Pub Date : 2021-09-05DOI: 10.1142/s0218196722500527
Alexandr Kazda, M. Kompatscher
For some Maltsev conditions Σ it is enough to check if a finite algebra A satisfies Σ locally on subsets of bounded size, in order to decide, whether A satisfies Σ (globally). This local-global property is the main known source of tractability results for deciding Maltsev conditions. In this paper we investigate the local-global property for the existence of a G-term, i.e. an n-ary term that is invariant under permuting its variables according to a permutation group G ≤ Sym(n). Our results imply in particular that all cyclic loop conditions (in the sense of Bodirsky, Starke, and Vucaj) have the local-global property (and thus can be decided in polynomial time), while symmetric terms of arity n > 2 fail to have it.
{"title":"The local-global property for G-invariant terms","authors":"Alexandr Kazda, M. Kompatscher","doi":"10.1142/s0218196722500527","DOIUrl":"https://doi.org/10.1142/s0218196722500527","url":null,"abstract":"For some Maltsev conditions Σ it is enough to check if a finite algebra A satisfies Σ locally on subsets of bounded size, in order to decide, whether A satisfies Σ (globally). This local-global property is the main known source of tractability results for deciding Maltsev conditions. In this paper we investigate the local-global property for the existence of a G-term, i.e. an n-ary term that is invariant under permuting its variables according to a permutation group G ≤ Sym(n). Our results imply in particular that all cyclic loop conditions (in the sense of Bodirsky, Starke, and Vucaj) have the local-global property (and thus can be decided in polynomial time), while symmetric terms of arity n > 2 fail to have it.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"96 1","pages":"1209-1231"},"PeriodicalIF":0.0,"publicationDate":"2021-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87695158","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 : 2021-09-03DOI: 10.1142/s0218196723500327
M. Bodirsky, A. Vucaj, Dmitriy Zhuk
There are continuum many clones on a three-element set even if they are considered up to emph{homomorphic equivalence}. The clones we use to prove this fact are clones consisting of emph{self-dual operations}, i.e., operations that preserve the relation ${(0,1),(1,2),(2,0)}$. However, there are only countably many such clones when considered up to equivalence with respect to emph{minor-preserving maps} instead of clone homomorphisms. We give a full description of the set of clones of self-dual operations, ordered by the existence of minor-preserving maps. Our result can also be phrased as a statement about structures on a three-element set, ordered by primitive positive constructability, because there is a minor-preserving map from the polymorphism clone of a finite structure $mathfrak A$ to the polymorphism clone of a finite structure $mathfrak B$ if and only if there is a primitive positive construction of $mathfrak B$ in $mathfrak A$.
{"title":"The lattice of clones of self-dual operations collapsed","authors":"M. Bodirsky, A. Vucaj, Dmitriy Zhuk","doi":"10.1142/s0218196723500327","DOIUrl":"https://doi.org/10.1142/s0218196723500327","url":null,"abstract":"There are continuum many clones on a three-element set even if they are considered up to emph{homomorphic equivalence}. The clones we use to prove this fact are clones consisting of emph{self-dual operations}, i.e., operations that preserve the relation ${(0,1),(1,2),(2,0)}$. However, there are only countably many such clones when considered up to equivalence with respect to emph{minor-preserving maps} instead of clone homomorphisms. We give a full description of the set of clones of self-dual operations, ordered by the existence of minor-preserving maps. Our result can also be phrased as a statement about structures on a three-element set, ordered by primitive positive constructability, because there is a minor-preserving map from the polymorphism clone of a finite structure $mathfrak A$ to the polymorphism clone of a finite structure $mathfrak B$ if and only if there is a primitive positive construction of $mathfrak B$ in $mathfrak A$.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"37 1","pages":"717-749"},"PeriodicalIF":0.0,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80950110","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 : 2021-08-08DOI: 10.1142/s0218196723500352
Zhicheng Zhu, Huhu Zhang, Xing Gao
In this paper, we obtain respectively some new linear bases of free unitary (modified) weighted differential algebras and free nonunitary (modified) Rota-Baxter algebras, in terms of the method of Gr"{o}bner-Shirshov bases.
{"title":"Free weighted (modified) differential algebras, free (modified) Rota-Baxter algebras and Gröbner-Shirshov bases","authors":"Zhicheng Zhu, Huhu Zhang, Xing Gao","doi":"10.1142/s0218196723500352","DOIUrl":"https://doi.org/10.1142/s0218196723500352","url":null,"abstract":"In this paper, we obtain respectively some new linear bases of free unitary (modified) weighted differential algebras and free nonunitary (modified) Rota-Baxter algebras, in terms of the method of Gr\"{o}bner-Shirshov bases.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"88 1","pages":"787-817"},"PeriodicalIF":0.0,"publicationDate":"2021-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83062907","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 : 2021-08-05DOI: 10.1142/s0218196723500194
Melissa Lee, Tomasz Popiel
Let $G$ be a permutation group on a set $Omega$. A base for $G$ is a subset of $Omega$ whose pointwise stabiliser is trivial, and the base size of $G$ is the minimal cardinality of a base. If $G$ has base size $2$, then the corresponding Saxl graph $Sigma(G)$ has vertex set $Omega$ and two vertices are adjacent if they form a base for $G$. A recent conjecture of Burness and Giudici states that if $G$ is a finite primitive permutation group with base size $2$, then $Sigma(G)$ has the property that every two vertices have a common neighbour. We investigate this conjecture when $G$ is an affine group and a point stabiliser is an almost quasisimple group whose unique quasisimple subnormal subgroup is a covering group of a sporadic simple group. We verify the conjecture under this assumption, in all but ten cases.
