Pub Date : 2023-11-08DOI: 10.1007/s10469-023-09716-z
S. V. Pchelintsev, O. V. Shashkov
It is proved that a singular superalgebra with a 2-dimensional even part is isomorphic to a superalgebra B2|3(φ, ξ, ψ). In particular, there do not exist infinite-dimensional simple singular superalgebras with a 2-dimensional even part. It is proved that if a singular superalgebra contains an odd left annihilator, then it contains a nondegenerate switch. Lastly, it is established that for any number N ≥ 5, except the numbers 6, 7, 8, 11, there exist singular superalgebras with a switch of dimension N. For the numbers N = 6, 7, 8, 11, there do not exist singular N -dimensional superalgebras with a switch.
研究证明，具有 2 维偶数部分的奇异超代数与超代数 B2|3(φ,ξ,ψ)同构。特别是，不存在具有 2 维偶数部分的无限维简单奇异超代数。证明了如果奇异超代数包含一个奇数左湮没器，那么它就包含一个非enerate 开关。最后，证明了对于任何 N ≥ 5 的数，除了 6、7、8、11 以外，都存在具有 N 维开关的奇异超代数；对于 N = 6、7、8、11 的数，不存在具有开关的 N 维奇异超代数。
{"title":"Structure of Singular Superalgebras with 2-Dimensional Even Part and New Examples of Singular Superalgebras","authors":"S. V. Pchelintsev, O. V. Shashkov","doi":"10.1007/s10469-023-09716-z","DOIUrl":"10.1007/s10469-023-09716-z","url":null,"abstract":"<p>It is proved that a singular superalgebra with a 2-dimensional even part is isomorphic to a superalgebra <i>B</i><sub>2|3</sub>(φ<i>, ξ, ψ</i>)<i>.</i> In particular, there do not exist infinite-dimensional simple singular superalgebras with a 2-dimensional even part. It is proved that if a singular superalgebra contains an odd left annihilator, then it contains a nondegenerate switch. Lastly, it is established that for any number <i>N ≥</i> 5, except the numbers 6, 7, 8, 11, there exist singular superalgebras with a switch of dimension <i>N</i>. For the numbers<i> N</i> = 6, 7, 8, 11, there do not exist singular <i>N</i> -dimensional superalgebras with a switch.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 6","pages":"506 - 523"},"PeriodicalIF":0.4,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135346370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-08DOI: 10.1007/s10469-023-09715-0
A. A. Onoprienko
The joint logic of problems and propositions QHC introduced by S. A. Melikhov, as well as intuitionistic modal logic QH4, is studied. An immersion of these logics into classical first-order predicate logic is considered. An analog of the Löwenheim–Skolem theorem on the existence of countable elementary submodels for QHC and QH4 is established.
{"title":"Cardinality Reduction Theorem for Logics QHC and QH4","authors":"A. A. Onoprienko","doi":"10.1007/s10469-023-09715-0","DOIUrl":"10.1007/s10469-023-09715-0","url":null,"abstract":"<p>The joint logic of problems and propositions QHC introduced by S. A. Melikhov, as well as intuitionistic modal logic QH4, is studied. An immersion of these logics into classical first-order predicate logic is considered. An analog of the Löwenheim–Skolem theorem on the existence of countable elementary submodels for QHC and QH4 is established.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 6","pages":"491 - 505"},"PeriodicalIF":0.4,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135390235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1007/s10469-023-09713-2
A. S. Zakharov
We consider a class of generalized derivations that arise in connection with the problem of adjoining unity to an algebra with generalized derivation, and of searching envelopes for Novikov–Poisson algebras. Conditions for the existence of the localization of an algebra with ternary derivation are specified, as well as conditions under which given an algebra with ternary derivation, we can construct a Novikov–Poisson algebra and a Jordan superalgebra. Finally, we show how the simplicity of an algebra with Brešar generalized derivation is connected with simplicity of the appropriate Novikov algebra.
