Pub Date : 2021-11-22DOI: 10.1142/s0218196722500060
C. Bisi, G. Chiaselotti, T. Gentile
In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.
本文研究了在抽象有序环境下的实子集组合问题。具体地说，设[Formula: see text]是一个有限偏序集，其中[Formula: see text]是一个顺序反转和对合映射，使得[Formula: see text]对于每个[Formula: see text]。设[公式:见文]是两个元素的布尔格，[公式:见文]是所有保序2值映射的族[公式:见文]，使得[公式:见文]对于所有[公式:见文]都是[公式:见文]。在本文中，我们基于[公式:见文本]为[公式:见文本]的特定子集建立了一个族[公式:见文本]，我们称之为[公式:见文本]，并确定族[公式:见文本]和族[公式:见文本]之间的双射。在这样的双射中，[公式:见文]上的[公式:见文]-基[公式:见文]对应于一个映射[公式:见文]，其[公式:见文]对[公式:见文]的限制是[公式:见文]上的最小2值部分映射，该映射在[公式:见文]中以[公式:见文]作为其唯一扩展。接下来，我们将展示每个[公式:见文]——基于[公式:见文]——如何在特定的环境中成为一个更大的线性不等式系统的一个子系统，其兼容性意味着整个系统的兼容性。
{"title":"Real subset sums and posets with an involution","authors":"C. Bisi, G. Chiaselotti, T. Gentile","doi":"10.1142/s0218196722500060","DOIUrl":"https://doi.org/10.1142/s0218196722500060","url":null,"abstract":"In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"1 1","pages":"127-157"},"PeriodicalIF":0.0,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88567443","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 : 2021-11-17DOI: 10.1142/s0218196722500023
D. Bessades, R. B. D. Santos, A. C. Vieira
Let [Formula: see text] be a field of characteristic zero and [Formula: see text] the algebra of [Formula: see text] matrices over [Formula: see text]. By the classical Amitsur–Levitzki theorem, it is well known that [Formula: see text] is the smallest degree of a standard polynomial identity of [Formula: see text]. A theorem due to Rowen shows that when the symplectic involution [Formula: see text] is considered, the standard polynomial of degree [Formula: see text] in symmetric variables is an identity of [Formula: see text]. This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of [Formula: see text] in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of [Formula: see text] with transpose involution.
{"title":"Minimal degree of standard identities of matrix algebras with symplectic graded involution","authors":"D. Bessades, R. B. D. Santos, A. C. Vieira","doi":"10.1142/s0218196722500023","DOIUrl":"https://doi.org/10.1142/s0218196722500023","url":null,"abstract":"Let [Formula: see text] be a field of characteristic zero and [Formula: see text] the algebra of [Formula: see text] matrices over [Formula: see text]. By the classical Amitsur–Levitzki theorem, it is well known that [Formula: see text] is the smallest degree of a standard polynomial identity of [Formula: see text]. A theorem due to Rowen shows that when the symplectic involution [Formula: see text] is considered, the standard polynomial of degree [Formula: see text] in symmetric variables is an identity of [Formula: see text]. This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of [Formula: see text] in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of [Formula: see text] with transpose involution.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"28 1","pages":"47-66"},"PeriodicalIF":0.0,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84226703","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 : 2021-11-17DOI: 10.1142/s0218196722500047
M. Dugas, D. Herden, Jack Rebrovich
Let [Formula: see text] denote the incidence algebra of a locally finite poset [Formula: see text] over a field [Formula: see text] and [Formula: see text] some equivalence relation on the set of generators of [Formula: see text]. Then [Formula: see text] is the subset of [Formula: see text] of all the elements that are constant on the equivalence classes of [Formula: see text]. If [Formula: see text] satisfies certain conditions, then [Formula: see text] is a subalgebra of [Formula: see text] called a reduced incidence algebra. We extend this notion to finitary incidence algebras [Formula: see text] for any poset [Formula: see text]. We investigate reduced finitary incidence algebras [Formula: see text] and determine their automorphisms in some special cases.
{"title":"Reduced finitary incidence algebras and their automorphisms","authors":"M. Dugas, D. Herden, Jack Rebrovich","doi":"10.1142/s0218196722500047","DOIUrl":"https://doi.org/10.1142/s0218196722500047","url":null,"abstract":"Let [Formula: see text] denote the incidence algebra of a locally finite poset [Formula: see text] over a field [Formula: see text] and [Formula: see text] some equivalence relation on the set of generators of [Formula: see text]. Then [Formula: see text] is the subset of [Formula: see text] of all the elements that are constant on the equivalence classes of [Formula: see text]. If [Formula: see text] satisfies certain conditions, then [Formula: see text] is a subalgebra of [Formula: see text] called a reduced incidence algebra. We extend this notion to finitary incidence algebras [Formula: see text] for any poset [Formula: see text]. We investigate reduced finitary incidence algebras [Formula: see text] and determine their automorphisms in some special cases.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"263 1","pages":"85-114"},"PeriodicalIF":0.0,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82802425","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 : 2021-10-22DOI: 10.1142/s0218196722500035
Layla Sorkatti
We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field [Formula: see text] that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension [Formula: see text] of rank [Formula: see text] over any field.
