Pub Date : 2022-12-03DOI: 10.1007/s10469-022-09694-8
{"title":"Sessions of the Seminar “Algebra i Logika”","authors":"","doi":"10.1007/s10469-022-09694-8","DOIUrl":"10.1007/s10469-022-09694-8","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50444828","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 : 2022-12-03DOI: 10.1007/s10469-022-09692-w
I. Sh. Kalimullin
{"title":"Degrees of Relative Computable Categoricity","authors":"I. Sh. Kalimullin","doi":"10.1007/s10469-022-09692-w","DOIUrl":"10.1007/s10469-022-09692-w","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50444829","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 : 2022-12-03DOI: 10.1007/s10469-022-09690-y
S. A. Badaev, S. S. Goncharov
{"title":"Rogers Semilattices with Least and Greatest Elements in the Ershov Hierarchy","authors":"S. A. Badaev, S. S. Goncharov","doi":"10.1007/s10469-022-09690-y","DOIUrl":"10.1007/s10469-022-09690-y","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50444830","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 : 2022-12-03DOI: 10.1007/s10469-022-09693-9
D. V. Churikov
{"title":"Closures of Finite Permutation Groups","authors":"D. V. Churikov","doi":"10.1007/s10469-022-09693-9","DOIUrl":"10.1007/s10469-022-09693-9","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50444827","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 : 2022-10-22DOI: 10.1007/s10469-022-09681-z
S. S. Korobkov
Associative rings are considered. By a lattice isomorphism (or projection) of a ring R onto a ring Rφ we mean an isomorphism φ of the subring lattice L(R) of a ring R onto the subring lattice L(Rφ) of a ring Rφ. Let Mn(GF(pk)) be the ring of all square matrices of order n over a finite field GF(pk), where n and k are natural numbers, p is a prime. A finite ring R with identity is called a semilocal (primary) ring if R/RadR ≅ Mn(GF(pk)). It is known that a finite ring R with identity is a semilocal ring iff R ≅ Mn(K) and K is a finite local ring. Here we study lattice isomorphisms of finite semilocal rings. It is proved that if φ is a projection of a ring R = Mn(K), where K is an arbitrary finite local ring, onto a ring Rφ, then Rφ = Mn(K′), in which case K′ is a local ring lattice-isomorphic to the ring K. We thus prove that the class of semilocal rings is lattice definable.
{"title":"Projections of Semilocal Rings","authors":"S. S. Korobkov","doi":"10.1007/s10469-022-09681-z","DOIUrl":"10.1007/s10469-022-09681-z","url":null,"abstract":"<div><div><p>Associative rings are considered. By a lattice isomorphism (or projection) of a ring <i>R</i> onto a ring <i>R</i><sup><i>φ</i></sup> we mean an isomorphism <i>φ</i> of the subring lattice L(<i>R</i>) of a ring <i>R</i> onto the subring lattice L(<i>R</i><sup><i>φ</i></sup>) of a ring <i>R</i><sup><i>φ</i></sup>. Let M<sub>n</sub>(GF(p<sup>k</sup>)) be the ring of all square matrices of order n over a finite field GF(<i>p</i><sup><i>k</i></sup>), where <i>n</i> and <i>k</i> are natural numbers, <i>p</i> is a prime. A finite ring R with identity is called a semilocal (primary) ring if R/RadR ≅ M<sub>n</sub>(GF(p<sup>k</sup>)). It is known that a finite ring R with identity is a semilocal ring iff <i>R</i> ≅ M<sub>n</sub>(<i>K</i>) and <i>K</i> is a finite local ring. Here we study lattice isomorphisms of finite semilocal rings. It is proved that if <i>φ</i> is a projection of a ring <i>R</i> = M<sub>n</sub>(<i>K</i>), where <i>K</i> is an arbitrary finite local ring, onto a ring <i>R</i><sup><i>φ</i></sup>, then <i>R</i><sup><i>φ</i></sup> = Mn(<i>K</i>′), in which case <i>K</i>′ is a local ring lattice-isomorphic to the ring <i>K</i>. We thus prove that the class of semilocal rings is lattice definable.</p></div></div>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50506596","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 : 2022-10-22DOI: 10.1007/s10469-022-09679-7
E. V. Aladova
Let an arbitrary variety of algebras and the category of all free finitely generated algebras in that variety be given. This paper is the second in a series of papers started in [Algebra and Logic, 61, No. 1, 1-15 (2022)] where we deal with automorphisms of the category of free finitely generated algebras. Here we describe in detail a method of verbal operations. The method provides a characterization of automorphisms of the category of all free finitely generated algebras in a given variety. The characterization plays a crucial role in universal algebraic geometry. We supply the reader with illuminating examples which clarify the method.
