Pub Date : 2022-05-06DOI: 10.1142/s0218196722500436
N. Martins-Ferreira, A. Montoli, A. Patchkoria, M. Sobral
We define the product of admissible abstract kernels of the form [Formula: see text], where [Formula: see text] is a monoid, [Formula: see text] is a group and [Formula: see text] is a monoid homomorphism. Identifying [Formula: see text]-equivalent abstract kernels, where [Formula: see text] is the center of [Formula: see text], we obtain that the set [Formula: see text] of [Formula: see text]-equivalence classes of admissible abstract kernels inducing the same action of [Formula: see text] on [Formula: see text] is a commutative monoid. Considering the submonoid [Formula: see text] of abstract kernels that are induced by special Schreier extensions, we prove that the factor monoid [Formula: see text] is an abelian group. Moreover, we show that this abelian group is isomorphic to the third cohomology group [Formula: see text].
{"title":"The third cohomology group of a monoid and admissible abstract kernels","authors":"N. Martins-Ferreira, A. Montoli, A. Patchkoria, M. Sobral","doi":"10.1142/s0218196722500436","DOIUrl":"https://doi.org/10.1142/s0218196722500436","url":null,"abstract":"We define the product of admissible abstract kernels of the form [Formula: see text], where [Formula: see text] is a monoid, [Formula: see text] is a group and [Formula: see text] is a monoid homomorphism. Identifying [Formula: see text]-equivalent abstract kernels, where [Formula: see text] is the center of [Formula: see text], we obtain that the set [Formula: see text] of [Formula: see text]-equivalence classes of admissible abstract kernels inducing the same action of [Formula: see text] on [Formula: see text] is a commutative monoid. Considering the submonoid [Formula: see text] of abstract kernels that are induced by special Schreier extensions, we prove that the factor monoid [Formula: see text] is an abelian group. Moreover, we show that this abelian group is isomorphic to the third cohomology group [Formula: see text].","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81750864","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}
Pub Date : 2022-05-04DOI: 10.1142/s0218196722500485
Trevor Nakamura
{"title":"The cohomology class of the mod 4 braid group","authors":"Trevor Nakamura","doi":"10.1142/s0218196722500485","DOIUrl":"https://doi.org/10.1142/s0218196722500485","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86980423","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}
Pub Date : 2022-04-30DOI: 10.1142/s021819672250045x
N. Nikolov
We prove that torsion in the abelianizations of open normal subgroups in finitely presented pro-[Formula: see text] groups can grow arbitrarily fast. By way of contrast in [Formula: see text]-adic analytic groups the torsion growth is at most polynomial.
{"title":"Homology torsion growth of finitely presented pro-p groups","authors":"N. Nikolov","doi":"10.1142/s021819672250045x","DOIUrl":"https://doi.org/10.1142/s021819672250045x","url":null,"abstract":"We prove that torsion in the abelianizations of open normal subgroups in finitely presented pro-[Formula: see text] groups can grow arbitrarily fast. By way of contrast in [Formula: see text]-adic analytic groups the torsion growth is at most polynomial.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72596596","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}
Pub Date : 2022-04-21DOI: 10.1142/s0218196722500709
A. Tuganbaev
In recent years, centrally essential rings have been intensively studied in ring theory. In particular, they find applications in homological algebra, group rings, and the structural theory of rings. The class of essentially central rings strongly extends the class of commutative rings. For such rings, a number of recent papers contain positive answers to some important questions from ring theory that previously had positive answers for commutative rings and negative answers in the general case. This work is devoted to a similar topic. A familiar description of right Noetherian, right distributive centrally essential rings is generalized on a larger class of rings. Let [Formula: see text] be a ring with prime radical [Formula: see text]. It is proved that [Formula: see text] is a right distributive, right invariant centrally essential ring and [Formula: see text] is a finitely generated right ideal such that the factor-ring [Formula: see text] does non contain an infinite direct sum of nonzero ideals if and only if [Formula: see text], where every ring [Formula: see text] is either a commutative Prüfer domain or an Artinian uniserial ring. The studies of Tuganbaev are supported by Russian scientific foundation project 22-11-00052.
