Pub Date : 2022-01-29DOI: 10.1142/s0218196722500370
Jacob Bamberger, D. Wise
The main result in this paper is the failure of the finitely generated intersection property (FGIP) of ascending HNN extensions of non-cyclic finite rank free groups. This class of groups consists of free-by-cyclic groups and properly ascending HNN extensions of free groups. We also give a sufficient condition for the failure of the FGIP in the context of relative hyperbolicity, we apply this to free-by-cyclic groups of exponential growth.
{"title":"Failure of the finitely generated intersection property for ascending HNN extensions of free groups","authors":"Jacob Bamberger, D. Wise","doi":"10.1142/s0218196722500370","DOIUrl":"https://doi.org/10.1142/s0218196722500370","url":null,"abstract":"The main result in this paper is the failure of the finitely generated intersection property (FGIP) of ascending HNN extensions of non-cyclic finite rank free groups. This class of groups consists of free-by-cyclic groups and properly ascending HNN extensions of free groups. We also give a sufficient condition for the failure of the FGIP in the context of relative hyperbolicity, we apply this to free-by-cyclic groups of exponential growth.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77557265","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 : 2022-01-28DOI: 10.1142/s0218196722500138
Sh. A. Ayupov, A. Khudoyberdiyev, Z. Shermatova
This paper is devoted to the so-called complete Leibniz algebras. We construct some complete Leibniz algebras with complete radical and prove that the direct sum of complete Leibniz algebras is also complete. It is known that a Lie algebra with a complete ideal is split. We discuss the analogs of this result for the Leibniz algebras and show that it is true for some special classes of Leibniz algebras. Finally, we consider derivations of Leibniz algebras and present some classes of Leibniz algebras which are not complete, since they admit outer derivation.
{"title":"On complete Leibniz algebras","authors":"Sh. A. Ayupov, A. Khudoyberdiyev, Z. Shermatova","doi":"10.1142/s0218196722500138","DOIUrl":"https://doi.org/10.1142/s0218196722500138","url":null,"abstract":"This paper is devoted to the so-called complete Leibniz algebras. We construct some complete Leibniz algebras with complete radical and prove that the direct sum of complete Leibniz algebras is also complete. It is known that a Lie algebra with a complete ideal is split. We discuss the analogs of this result for the Leibniz algebras and show that it is true for some special classes of Leibniz algebras. Finally, we consider derivations of Leibniz algebras and present some classes of Leibniz algebras which are not complete, since they admit outer derivation.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74805081","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 : 2022-01-04DOI: 10.1142/s021819672350011x
Yanan Jiang, Bangzheng Li, So-Fan Zhu
A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number $alpha$, the additive monoid $M_alpha$ of the evaluation semiring $mathbb{N}_0[alpha]$ is atomic. The atomic structure of both the additive and the multiplicative monoids of $mathbb{N}_0[alpha]$ has been the subject of several recent papers. Here we focus on the monoids $M_alpha$, and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when $alpha$ is less than 1, the atoms of $M_alpha$ are as far from being prime as they can possibly be. Then we establish some results about the elasticity of $M_alpha$, including that when $alpha$ is rational, the elasticity of $M_alpha$ is full (this was previously conjectured by S. T. Chapman, F. Gotti, and M. Gotti).
如果每个不可逆的元素因子化为不可约的，则可消交换单群是原子的。在一定温和条件下，求值半环$mathbb{N}_0[alpha]$的加性单群$M_alpha$是原子的。$mathbb{N}_0[alpha]$的加性和乘性单群的原子结构是最近几篇论文的主题。本文主要研究一元群$M_ α $，并研究其ω -原数和弹性，旨在更好地理解它们的原子分解的一些基本问题。我们证明了当$ α $小于1时，$M_ α $的原子离素数的距离是尽可能远的。然后我们建立了关于$M_alpha$弹性的一些结果，包括当$alpha$是有理时，$M_alpha$的弹性是满的(这是S. T. Chapman, F. Gotti和M. Gotti先前推测的)。