{"title":"Saxl graphs of primitive affine groups with sporadic point stabilizers","authors":"Melissa Lee, Tomasz Popiel","doi":"10.1142/s0218196723500194","DOIUrl":"https://doi.org/10.1142/s0218196723500194","url":null,"abstract":"Let $G$ be a permutation group on a set $Omega$. A base for $G$ is a subset of $Omega$ whose pointwise stabiliser is trivial, and the base size of $G$ is the minimal cardinality of a base. If $G$ has base size $2$, then the corresponding Saxl graph $Sigma(G)$ has vertex set $Omega$ and two vertices are adjacent if they form a base for $G$. A recent conjecture of Burness and Giudici states that if $G$ is a finite primitive permutation group with base size $2$, then $Sigma(G)$ has the property that every two vertices have a common neighbour. We investigate this conjecture when $G$ is an affine group and a point stabiliser is an almost quasisimple group whose unique quasisimple subnormal subgroup is a covering group of a sporadic simple group. We verify the conjecture under this assumption, in all but ten cases.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"54 1","pages":"369-389"},"PeriodicalIF":0.0,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83846612","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 : 2021-07-31DOI: 10.1142/s0218196721500545
G. Fonseca, G. Martini, Leonardo Silva
In this paper, we determine all partial actions and partial coactions of Taft and Nichols Hopf algebras on their base fields. Furthermore, we prove that all such partial (co)actions are symmetric.
{"title":"Partial (co)actions of Taft and Nichols Hopf algebras on their base fields","authors":"G. Fonseca, G. Martini, Leonardo Silva","doi":"10.1142/s0218196721500545","DOIUrl":"https://doi.org/10.1142/s0218196721500545","url":null,"abstract":"In this paper, we determine all partial actions and partial coactions of Taft and Nichols Hopf algebras on their base fields. Furthermore, we prove that all such partial (co)actions are symmetric.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"140 1","pages":"1471-1496"},"PeriodicalIF":0.0,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80008490","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 : 2021-07-26DOI: 10.1142/s0218196721500569
Z. B'acskai, D. Flannery, E. O'Brien
Let [Formula: see text] be a prime and let [Formula: see text] be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of [Formula: see text] up to conjugacy. That is, we give a complete and irredundant list of [Formula: see text]-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of [Formula: see text] is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For [Formula: see text], we classify all finite irreducible subgroups of [Formula: see text]. Our classifications are available publicly in Magma.
{"title":"Classifying finite monomial linear groups of prime degree in characteristic zero","authors":"Z. B'acskai, D. Flannery, E. O'Brien","doi":"10.1142/s0218196721500569","DOIUrl":"https://doi.org/10.1142/s0218196721500569","url":null,"abstract":"Let [Formula: see text] be a prime and let [Formula: see text] be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of [Formula: see text] up to conjugacy. That is, we give a complete and irredundant list of [Formula: see text]-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of [Formula: see text] is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For [Formula: see text], we classify all finite irreducible subgroups of [Formula: see text]. Our classifications are available publicly in Magma.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"16 1","pages":"1547-1585"},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74750951","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 : 2021-07-16DOI: 10.1142/s0218196722500084
M. Khrypchenko
Let [Formula: see text] and [Formula: see text] be finite posets and [Formula: see text] a commutative unital ring. In the case where [Formula: see text] is indecomposable, we prove that the [Formula: see text]-linear isomorphisms between partial flag incidence algebras [Formula: see text] and [Formula: see text] are exactly those induced by poset isomorphisms between [Formula: see text] and [Formula: see text]. We also show that the [Formula: see text]-linear derivations of [Formula: see text] are trivial.
{"title":"Isomorphisms and derivations of partial flag incidence algebras","authors":"M. Khrypchenko","doi":"10.1142/s0218196722500084","DOIUrl":"https://doi.org/10.1142/s0218196722500084","url":null,"abstract":"Let [Formula: see text] and [Formula: see text] be finite posets and [Formula: see text] a commutative unital ring. In the case where [Formula: see text] is indecomposable, we prove that the [Formula: see text]-linear isomorphisms between partial flag incidence algebras [Formula: see text] and [Formula: see text] are exactly those induced by poset isomorphisms between [Formula: see text] and [Formula: see text]. We also show that the [Formula: see text]-linear derivations of [Formula: see text] are trivial.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"05 1","pages":"193-209"},"PeriodicalIF":0.0,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85973287","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 : 2021-07-09DOI: 10.1142/s0218196722500151
Jennifer Biermann, Selvi Kara, Kuei-Nuan Lin, Augustine O’Keefe
Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph.