{"title":"A bound for the class of nilpotent symplectic alternating algebras","authors":"Layla Sorkatti","doi":"10.1142/s0218196722500035","DOIUrl":"https://doi.org/10.1142/s0218196722500035","url":null,"abstract":"We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field [Formula: see text] that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension [Formula: see text] of rank [Formula: see text] over any field.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"88 15 1","pages":"67-84"},"PeriodicalIF":0.0,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84064376","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 : 2021-10-20DOI: 10.1142/s0218196722500345
S. Bouarroudj, D. Leites, I. Shchepochkina
The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the [Formula: see text]-dimensional supermanifold (superstring) [Formula: see text], we classify analogs of Bol operators invariant under the simple maximal subalgebra [Formula: see text] of the same rank as its simple ambient superalgebra [Formula: see text] of vector fields on [Formula: see text] and containing all elements of negative degree of [Formula: see text] in a [Formula: see text]-grading. We also consider the Lie superalgebras of vector fields [Formula: see text] preserving a contact structure on the superstring [Formula: see text]. We have discovered many new operators.
{"title":"Analogs of Bol operators on superstrings","authors":"S. Bouarroudj, D. Leites, I. Shchepochkina","doi":"10.1142/s0218196722500345","DOIUrl":"https://doi.org/10.1142/s0218196722500345","url":null,"abstract":"The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the [Formula: see text]-dimensional supermanifold (superstring) [Formula: see text], we classify analogs of Bol operators invariant under the simple maximal subalgebra [Formula: see text] of the same rank as its simple ambient superalgebra [Formula: see text] of vector fields on [Formula: see text] and containing all elements of negative degree of [Formula: see text] in a [Formula: see text]-grading. We also consider the Lie superalgebras of vector fields [Formula: see text] preserving a contact structure on the superstring [Formula: see text]. We have discovered many new operators.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"26 1","pages":"807-835"},"PeriodicalIF":0.0,"publicationDate":"2021-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84885163","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 : 2021-10-13DOI: 10.1142/S0218196722500588
V. Gubarev
Applying the Poincare-Birkhoff-Witt property and the Groebner-Shirshov bases technique, we find the linear basis of the associative universal enveloping algebra in the sense of V. Ginzburg and M. Kapranov of a pair of compatible Lie brackets. We state that the growth rate of this universal enveloping over $n$-dimensional compatible Lie algebra equals $n+1$.
{"title":"Universal enveloping algebra of a pair of compatible Lie brackets","authors":"V. Gubarev","doi":"10.1142/S0218196722500588","DOIUrl":"https://doi.org/10.1142/S0218196722500588","url":null,"abstract":"Applying the Poincare-Birkhoff-Witt property and the Groebner-Shirshov bases technique, we find the linear basis of the associative universal enveloping algebra in the sense of V. Ginzburg and M. Kapranov of a pair of compatible Lie brackets. We state that the growth rate of this universal enveloping over $n$-dimensional compatible Lie algebra equals $n+1$.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"1 1","pages":"1335-1344"},"PeriodicalIF":0.0,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79930835","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 : 2021-09-30DOI: 10.1142/S0218196723500133
J. Bell, Haggai Liu, M. Mishna
Given a finitely generated group with generating set $S$, we study the cogrowth sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks the corresponding Cayley graph. Muller and Schupp proved the generating function of the sequence is algebraic when $G$ has a finite-index free subgroup (using a result of Dunwoody). In this work we make this result effective for free products of finite groups: we determine bounds for the degree and height of the minimal polynomial of the generating function, and determine the minimal polynomial explicitly for some families of free products. Using these results we are able to prove that a gap theorem holds: if $S$ is a finite symmetric generating set for a group $G$ and if $a_n$ denotes the number of words of length $n$ over the alphabet $S$ that are equal to $1$ then $limsup_n a_n^{1/n}$ exists and is either $1$, $2$, or at least $2sqrt{2}$.