{"title":"On the primality and elasticity of algebraic valuations of cyclic free semirings","authors":"Yanan Jiang, Bangzheng Li, So-Fan Zhu","doi":"10.1142/s021819672350011x","DOIUrl":"https://doi.org/10.1142/s021819672350011x","url":null,"abstract":"A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number $alpha$, the additive monoid $M_alpha$ of the evaluation semiring $mathbb{N}_0[alpha]$ is atomic. The atomic structure of both the additive and the multiplicative monoids of $mathbb{N}_0[alpha]$ has been the subject of several recent papers. Here we focus on the monoids $M_alpha$, and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when $alpha$ is less than 1, the atoms of $M_alpha$ are as far from being prime as they can possibly be. Then we establish some results about the elasticity of $M_alpha$, including that when $alpha$ is rational, the elasticity of $M_alpha$ is full (this was previously conjectured by S. T. Chapman, F. Gotti, and M. Gotti).","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77102125","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 : 2022-01-01DOI: 10.1142/S0218196722500175
K. Adaricheva, R. Freese, J. B. Nation
{"title":"Notes on join semidistributive lattices","authors":"K. Adaricheva, R. Freese, J. B. Nation","doi":"10.1142/S0218196722500175","DOIUrl":"https://doi.org/10.1142/S0218196722500175","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77201563","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 : 2022-01-01DOI: 10.1142/S0218196722500461
T. Aird
{"title":"Identities of tropical matrix semigroups and the plactic monoid of rank 4","authors":"T. Aird","doi":"10.1142/S0218196722500461","DOIUrl":"https://doi.org/10.1142/S0218196722500461","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75326337","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 : 2022-01-01DOI: 10.1142/S0218196722500515
M. Kepczyk
{"title":"On rings which are sums of subrings and additive subgroups","authors":"M. Kepczyk","doi":"10.1142/S0218196722500515","DOIUrl":"https://doi.org/10.1142/S0218196722500515","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73940827","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 : 2021-12-21DOI: 10.1142/s0218196723500388
Lu'is Oliveira
A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idempotents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation $langle G(X),rho_ecuprho_srangle$ and its structure is studied. Although each of the sets $G(X)$, $rho_e$, and $rho_s$ is infinite for $|X|geq 2$, we show that the word problem is decidable as each congruence class has a canonical form. If $FI_n$ denotes FI(X) for $|X|=n$, we prove also that $FI_2$ contains copies of all $FI_n$ as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide $FI_2$; and (ii) all finite semigroups divide $FI_2$.
{"title":"Regular semigroups weakly generated by idempotents","authors":"Lu'is Oliveira","doi":"10.1142/s0218196723500388","DOIUrl":"https://doi.org/10.1142/s0218196723500388","url":null,"abstract":"A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroups containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI(X) weakly generated by |X| idempotents such that all other regular semigroups weakly generated by |X| idempotents are homomorphic images of FI(X). The semigroup FI(X) is defined by a presentation $langle G(X),rho_ecuprho_srangle$ and its structure is studied. Although each of the sets $G(X)$, $rho_e$, and $rho_s$ is infinite for $|X|geq 2$, we show that the word problem is decidable as each congruence class has a canonical form. If $FI_n$ denotes FI(X) for $|X|=n$, we prove also that $FI_2$ contains copies of all $FI_n$ as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide $FI_2$; and (ii) all finite semigroups divide $FI_2$.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82238860","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 : 2021-12-18DOI: 10.1142/s0218196722500667
V. Guba
In this paper we introduce a concept of an evacuation scheme on the Cayley graph of an infinite finitely generated group. This is a collection of infinite simple paths bringing all vertices to infinity. We impose a restriction that every edge can be used a uniformly bounded number of times in this scheme. An easy observation shows that existing of such a scheme is equivalent to non-amenability of the group. A special case happens if every edge can be used only once. These scheme are called pure. We obtain a criterion for existing of such a scheme in terms of isoperimetric constant of the graph. We analyze R.,Thompson's group $F$, for which the amenability property is a famous open problem. We show that pure evacuation schemes do not exist for the set of generators ${x_0,x_1,bar{x}_1}$, where $bar{x}_1=x_1x_0^{-1}$. However, the question becomes open if edges with labels $x_0^{pm1}$ can be used twice. Existing of pure evacuation scheme for this version is implied by some natural conjectures.