{"title":"A Class of Generalized Derivations","authors":"A. S. Zakharov","doi":"10.1007/s10469-023-09713-2","DOIUrl":"10.1007/s10469-023-09713-2","url":null,"abstract":"<p>We consider a class of generalized derivations that arise in connection with the problem of adjoining unity to an algebra with generalized derivation, and of searching envelopes for Novikov–Poisson algebras. Conditions for the existence of the localization of an algebra with ternary derivation are specified, as well as conditions under which given an algebra with ternary derivation, we can construct a Novikov–Poisson algebra and a Jordan superalgebra. Finally, we show how the simplicity of an algebra with Brešar generalized derivation is connected with simplicity of the appropriate Novikov algebra.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 6","pages":"466 - 480"},"PeriodicalIF":0.4,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135867992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-02DOI: 10.1007/s10469-023-09719-w
{"title":"Sessions of the Seminar “Algebra i Logika”","authors":"","doi":"10.1007/s10469-023-09719-w","DOIUrl":"10.1007/s10469-023-09719-w","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 6","pages":"545 - 545"},"PeriodicalIF":0.4,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135933123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1007/s10469-023-09718-x
A. A. Stepanova
The concept of P-stability is a particular case of generalized stability of complete theories. We study injective S-acts with a P-stable theory. It is proved that the class of injective S-acts is (P, 1)-stable only if S is a one-element monoid. Also we describe commutative and linearly ordered monoids S the class of injective S-acts over which is (P, s)-, (P, a)-, and (P, e)-stable.
P-stability 概念是完整理论广义稳定性的一种特殊情况。我们研究具有 P 稳定性理论的注入式 S 行为。研究证明，只有当 S 是单元素单元时，注入式 S 作用类才是（P，1）稳定的。此外，我们还描述了交换和线性有序单元 S，其上的注入式 S-行为类是（P，s）-、（P，a）-和（P，e）-稳定的。
{"title":"Generalized Stability of the Class of Injective S-Acts","authors":"A. A. Stepanova","doi":"10.1007/s10469-023-09718-x","DOIUrl":"10.1007/s10469-023-09718-x","url":null,"abstract":"<p>The concept of P-stability is a particular case of generalized stability of complete theories. We study injective <i>S</i>-acts with a <i>P</i>-stable theory. It is proved that the class of injective <i>S</i>-acts is (<i>P</i>, 1)-stable only if <i>S</i> is a one-element monoid. Also we describe commutative and linearly ordered monoids <i>S</i> the class of injective <i>S</i>-acts over which is (<i>P</i>, <i>s</i>)-, (<i>P</i>, <i>a</i>)-, and (<i>P</i>, <i>e</i>)-stable.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 6","pages":"537 - 544"},"PeriodicalIF":0.4,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135321441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-06DOI: 10.1007/s10469-023-09709-y
I. P. Shestakov
Yu. A. Medvedev [Algebra and Logic, 19, No. 3, 191—201 (1980)] constructed an example of alternative algebra that he used to prove that a certain variety of alternative algebras possess the non-Specht property over a field of characteristic 2. Though his result concerned the characteristic 2 case, the example was claimed to be alternative over an arbitrary field, and it was later used by V. T. Filippov in a series of papers. Unfortunately, Medvedev’s example is in fact not alternative in any characteristic. Therefore, whether the variety considered by Medvedev has the non-Specht property is still not clear. Moreover, the results of Filippov’s papers, in which Medvedev’s example was used, also become questionable. We construct new examples and employ them to prove that the results of Filippov remain true.
{"title":"Modification and Correction of Medvedev’s Example of a Solvable Alternative Algebra","authors":"I. P. Shestakov","doi":"10.1007/s10469-023-09709-y","DOIUrl":"10.1007/s10469-023-09709-y","url":null,"abstract":"<div><div><p>Yu. A. Medvedev [Algebra and Logic, <b>19</b>, No. 3, 191—201 (1980)] constructed an example of alternative algebra that he used to prove that a certain variety of alternative algebras possess the non-Specht property over a field of characteristic 2. Though his result concerned the characteristic 2 case, the example was claimed to be alternative over an arbitrary field, and it was later used by V. T. Filippov in a series of papers. Unfortunately, Medvedev’s example is in fact not alternative in any characteristic. Therefore, whether the variety considered by Medvedev has the non-Specht property is still not clear. Moreover, the results of Filippov’s papers, in which Medvedev’s example was used, also become questionable. We construct new examples and employ them to prove that the results of Filippov remain true.</p></div></div>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 5","pages":"420 - 436"},"PeriodicalIF":0.5,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50457125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-04DOI: 10.1007/s10469-023-09707-0
F. Castro, D. Freitas, A. Paques, G. Quadros, T. Tamusiunas
A partial Hopf–Galois theory is developed for partial H-module algebras, and we recover analogs of classical results for Hopf algebras.
