nr}_n{\\\\sf CA}_{n+3}\\\\) is not first order definable, where \\\\(\\\\mathbf{S_d}\\\\) is the operation of forming dense subalgebras, and that for any \\\\(2<n<m\\\\), any \\\\(l\\\\geq n+3\\\\) any any class \\\\(\\\\mathbf{K}\\\\) (such that \\\\({\\\\sf At}({\\\\sf Nr}_n{\\\\sf CA}_{m})\\\\cap {\\\\sf CRCA}_n\\\\subseteq \\\\mathbf{K}\\\\subseteq {\\\\sf At}\\\\mathbf{S_c}{\\\\sf Nr}_n{\\\\sf CA}_{l}\\\\), \\\\(\\\\mathbf{K}\\\\) is not not first order definable either.\",\"PeriodicalId\":38667,\"journal\":{\"name\":\"Bulletin of the Section of Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-09-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the Section of Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.18778/0138-0680.2022.17\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Arts and Humanities\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Section of Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.18778/0138-0680.2022.17","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Arts and Humanities","Score":null,"Total":0}