We discuss new problems in universal algebraic geometry and explain them by boolean equations.
讨论了通用代数几何中的一些新问题，并用布尔方程进行了解释。
{"title":"New problems in universal algebraic geometry\u0000 illustrated by Boolean equations","authors":"A. Shevlyakov","doi":"10.1090/conm/726/14612","DOIUrl":"https://doi.org/10.1090/conm/726/14612","url":null,"abstract":"We discuss new problems in universal algebraic geometry and explain them by boolean equations.","PeriodicalId":355806,"journal":{"name":"Groups, Algebras and Identities","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124686519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Atkarskaya, A. Kanel-Belov, E. Plotkin, E. Rips
{"title":"Construction of a quotient ring of ℤ₂ℱ in\u0000 which a binomial 1+𝓌 is invertible using small\u0000 cancellation methods","authors":"A. Atkarskaya, A. Kanel-Belov, E. Plotkin, E. Rips","doi":"10.1090/conm/726/14605","DOIUrl":"https://doi.org/10.1090/conm/726/14605","url":null,"abstract":"","PeriodicalId":355806,"journal":{"name":"Groups, Algebras and Identities","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121515959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Syntax versus semantics in knowledge bases\u0000 II","authors":"E. Aladova, T. Plotkin","doi":"10.1090/CONM/726/14607","DOIUrl":"https://doi.org/10.1090/CONM/726/14607","url":null,"abstract":"","PeriodicalId":355806,"journal":{"name":"Groups, Algebras and Identities","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114522712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A number of geometric properties of Ω-groups from a given variety of Ω-groups can be characterized using the notions of domain and equational domain. An Ω-group H of a variety Θ is an equational domain in Θ if the union of algebraic varieties over H is an algebraic variety. We give necessary and sufficient conditions for an Ω-group H in Θ to be an equational domain in this variety. Let F = F (X) be a finitely generated by X free Ω-group in a variety of Ωgroups Θ and H be an Ω-group in Θ. One of the important questions in the algebraic geometry of varieties of Ω-groups is whether it is possible to equip the space of points Hom(F,H) with the Zariski topology, whose closed sets are precisely algebraic sets. If this is so, then the Ω-group H is called the equational domain (or stable in the terminology of the authors of the papers [P], [BPP]) in the variety Θ (see [BMR]). The same problem arises for a variety Θ(G) of Ω-groups with the given Ω-group of constants G. An important role in the study of equational domains in varieties of Ω-groups is played by the notion of domain. Necessary and sufficient conditions for linear algebras over an algebra of constants and for groups over a group of constants to be equational domains in terms of domains are given in [BMR], [BPP], [DMR] and [P]. Here we continue the study of equational domains in varieties of Ω-groups (without of the Ω-group of constants). We give necessary and sufficient conditions for an Ω-group H in Θ to be an equational domain in the variety Θ. The results presented in this paper were partially announced earlier in [L].
{"title":"On the Zariski topology of Ω-groups","authors":"R. Lipyanski","doi":"10.1090/conm/726/14610","DOIUrl":"https://doi.org/10.1090/conm/726/14610","url":null,"abstract":"A number of geometric properties of Ω-groups from a given variety of Ω-groups can be characterized using the notions of domain and equational domain. An Ω-group H of a variety Θ is an equational domain in Θ if the union of algebraic varieties over H is an algebraic variety. We give necessary and sufficient conditions for an Ω-group H in Θ to be an equational domain in this variety. Let F = F (X) be a finitely generated by X free Ω-group in a variety of Ωgroups Θ and H be an Ω-group in Θ. One of the important questions in the algebraic geometry of varieties of Ω-groups is whether it is possible to equip the space of points Hom(F,H) with the Zariski topology, whose closed sets are precisely algebraic sets. If this is so, then the Ω-group H is called the equational domain (or stable in the terminology of the authors of the papers [P], [BPP]) in the variety Θ (see [BMR]). The same problem arises for a variety Θ(G) of Ω-groups with the given Ω-group of constants G. An important role in the study of equational domains in varieties of Ω-groups is played by the notion of domain. Necessary and sufficient conditions for linear algebras over an algebra of constants and for groups over a group of constants to be equational domains in terms of domains are given in [BMR], [BPP], [DMR] and [P]. Here we continue the study of equational domains in varieties of Ω-groups (without of the Ω-group of constants). We give necessary and sufficient conditions for an Ω-group H in Θ to be an equational domain in the variety Θ. The results presented in this paper were partially announced earlier in [L].","PeriodicalId":355806,"journal":{"name":"Groups, Algebras and Identities","volume":"544 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117322047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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