Complete Representations and Neat Embeddings

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":"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}