{"title":"Method of Verbal Operations and Automorphisms of the Category of Free Algebras","authors":"E. V. Aladova","doi":"10.1007/s10469-022-09679-7","DOIUrl":"10.1007/s10469-022-09679-7","url":null,"abstract":"<div><div><p>Let an arbitrary variety of algebras and the category of all free finitely generated algebras in that variety be given. This paper is the second in a series of papers started in [Algebra and Logic, <b>61</b>, No. 1, 1-15 (2022)] where we deal with automorphisms of the category of free finitely generated algebras. Here we describe in detail a method of verbal operations. The method provides a characterization of automorphisms of the category of all free finitely generated algebras in a given variety. The characterization plays a crucial role in universal algebraic geometry. We supply the reader with illuminating examples which clarify the method.</p></div></div>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50506597","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 : 2022-10-20DOI: 10.1007/s10469-022-09684-w
V. N. Zhelyabin, A. P. Pozhidaev, U. U. Umirbaev
{"title":"Simple Lie-Solvable Pre-Lie Algebras","authors":"V. N. Zhelyabin, A. P. Pozhidaev, U. U. Umirbaev","doi":"10.1007/s10469-022-09684-w","DOIUrl":"10.1007/s10469-022-09684-w","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50501016","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 : 2022-10-15DOI: 10.1007/s10469-022-09683-x
B. Khoussainov, A. G. Melnikov
We present a new construction of indecomposable type 0 Abelian groups of rank 2. The new construction is used to study degree spectra of such groups. As a corollary, we obtain a new computability-theoretic proof showing that there exist continuum many nonisomorphic type 0 indecomposable Abelian groups of rank 2.
{"title":"Decomposability and Computability","authors":"B. Khoussainov, A. G. Melnikov","doi":"10.1007/s10469-022-09683-x","DOIUrl":"10.1007/s10469-022-09683-x","url":null,"abstract":"<div><div><p>We present a new construction of indecomposable type <b>0</b> Abelian groups of rank 2. The new construction is used to study degree spectra of such groups. As a corollary, we obtain a new computability-theoretic proof showing that there exist continuum many nonisomorphic type <b>0</b> indecomposable Abelian groups of rank 2.</p></div></div>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50485580","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 : 2022-10-15DOI: 10.1007/s10469-022-09680-0
A. S. Vasil’ev, D. O. Revin
Let 𝖃 be a class of finite groups which contains a group of order 2 and is closed under subgroups, homomorphic images, and extensions. We define the concept of an 𝖃-admissible diagram representing a natural number n. Associated with each n are finitely many such diagrams, and they all can be found easily. Admissible diagrams representing a number n are used to uniquely parametrize conjugacy classes of maximal 𝖃-subgroups of odd index in the symmetric group Symn, and we define the structure of such groups. As a consequence, we obtain a complete classification of submaximal 𝖃-subgroups of odd index in alternating groups.
{"title":"Relatively Maximal Subgroups of Odd Index in Symmetric Groups","authors":"A. S. Vasil’ev, D. O. Revin","doi":"10.1007/s10469-022-09680-0","DOIUrl":"10.1007/s10469-022-09680-0","url":null,"abstract":"<div><div><p>Let 𝖃 be a class of finite groups which contains a group of order 2 and is closed under subgroups, homomorphic images, and extensions. We define the concept of an 𝖃-admissible diagram representing a natural number n. Associated with each n are finitely many such diagrams, and they all can be found easily. Admissible diagrams representing a number n are used to uniquely parametrize conjugacy classes of maximal 𝖃-subgroups of odd index in the symmetric group Sym<sub>n</sub>, and we define the structure of such groups. As a consequence, we obtain a complete classification of submaximal 𝖃-subgroups of odd index in alternating groups.</p></div></div>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50485579","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 : 2022-10-14DOI: 10.1007/s10469-022-09682-y
E. I. Timoshenko
Given a finite undirected graph Γ without loops, we define a sentence Φ(Γ) of group theory. A sequence of graphs Γi is used to obtain a sequence of sentences Φ(Γi). These are employed to determine the Γ-dimension of a group and to study properties of the dimension. Under certain restrictions on a group, the known centralizer dimension is the Γ-dimension for some sequence of graphs. We mostly focus on dimensions defined by using linear graphs and cycles. Dimensions for a number of partially commutative metabelian groups are computed.
{"title":"Group Signature Formulas Constructed from Graphs","authors":"E. I. Timoshenko","doi":"10.1007/s10469-022-09682-y","DOIUrl":"10.1007/s10469-022-09682-y","url":null,"abstract":"<div><div><p>Given a finite undirected graph Γ without loops, we define a sentence Φ(Γ) of group theory. A sequence of graphs Γ<sub><i>i</i></sub> is used to obtain a sequence of sentences Φ(Γ<sub><i>i</i></sub>). These are employed to determine the Γ-dimension of a group and to study properties of the dimension. Under certain restrictions on a group, the known centralizer dimension is the Γ-dimension for some sequence of graphs. We mostly focus on dimensions defined by using linear graphs and cycles. Dimensions for a number of partially commutative metabelian groups are computed.</p></div></div>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50481829","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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