Pub Date : 2021-06-21DOI: 10.1142/s0218196723500236
F. M. Abdelmalek, Priyavrat Deshpande, Shuchita Goyal, A. Roy, Anurag Singh
Given a simple undirected graph $G$ there is a simplicial complex $mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. One of the line of research pursued by many authors is to determine the graph classes for which the associated independence complex is Cohen-Macaulay. For example, it is known that when $G$ is a chordal graph the complex $mathrm{Ind}(G)$ is in fact vertex decomposable, the strongest condition in the Cohen-Macaulay ladder. In this article we consider a generalization of independence complex. Given $rgeq 1$, a subset of the vertex set is called $r$-independent if the connected components of the induced subgraph have cardinality at most $r$. The collection of all $r$-independent subsets of $G$ form a simplicial complex called the $r$-independence complex and is denoted by $mathrm{Ind}_r(G)$. It is known that when $G$ is a chordal graph the complex $mathrm{Ind}_r(G)$ has the homotopy type of a wedge of spheres. Hence it is natural to ask which of these complexes are shellable or even vertex decomposable. We prove, using Woodroofe's chordal hypergraph notion, that these complexes are always shellable when the underlying chordal graph is a tree. Further, using the notion of vertex splittable ideals we show that for caterpillar graphs the associated $r$-independence complex is vertex decomposable for all values of $r$. We also construct chordal graphs on $2r+2$ vertices such that their $r$-independence complexes are not sequentially Cohen-Macaulay for any $r ge 2$.
给定一个简单无向图$G$，存在一个简单复形$mathrm{Ind}(G)$，称为独立复形，其面对应于$G$的独立集。这是一个研究得很好的概念，因为它为交换代数、图论和代数拓扑之间的相互作用提供了肥沃的土壤。许多作者所追求的研究方向之一是确定相关独立复合体为Cohen-Macaulay的图类。例如，已知当$G$是弦图时，复体$mathrm{Ind}(G)$实际上是顶点可分解的，这是Cohen-Macaulay阶梯中最强的条件。在本文中，我们考虑独立复合体的推广。给定$rgeq 1$，如果诱导子图的连通分量的基数不超过$r$，则顶点集的子集称为$r$独立的。所有$G$的$r$独立子集的集合形成一个简单复合体，称为$r$独立复合体，用$mathrm{Ind}_r(G)$表示。我们知道，当$G$是弦图时，复$mathrm{Ind}_r(G)$具有球楔的同伦类型。因此，很自然地要问这些复合体中哪些是可壳化的，甚至是顶点可分解的。利用Woodroofe的弦超图概念，证明了当底层弦图为树时，这些复合体总是可壳的。进一步，利用顶点可分理想的概念，我们证明了对于毛虫图，相关的$r$无关复数对于$r$的所有值都是顶点可分解的。我们还构造了$2r+2$顶点上的弦图，使得它们的$r$无关复合体对于任何$r ge 2$都不是连续的Cohen-Macaulay。
{"title":"Chordal graphs, higher independence and vertex decomposable complexes","authors":"F. M. Abdelmalek, Priyavrat Deshpande, Shuchita Goyal, A. Roy, Anurag Singh","doi":"10.1142/s0218196723500236","DOIUrl":"https://doi.org/10.1142/s0218196723500236","url":null,"abstract":"Given a simple undirected graph $G$ there is a simplicial complex $mathrm{Ind}(G)$, called the independence complex, whose faces correspond to the independent sets of $G$. This is a well studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. One of the line of research pursued by many authors is to determine the graph classes for which the associated independence complex is Cohen-Macaulay. For example, it is known that when $G$ is a chordal graph the complex $mathrm{Ind}(G)$ is in fact vertex decomposable, the strongest condition in the Cohen-Macaulay ladder. In this article we consider a generalization of independence complex. Given $rgeq 1$, a subset of the vertex set is called $r$-independent if the connected components of the induced subgraph have cardinality at most $r$. The collection of all $r$-independent subsets of $G$ form a simplicial complex called the $r$-independence complex and is denoted by $mathrm{Ind}_r(G)$. It is known that when $G$ is a chordal graph the complex $mathrm{Ind}_r(G)$ has the homotopy type of a wedge of spheres. Hence it is natural to ask which of these complexes are shellable or even vertex decomposable. We prove, using Woodroofe's chordal hypergraph notion, that these complexes are always shellable when the underlying chordal graph is a tree. Further, using the notion of vertex splittable ideals we show that for caterpillar graphs the associated $r$-independence complex is vertex decomposable for all values of $r$. We also construct chordal graphs on $2r+2$ vertices such that their $r$-independence complexes are not sequentially Cohen-Macaulay for any $r ge 2$.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79362761","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 : 2021-06-09DOI: 10.1142/S021819672102001X