{"title":"Distributive invariant centrally essential rings","authors":"A. Tuganbaev","doi":"10.1142/s0218196722500709","DOIUrl":"https://doi.org/10.1142/s0218196722500709","url":null,"abstract":"In recent years, centrally essential rings have been intensively studied in ring theory. In particular, they find applications in homological algebra, group rings, and the structural theory of rings. The class of essentially central rings strongly extends the class of commutative rings. For such rings, a number of recent papers contain positive answers to some important questions from ring theory that previously had positive answers for commutative rings and negative answers in the general case. This work is devoted to a similar topic. A familiar description of right Noetherian, right distributive centrally essential rings is generalized on a larger class of rings. Let [Formula: see text] be a ring with prime radical [Formula: see text]. It is proved that [Formula: see text] is a right distributive, right invariant centrally essential ring and [Formula: see text] is a finitely generated right ideal such that the factor-ring [Formula: see text] does non contain an infinite direct sum of nonzero ideals if and only if [Formula: see text], where every ring [Formula: see text] is either a commutative Prüfer domain or an Artinian uniserial ring. The studies of Tuganbaev are supported by Russian scientific foundation project 22-11-00052.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87785152","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}
Pub Date : 2022-04-18DOI: 10.1142/s0218196722500382
A. Myasnikov, A. Weiss
Recently, Macdonald et al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in [Formula: see text]. Here, we follow their approach and show that all these problems are complete for the uniform circuit class [Formula: see text] — even if an [Formula: see text]-generated nilpotent group of class at most [Formula: see text] is part of the input but [Formula: see text] and [Formula: see text] are fixed constants. In particular, unary encoded systems of a bounded number of linear equations over the integers can be solved in [Formula: see text]. In order to solve these problems in [Formula: see text], we show that the unary version of the extended gcd problem (compute greatest common divisors and express them as linear combinations) is in [Formula: see text]. Moreover, if we allow a certain binary representation of the inputs, then the word problem and computation of normal forms is still in uniform [Formula: see text], while all the other problems we examine are shown to be [Formula: see text]-Turing-reducible to the binary extended gcd problem.
{"title":"Parallel complexity for nilpotent groups","authors":"A. Myasnikov, A. Weiss","doi":"10.1142/s0218196722500382","DOIUrl":"https://doi.org/10.1142/s0218196722500382","url":null,"abstract":"Recently, Macdonald et al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in [Formula: see text]. Here, we follow their approach and show that all these problems are complete for the uniform circuit class [Formula: see text] — even if an [Formula: see text]-generated nilpotent group of class at most [Formula: see text] is part of the input but [Formula: see text] and [Formula: see text] are fixed constants. In particular, unary encoded systems of a bounded number of linear equations over the integers can be solved in [Formula: see text]. In order to solve these problems in [Formula: see text], we show that the unary version of the extended gcd problem (compute greatest common divisors and express them as linear combinations) is in [Formula: see text]. Moreover, if we allow a certain binary representation of the inputs, then the word problem and computation of normal forms is still in uniform [Formula: see text], while all the other problems we examine are shown to be [Formula: see text]-Turing-reducible to the binary extended gcd problem.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90643327","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}
Pub Date : 2022-03-28DOI: 10.1142/s021819672250031x
P. Jipsen, J. B. Nation
A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition [Formula: see text]. For example, if every finite subdirectly irreducible lattice in a locally finite variety [Formula: see text] satisfies Whitman’s condition [Formula: see text], then [Formula: see text] is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are [Formula: see text] such varieties. Some lattices that fail [Formula: see text] also generate primitive varieties. But if [Formula: see text] is a [Formula: see text]-failure interval in a finite subdirectly irreducible lattice [Formula: see text], and [Formula: see text] denotes the lattice with [Formula: see text] doubled, then [Formula: see text] is never primitive.