{"title":"Toric ideals of weighted oriented graphs","authors":"Jennifer Biermann, Selvi Kara, Kuei-Nuan Lin, Augustine O’Keefe","doi":"10.1142/s0218196722500151","DOIUrl":"https://doi.org/10.1142/s0218196722500151","url":null,"abstract":"Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"108 1","pages":"307-325"},"PeriodicalIF":0.0,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87584032","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 : 2021-07-05DOI: 10.1142/S0218196721500508
Qinghong Guo, Xuanli He, Muhong Huang
Let [Formula: see text] be a finite group. How minimal subgroups can be embedded in [Formula: see text] is a question of particular interest in studying the structure of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] for all Sylow subgroups [Formula: see text] of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] with [Formula: see text]-embedded subgroups.
设[公式:见文本]是一个有限群。如何在[公式:见文本]中嵌入最小的子组是研究[公式:见文本]结构时特别感兴趣的问题。如果[Formula: see text]的所有Sylow子组[Formula: see text]的[Formula: see text]都存在[Formula: see text]，则[Formula: see text]中的子组[Formula: see text]称为[Formula: see text]-在[Formula: see text]中是可变的。[公式:见文]的子群[公式:见文]被称为[公式:见文]-嵌入在[公式:见文]中，如果存在[公式:见文]的正常子群[公式:见文]使得[公式:见文]和[公式:见文]，其中[公式:见文]是[公式:见文]的所有子群生成的[公式:见文]的子群[公式:见文]-在[公式:见文]中是可变的[公式:见文]。本文研究了嵌入子群的有限群[公式:见文]的结构。
{"title":"Finite groups with n-embedded subgroups","authors":"Qinghong Guo, Xuanli He, Muhong Huang","doi":"10.1142/S0218196721500508","DOIUrl":"https://doi.org/10.1142/S0218196721500508","url":null,"abstract":"Let [Formula: see text] be a finite group. How minimal subgroups can be embedded in [Formula: see text] is a question of particular interest in studying the structure of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] for all Sylow subgroups [Formula: see text] of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] with [Formula: see text]-embedded subgroups.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"66 1","pages":"1419-1428"},"PeriodicalIF":0.0,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79598816","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 : 2021-06-28DOI: 10.1142/S0218196721500521
Şehmus Fındık, O. Kelekci
Let [Formula: see text] and [Formula: see text] be two generic traceless matrices of size [Formula: see text] with entries from a commutative associative polynomial algebra over a field [Formula: see text] of characteristic zero. Consider the associative unitary algebra [Formula: see text], and its Lie subalgebra [Formula: see text] generated by [Formula: see text] and [Formula: see text] over the field [Formula: see text]. It is well known that the center [Formula: see text] of [Formula: see text] is the polynomial algebra generated by the algebraically independent commuting elements [Formula: see text], [Formula: see text], [Formula: see text]. We call a polynomial [Formula: see text] symmetric, if [Formula: see text]. The set of symmetric polynomials is equal to the algebra [Formula: see text] of invariants of symmetric group [Formula: see text]. Similarly, we define the Lie algebra [Formula: see text] of symmetric polynomials in the Lie algebra [Formula: see text]. We give the description of the algebras [Formula: see text] and [Formula: see text], and we provide finite sets of free generators for [Formula: see text], and [Formula: see text] as [Formula: see text]-modules.
{"title":"Symmetric polynomials of algebras related with 2 × 2 generic traceless matrices","authors":"Şehmus Fındık, O. Kelekci","doi":"10.1142/S0218196721500521","DOIUrl":"https://doi.org/10.1142/S0218196721500521","url":null,"abstract":"Let [Formula: see text] and [Formula: see text] be two generic traceless matrices of size [Formula: see text] with entries from a commutative associative polynomial algebra over a field [Formula: see text] of characteristic zero. Consider the associative unitary algebra [Formula: see text], and its Lie subalgebra [Formula: see text] generated by [Formula: see text] and [Formula: see text] over the field [Formula: see text]. It is well known that the center [Formula: see text] of [Formula: see text] is the polynomial algebra generated by the algebraically independent commuting elements [Formula: see text], [Formula: see text], [Formula: see text]. We call a polynomial [Formula: see text] symmetric, if [Formula: see text]. The set of symmetric polynomials is equal to the algebra [Formula: see text] of invariants of symmetric group [Formula: see text]. Similarly, we define the Lie algebra [Formula: see text] of symmetric polynomials in the Lie algebra [Formula: see text]. We give the description of the algebras [Formula: see text] and [Formula: see text], and we provide finite sets of free generators for [Formula: see text], and [Formula: see text] as [Formula: see text]-modules.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"83 1","pages":"1433-1442"},"PeriodicalIF":0.0,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89435967","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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