A. Frigeri, V. Gould, P. Longobardi, E. Rodaro
{"title":"Preface: Special issue on papers from the conference \"Semigroups and Groups, Automata, Logics\" SandGAL 2019","authors":"A. Frigeri, V. Gould, P. Longobardi, E. Rodaro","doi":"10.1142/S021819672102001X","DOIUrl":"https://doi.org/10.1142/S021819672102001X","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87948161","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 : 2021-06-09DOI: 10.1142/S0218196721400026
Timothy J. Koussas
We determine all quasivarieties of aperiodic semigroups that are contained in some residually finite variety. This endeavor was initially motivated by a problem in natural dualities, but our work here also serves as a partial correction to an error found in a result of Sapir from the 1980s.
{"title":"Semigroup quasivarieties: Two lattices and a reopened problem","authors":"Timothy J. Koussas","doi":"10.1142/S0218196721400026","DOIUrl":"https://doi.org/10.1142/S0218196721400026","url":null,"abstract":"We determine all quasivarieties of aperiodic semigroups that are contained in some residually finite variety. This endeavor was initially motivated by a problem in natural dualities, but our work here also serves as a partial correction to an error found in a result of Sapir from the 1980s.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80554999","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 : 2021-06-01DOI: 10.1142/S0218196721500326
S. Zhilina
We study zero divisors whose components alternate strongly pairwise and construct oriented hexagons in the zero divisor graph of an arbitrary real Cayley–Dickson algebra. In case of the algebras of the main sequence, the zero divisor graph coincides with the orthogonality graph, and any hexagon can be extended to a double hexagon. We determine the multiplication table of the vertices of a double hexagon. Then we find a sufficient condition for three elements to generate an alternative subalgebra of an arbitrary Cayley–Dickson algebra. Finally, we consider those zero divisors whose components are both standard basis elements up to sign. We classify them and determine necessary and sufficient conditions under which two such elements are orthogonal.
{"title":"Orthogonality graphs of real Cayley-Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons","authors":"S. Zhilina","doi":"10.1142/S0218196721500326","DOIUrl":"https://doi.org/10.1142/S0218196721500326","url":null,"abstract":"We study zero divisors whose components alternate strongly pairwise and construct oriented hexagons in the zero divisor graph of an arbitrary real Cayley–Dickson algebra. In case of the algebras of the main sequence, the zero divisor graph coincides with the orthogonality graph, and any hexagon can be extended to a double hexagon. We determine the multiplication table of the vertices of a double hexagon. Then we find a sufficient condition for three elements to generate an alternative subalgebra of an arbitrary Cayley–Dickson algebra. Finally, we consider those zero divisors whose components are both standard basis elements up to sign. We classify them and determine necessary and sufficient conditions under which two such elements are orthogonal.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86328736","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 : 2021-06-01DOI: 10.1142/S0218196721500338
S. Zhilina
We consider zero divisors of an arbitrary real Cayley–Dickson algebra such that their components are both standard basis elements. We construct inductively the orthogonality graph on these elements. Then we show that, if we restrict our attention to at least [Formula: see text]-dimensional algebras, two algebras are isomorphic if and only if their graphs are isomorphic. We also provide an algorithm to retrieve the Cayley–Dickson parameters of an algebra from its graph.