{"title":"Primitive lattice varieties","authors":"P. Jipsen, J. B. Nation","doi":"10.1142/s021819672250031x","DOIUrl":"https://doi.org/10.1142/s021819672250031x","url":null,"abstract":"A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition [Formula: see text]. For example, if every finite subdirectly irreducible lattice in a locally finite variety [Formula: see text] satisfies Whitman’s condition [Formula: see text], then [Formula: see text] is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are [Formula: see text] such varieties. Some lattices that fail [Formula: see text] also generate primitive varieties. But if [Formula: see text] is a [Formula: see text]-failure interval in a finite subdirectly irreducible lattice [Formula: see text], and [Formula: see text] denotes the lattice with [Formula: see text] doubled, then [Formula: see text] is never primitive.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75975564","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}
Pub Date : 2022-03-17DOI: 10.1142/s0218196722500680
E. Aichinger, Nebojvsa Mudrinski
Several structural properties of a universal algebra can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In the present paper, we exhibit finite representations of this sequence.
{"title":"Finite representation of commutator sequences","authors":"E. Aichinger, Nebojvsa Mudrinski","doi":"10.1142/s0218196722500680","DOIUrl":"https://doi.org/10.1142/s0218196722500680","url":null,"abstract":"Several structural properties of a universal algebra can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In the present paper, we exhibit finite representations of this sequence.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75654191","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":"Computable paradoxical decompositions","authors":"Karol Duda, A. Ivanov","doi":"10.1142/s0218196722500400","DOIUrl":"https://doi.org/10.1142/s0218196722500400","url":null,"abstract":"We prove a computable version of Hall’s Harem theorem and apply it to computable versions of Tarski’s alternative theorem.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75459856","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}
Pub Date : 2022-03-14DOI: 10.1142/s0218196722500357
Luca Amata, M. Crupi, A. Ficarra
In this paper, we study some algebraic invariants of [Formula: see text]-spread ideals, [Formula: see text], such as the projective dimension and the Castelnuovo–Mumford regularity, by means of well-known graded resolutions. We state upper bounds for these invariants and, furthermore, we identify a special class of [Formula: see text]-spread ideals for which such bounds are optimal.
{"title":"Projective dimension and Castelnuovo-Mumford regularity of t-spread ideals","authors":"Luca Amata, M. Crupi, A. Ficarra","doi":"10.1142/s0218196722500357","DOIUrl":"https://doi.org/10.1142/s0218196722500357","url":null,"abstract":"In this paper, we study some algebraic invariants of [Formula: see text]-spread ideals, [Formula: see text], such as the projective dimension and the Castelnuovo–Mumford regularity, by means of well-known graded resolutions. We state upper bounds for these invariants and, furthermore, we identify a special class of [Formula: see text]-spread ideals for which such bounds are optimal.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87013515","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}
Pub Date : 2022-03-07DOI: 10.1142/s0218196722500564
A. Guterman, D. Kudryavtsev
We investigate the class of finite dimensional not necessary associative algebras that have slowly growing length, that is, for any algebra in this class its length is less than or equal to its dimension. We show that this class is considerably big, in particular, finite dimensional Lie algebras as well as many other important classical finite dimensional algebras belong to this class, for example, Leibniz algebras, Novikov algebras, and Zinbiel algebras. An exact upper bounds for the length of these algebras is proved. To do this we transfer the method of characteristic sequences to non-unital algebras and find certain polynomial conditions on the algebra elements that guarantee the slow growth of the length function. MSC: 15A03,17A99,15A78
{"title":"Algebras of slowly growing length","authors":"A. Guterman, D. Kudryavtsev","doi":"10.1142/s0218196722500564","DOIUrl":"https://doi.org/10.1142/s0218196722500564","url":null,"abstract":"We investigate the class of finite dimensional not necessary associative algebras that have slowly growing length, that is, for any algebra in this class its length is less than or equal to its dimension. We show that this class is considerably big, in particular, finite dimensional Lie algebras as well as many other important classical finite dimensional algebras belong to this class, for example, Leibniz algebras, Novikov algebras, and Zinbiel algebras. An exact upper bounds for the length of these algebras is proved. To do this we transfer the method of characteristic sequences to non-unital algebras and find certain polynomial conditions on the algebra elements that guarantee the slow growth of the length function. MSC: 15A03,17A99,15A78","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72876507","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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