{"title":"Evacuation schemes on Cayley graphs and non-amenability of groups","authors":"V. Guba","doi":"10.1142/s0218196722500667","DOIUrl":"https://doi.org/10.1142/s0218196722500667","url":null,"abstract":"In this paper we introduce a concept of an evacuation scheme on the Cayley graph of an infinite finitely generated group. This is a collection of infinite simple paths bringing all vertices to infinity. We impose a restriction that every edge can be used a uniformly bounded number of times in this scheme. An easy observation shows that existing of such a scheme is equivalent to non-amenability of the group. A special case happens if every edge can be used only once. These scheme are called pure. We obtain a criterion for existing of such a scheme in terms of isoperimetric constant of the graph. We analyze R.,Thompson's group $F$, for which the amenability property is a famous open problem. We show that pure evacuation schemes do not exist for the set of generators ${x_0,x_1,bar{x}_1}$, where $bar{x}_1=x_1x_0^{-1}$. However, the question becomes open if edges with labels $x_0^{pm1}$ can be used twice. Existing of pure evacuation scheme for this version is implied by some natural conjectures.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89589854","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 : 2021-11-30DOI: 10.1142/s0218196722500618
Carl-Fredrik Nyberg Brodda
This note investigates and clarifies some connections between the theory of one-relator groups and special one-relation inverse monoids, i.e. those inverse monoids with a presentation of the form $operatorname{Inv}langle A mid w=1 rangle$. We show that every one-relator group admits a special one-relation inverse monoid presentation. We subsequently consider the classes ${rm {small ANY}}, {rm {small RED}}, {rm {small CRED}},$ and ${rm {small POS}}$ of one-relator groups which can be defined by special one-relation inverse monoid presentations in which the defining word is arbitrary; reduced; cyclically reduced; or positive, respectively. We show that the inclusions ${rm {small ANY}} supset {rm {small CRED}} supset {rm {small POS}}$ are all strict. Conditional on a natural conjecture, we prove ${rm {small ANY}} supset {rm {small RED}}$. Following this, we use the Benois algorithm recently devised by Gray&Ruskuc to produce an infinite family of special one-relation inverse monoids which exhibit similar pathological behaviour (which we term O'Haresque) to the O'Hare monoid with respect to computing the minimal invertible pieces of the defining word. Finally, we provide a counterexample to a conjecture by Gray&Ruskuc that the Benois algorithm always correctly computes the minimal invertible pieces of a special one-relation inverse monoid.
{"title":"On one-relator groups and units of special one-relation inverse monoids","authors":"Carl-Fredrik Nyberg Brodda","doi":"10.1142/s0218196722500618","DOIUrl":"https://doi.org/10.1142/s0218196722500618","url":null,"abstract":"This note investigates and clarifies some connections between the theory of one-relator groups and special one-relation inverse monoids, i.e. those inverse monoids with a presentation of the form $operatorname{Inv}langle A mid w=1 rangle$. We show that every one-relator group admits a special one-relation inverse monoid presentation. We subsequently consider the classes ${rm {small ANY}}, {rm {small RED}}, {rm {small CRED}},$ and ${rm {small POS}}$ of one-relator groups which can be defined by special one-relation inverse monoid presentations in which the defining word is arbitrary; reduced; cyclically reduced; or positive, respectively. We show that the inclusions ${rm {small ANY}} supset {rm {small CRED}} supset {rm {small POS}}$ are all strict. Conditional on a natural conjecture, we prove ${rm {small ANY}} supset {rm {small RED}}$. Following this, we use the Benois algorithm recently devised by Gray&Ruskuc to produce an infinite family of special one-relation inverse monoids which exhibit similar pathological behaviour (which we term O'Haresque) to the O'Hare monoid with respect to computing the minimal invertible pieces of the defining word. Finally, we provide a counterexample to a conjecture by Gray&Ruskuc that the Benois algorithm always correctly computes the minimal invertible pieces of a special one-relation inverse monoid.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82870868","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 : 2021-11-26DOI: 10.1142/s0218196723500224
I. F. Guimaraes
Birget, Margolis, Meakin and Weil proved that a finitely generated subgroup $K$ of a free group is pure if and only if the transition monoid $M(K)$ of its Stallings automaton is aperiodic. In this paper, we establish further connections between algebraic properties of $K$ and algebraic properties of $M(K)$. We mainly focus on the cases where $M(K)$ belongs to the pseudovariety $overline{boldsymbol{mathbf{{H}}}}$ of finite monoids all of whose subgroups lie in a given pseudovariety $overline{boldsymbol{mathbf{{H}}}}$ of finite groups. We also discuss normal, malnormal and cyclonormal subgroups of $F_A$ using the transition monoid of the corresponding Stallings automaton.
{"title":"On the transition monoid of the Stallings automaton of a subgroup of a free group","authors":"I. F. Guimaraes","doi":"10.1142/s0218196723500224","DOIUrl":"https://doi.org/10.1142/s0218196723500224","url":null,"abstract":"Birget, Margolis, Meakin and Weil proved that a finitely generated subgroup $K$ of a free group is pure if and only if the transition monoid $M(K)$ of its Stallings automaton is aperiodic. In this paper, we establish further connections between algebraic properties of $K$ and algebraic properties of $M(K)$. We mainly focus on the cases where $M(K)$ belongs to the pseudovariety $overline{boldsymbol{mathbf{{H}}}}$ of finite monoids all of whose subgroups lie in a given pseudovariety $overline{boldsymbol{mathbf{{H}}}}$ of finite groups. We also discuss normal, malnormal and cyclonormal subgroups of $F_A$ using the transition monoid of the corresponding Stallings automaton.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74274952","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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