{"title":"Cogrowth series for free products of finite groups","authors":"J. Bell, Haggai Liu, M. Mishna","doi":"10.1142/S0218196723500133","DOIUrl":"https://doi.org/10.1142/S0218196723500133","url":null,"abstract":"Given a finitely generated group with generating set $S$, we study the cogrowth sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks the corresponding Cayley graph. Muller and Schupp proved the generating function of the sequence is algebraic when $G$ has a finite-index free subgroup (using a result of Dunwoody). In this work we make this result effective for free products of finite groups: we determine bounds for the degree and height of the minimal polynomial of the generating function, and determine the minimal polynomial explicitly for some families of free products. Using these results we are able to prove that a gap theorem holds: if $S$ is a finite symmetric generating set for a group $G$ and if $a_n$ denotes the number of words of length $n$ over the alphabet $S$ that are equal to $1$ then $limsup_n a_n^{1/n}$ exists and is either $1$, $2$, or at least $2sqrt{2}$.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"13 1","pages":"237-260"},"PeriodicalIF":0.0,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87277821","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 : 2021-09-28DOI: 10.1142/s0218196721500582
Dayan Liu, Fumei Liu, Xiaosong Sun
The investigation of co-tame automorphisms of the affine space [Formula: see text] is helpful to understand the structure of its automorphisms group. In this paper, we show the co-tameness of several classes of automorphisms, including some 3-parabolic automorphisms, power-linear automorphisms, homogeneous automorphisms in small dimension or small transcendence degree. We also classify all additive-nilpotent automorphisms in dimension four and show that they are co-tame.
{"title":"Some co-tame automorphisms of affine spaces","authors":"Dayan Liu, Fumei Liu, Xiaosong Sun","doi":"10.1142/s0218196721500582","DOIUrl":"https://doi.org/10.1142/s0218196721500582","url":null,"abstract":"The investigation of co-tame automorphisms of the affine space [Formula: see text] is helpful to understand the structure of its automorphisms group. In this paper, we show the co-tameness of several classes of automorphisms, including some 3-parabolic automorphisms, power-linear automorphisms, homogeneous automorphisms in small dimension or small transcendence degree. We also classify all additive-nilpotent automorphisms in dimension four and show that they are co-tame.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"136 1","pages":"1601-1612"},"PeriodicalIF":0.0,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81425817","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 : 2021-09-24DOI: 10.1142/S0218196722500473
Herman Goulet-Ouellet
. Our main goal is to study the freeness of Schützenberger groups defined by primitive substitutions. Our findings include a simple freeness test for these groups, which is applied to exhibit a primitive invertible substitution with corresponding non-free Schützenberger group. This constitutes a coun- terexample to a result of Almeida dating back to 2005. We also give some early results concerning relative freeness of Schützenberger groups, a question which remains largely unexplored.
{"title":"Freeness of Schützenberger groups of primitive substitutions","authors":"Herman Goulet-Ouellet","doi":"10.1142/S0218196722500473","DOIUrl":"https://doi.org/10.1142/S0218196722500473","url":null,"abstract":". Our main goal is to study the freeness of Schützenberger groups defined by primitive substitutions. Our findings include a simple freeness test for these groups, which is applied to exhibit a primitive invertible substitution with corresponding non-free Schützenberger group. This constitutes a coun- terexample to a result of Almeida dating back to 2005. We also give some early results concerning relative freeness of Schützenberger groups, a question which remains largely unexplored.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"83 1","pages":"1101-1123"},"PeriodicalIF":0.0,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86576647","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 : 2021-09-15DOI: 10.1142/s0218196721500570
A. Zhuchok
This paper is devoted to the study of the problem of adjoining a set of bar-units to dimonoids. We give necessary and sufficient conditions for adjoining a set of bar-units to the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), and prove that it is impossible to adjoin a set of bar-units to the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid. As consequences, we establish that it is impossible to extend by a set of bar-units the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid to a generalized digroup. We also count the cardinalities of the free left [Formula: see text]-dinilpotent dimonoid and the free [Formula: see text]-dimonoid for a finite case.
{"title":"Relatively free dimonoids and bar-units","authors":"A. Zhuchok","doi":"10.1142/s0218196721500570","DOIUrl":"https://doi.org/10.1142/s0218196721500570","url":null,"abstract":"This paper is devoted to the study of the problem of adjoining a set of bar-units to dimonoids. We give necessary and sufficient conditions for adjoining a set of bar-units to the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), and prove that it is impossible to adjoin a set of bar-units to the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid. As consequences, we establish that it is impossible to extend by a set of bar-units the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid to a generalized digroup. We also count the cardinalities of the free left [Formula: see text]-dinilpotent dimonoid and the free [Formula: see text]-dimonoid for a finite case.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"33 1","pages":"1587-1599"},"PeriodicalIF":0.0,"publicationDate":"2021-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85205789","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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