{"title":"Orthogonality graphs of real Cayley-Dickson algebras. Part II: The subgraph on pairs of basis elements","authors":"S. Zhilina","doi":"10.1142/S0218196721500338","DOIUrl":"https://doi.org/10.1142/S0218196721500338","url":null,"abstract":"We consider zero divisors of an arbitrary real Cayley–Dickson algebra such that their components are both standard basis elements. We construct inductively the orthogonality graph on these elements. Then we show that, if we restrict our attention to at least [Formula: see text]-dimensional algebras, two algebras are isomorphic if and only if their graphs are isomorphic. We also provide an algorithm to retrieve the Cayley–Dickson parameters of an algebra from its graph.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82589799","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 : 2021-05-30DOI: 10.1142/s0218196722500242
A. Reyes, Cristian Sarmiento
In this paper, we study the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras.
本文研究了三维偏多项式代数和扩散代数的微分光滑性。
{"title":"On the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras","authors":"A. Reyes, Cristian Sarmiento","doi":"10.1142/s0218196722500242","DOIUrl":"https://doi.org/10.1142/s0218196722500242","url":null,"abstract":"In this paper, we study the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89922672","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 : 2021-05-28DOI: 10.1142/S0218196723500017
P. Lipparini
We devise a condition strictly between the existence of an [Formula: see text]-ary and an [Formula: see text]-ary near-unanimity term. We evaluate exactly the distributivity and modularity levels implied by such a condition.
{"title":"Between an n-ary and an n + 1-ary near-unanimity term","authors":"P. Lipparini","doi":"10.1142/S0218196723500017","DOIUrl":"https://doi.org/10.1142/S0218196723500017","url":null,"abstract":"We devise a condition strictly between the existence of an [Formula: see text]-ary and an [Formula: see text]-ary near-unanimity term. We evaluate exactly the distributivity and modularity levels implied by such a condition.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80216518","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 : 2021-05-21DOI: 10.1142/S0218196721730019
J. Meakin
{"title":"Tribute to Alessandra Cherubini","authors":"J. Meakin","doi":"10.1142/S0218196721730019","DOIUrl":"https://doi.org/10.1142/S0218196721730019","url":null,"abstract":"","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75253311","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 : 2021-05-18DOI: 10.1142/S0218196721500429
M. E. Kahoui, Najoua Essamaoui, M. Ouali
Let [Formula: see text] be a reduced ring containing [Formula: see text] and let [Formula: see text] be commuting locally nilpotent derivations of [Formula: see text]. In this paper, we give an algorithm to decide the local nilpotency of derivations of the form [Formula: see text], where [Formula: see text] are elements in [Formula: see text].
{"title":"A nilpotency criterion for derivations over reduced ℚ-algebras","authors":"M. E. Kahoui, Najoua Essamaoui, M. Ouali","doi":"10.1142/S0218196721500429","DOIUrl":"https://doi.org/10.1142/S0218196721500429","url":null,"abstract":"Let [Formula: see text] be a reduced ring containing [Formula: see text] and let [Formula: see text] be commuting locally nilpotent derivations of [Formula: see text]. In this paper, we give an algorithm to decide the local nilpotency of derivations of the form [Formula: see text], where [Formula: see text] are elements in [Formula: see text].","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78378480","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 : 2021-05-17DOI: 10.1142/S0218196722500394
Nohra Hage
This paper presents a combinatorial study of the super plactic monoid of type A, which is related to the representations of the general linear Lie superalgebra. We introduce the analogue of the Schützenberger’s jeu de taquin on the structure of super tableaux over a signed alphabet. We show that this procedure which transforms super skew tableaux into super Young tableaux is compatible with the super plactic congruence and it is confluent. We deduce properties relating the super jeu de taquin to insertion algorithms on super tableaux. Moreover, we introduce the super evacuation procedure as an involution on super tableaux and we show its compatibility with the super plactic congruence. Finally, we describe the super jeu de taquin in terms of Fomin’s growth diagrams in order to give a combinatorial version of the super Littlewood–Richardson rule.
本文给出了与一般线性李超代数表示有关的a型超单调群的组合研究。我们介绍了sch曾伯格的jeu de taquin在符号字母表上的超级表的结构上的类比。我们证明了将超斜图转化为超杨图的过程与超平同余是相容的，并且是合流的。我们推导了超级表上插入算法与超级表上插入算法之间的关系。此外，我们还引入了超疏散过程作为超场景的一种对合，并证明了它与超plactic同余的相容性。最后，我们根据福明的增长图描述超级巨巨巨巨，以便给出超级Littlewood-Richardson规则的组合版本。
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