针对部分H-模代数，提出了部分Hopf–Galois理论，并恢复了Hopf代数经典结果的相似性。
{"title":"Partial Hopf–Galois Theory","authors":"F. Castro, D. Freitas, A. Paques, G. Quadros, T. Tamusiunas","doi":"10.1007/s10469-023-09707-0","DOIUrl":"10.1007/s10469-023-09707-0","url":null,"abstract":"<div><div><p>A partial Hopf–Galois theory is developed for partial H-module algebras, and we recover analogs of classical results for Hopf algebras.</p></div></div>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 5","pages":"385 - 406"},"PeriodicalIF":0.5,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50451043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-30DOI: 10.1007/s10469-023-09705-2
Yu. L. Ershov
We offer for consideration and study the concept of a pre-minimal pair. Also we examine extensions of valuations of a field F to valuations of the rational function field F(x) in one variable.
{"title":"Pre-Minimal Pairs and Homogeneous Valuations","authors":"Yu. L. Ershov","doi":"10.1007/s10469-023-09705-2","DOIUrl":"10.1007/s10469-023-09705-2","url":null,"abstract":"<div><div><p>We offer for consideration and study the concept of a pre-minimal pair. Also we examine extensions of valuations of a field <i>F</i> to valuations of the rational function field <i>F</i>(<i>x</i>) in one variable.</p></div></div>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 5","pages":"363 - 371"},"PeriodicalIF":0.5,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50524839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-30DOI: 10.1007/s10469-023-09708-z
V. V. Rybakov
We study temporal multi-agent logics using a new approach to defining time for individual agents. It is assumed that in any time state each agent (in a sense) generates its own future time, which will only be available for analysis in the future. That is, the defined time interval depends both on the agent and on the initial state where the agent starts to act. It is also assumed that the agent may have intervals of forgotten (lost) time. We investigate problems of unification and problems of computable recognizing admissible inference rules. An algorithm is found for solving these problems based on the construction of a finite computable set of formulas which is a complete set of unifiers. We use the technique of projective formulas developed by S. Ghilardi. It is proved that any unifiable formula is in fact projective and an algorithm is constructed which creates its projective unifier. Thereby we solve the unification problem, and based at this, find the solution to the open problem of computable recognizing admissible inference rules.
{"title":"Dynamic Temporal Logical Operations in Multi-Agent Logics","authors":"V. V. Rybakov","doi":"10.1007/s10469-023-09708-z","DOIUrl":"10.1007/s10469-023-09708-z","url":null,"abstract":"<div><div><p>We study temporal multi-agent logics using a new approach to defining time for individual agents. It is assumed that in any time state each agent (in a sense) generates its own future time, which will only be available for analysis in the future. That is, the defined time interval depends both on the agent and on the initial state where the agent starts to act. It is also assumed that the agent may have intervals of forgotten (lost) time. We investigate problems of unification and problems of computable recognizing admissible inference rules. An algorithm is found for solving these problems based on the construction of a finite computable set of formulas which is a complete set of unifiers. We use the technique of projective formulas developed by S. Ghilardi. It is proved that any unifiable formula is in fact projective and an algorithm is constructed which creates its projective unifier. Thereby we solve the unification problem, and based at this, find the solution to the open problem of computable recognizing admissible inference rules.</p></div></div>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 5","pages":"407 - 419"},"PeriodicalIF":0.5,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50524838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-30DOI: 10.1007/s10469-023-09710-5
P. E. Alaev
{"title":"A Criterion for P-Computability of Structures","authors":"P. E. Alaev","doi":"10.1007/s10469-023-09710-5","DOIUrl":"10.1007/s10469-023-09710-5","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"61 5","pages":"437 - 441"},"PeriodicalIF":0.5,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50524837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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