Pub Date : 2020-02-24DOI: 10.1142/9789811215476_0011
G. Katona
The underlying set will be {1, 2, . . . , n}. The family of all k-element subsets of [n] is denoted by ( [n] k ) . Its subfamilies are called uniform. A family F of some subsets of [n] is called intersecting if F ∩ G 6= ∅ holds for every pair F,G ∈ F . The whole story has started with the seminal paper of Erdős, Ko and Rado [7]. Their first observation was that an intersecting family in 2 can contain at most one of the complementing pairs, therefore the size of an intersecting family cannot exceed the half of the number of all subsets of [n].
{"title":"Results on intersecting families of subsets, a survey","authors":"G. Katona","doi":"10.1142/9789811215476_0011","DOIUrl":"https://doi.org/10.1142/9789811215476_0011","url":null,"abstract":"The underlying set will be {1, 2, . . . , n}. The family of all k-element subsets of [n] is denoted by ( [n] k ) . Its subfamilies are called uniform. A family F of some subsets of [n] is called intersecting if F ∩ G 6= ∅ holds for every pair F,G ∈ F . The whole story has started with the seminal paper of Erdős, Ko and Rado [7]. Their first observation was that an intersecting family in 2 can contain at most one of the complementing pairs, therefore the size of an intersecting family cannot exceed the half of the number of all subsets of [n].","PeriodicalId":106509,"journal":{"name":"New Trends in Algebras and Combinatorics","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132016125","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 : 2020-02-24DOI: 10.1142/9789811215476_fmatter
K. Shum, E. Zelmanov, P. Kolesnikov, S. M. Anita Wong
{"title":"FRONT MATTER","authors":"K. Shum, E. Zelmanov, P. Kolesnikov, S. M. Anita Wong","doi":"10.1142/9789811215476_fmatter","DOIUrl":"https://doi.org/10.1142/9789811215476_fmatter","url":null,"abstract":"","PeriodicalId":106509,"journal":{"name":"New Trends in Algebras and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129597367","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 : 2020-02-24DOI: 10.1142/9789811215476_bmatter
K. Shum, E. Zelmanov, P. Kolesnikov, S. M. Anita Wong
{"title":"BACK MATTER","authors":"K. Shum, E. Zelmanov, P. Kolesnikov, S. M. Anita Wong","doi":"10.1142/9789811215476_bmatter","DOIUrl":"https://doi.org/10.1142/9789811215476_bmatter","url":null,"abstract":"","PeriodicalId":106509,"journal":{"name":"New Trends in Algebras and Combinatorics","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124403951","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 : 2020-02-24DOI: 10.1142/9789811215476_0025
Yanhui Wang, X. Ren, K. Shum
The purpose of this paper is to investigate restriction ω -semigroups. Here a restriction ω -semigroup is a generalisation of an inverse ω -semigroup. We give a description of a class of restriction ω -semigroups, namely, restriction ω -semigroups with an inverse skeleton. We show that a restriction ω -semigroup with an inverse skeleton is an ideal extension of a (cid:2) J -simple restriction ω -semigroup by a restriction semigroup with a finite chain of projections with a zero adjoined. This result is analogous to Munn’s result for inverse ω -semigroups. In addition, we show that the Bruck–Reilly semigroup of a strong semilattice of monoids indexed by a finite chain is a (cid:2) J -simple restriction ω -semigroup with an inverse skeleton, conversely, every (cid:2) J -simple restriction ω -semigroup with an inverse skeleton arises in this way.
{"title":"Restriction semigroups","authors":"Yanhui Wang, X. Ren, K. Shum","doi":"10.1142/9789811215476_0025","DOIUrl":"https://doi.org/10.1142/9789811215476_0025","url":null,"abstract":"The purpose of this paper is to investigate restriction ω -semigroups. Here a restriction ω -semigroup is a generalisation of an inverse ω -semigroup. We give a description of a class of restriction ω -semigroups, namely, restriction ω -semigroups with an inverse skeleton. We show that a restriction ω -semigroup with an inverse skeleton is an ideal extension of a (cid:2) J -simple restriction ω -semigroup by a restriction semigroup with a finite chain of projections with a zero adjoined. This result is analogous to Munn’s result for inverse ω -semigroups. In addition, we show that the Bruck–Reilly semigroup of a strong semilattice of monoids indexed by a finite chain is a (cid:2) J -simple restriction ω -semigroup with an inverse skeleton, conversely, every (cid:2) J -simple restriction ω -semigroup with an inverse skeleton arises in this way.","PeriodicalId":106509,"journal":{"name":"New Trends in Algebras and Combinatorics","volume":"316 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115372853","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 : 2018-09-20DOI: 10.1142/9789811215476_0012
L. T. Hoa
This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial complexes, integral points in polytopes and graphs.
{"title":"Powers of Monomial Ideals and Combinatorics","authors":"L. T. Hoa","doi":"10.1142/9789811215476_0012","DOIUrl":"https://doi.org/10.1142/9789811215476_0012","url":null,"abstract":"This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial complexes, integral points in polytopes and graphs.","PeriodicalId":106509,"journal":{"name":"New Trends in Algebras and Combinatorics","volume":"128 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115769390","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 : 2018-08-27DOI: 10.1142/9789811215476_0007
V. Gubarev
Applying Groebner-Shirshov technique, we prove that any post-Lie algebra injectively embeds into its universal enveloping postassociative algebra.
利用Groebner-Shirshov技术，证明了任何后李代数都可以内嵌入到其包络的后联想代数中。
{"title":"Embedding of post-Lie algebras into postassociative algebras","authors":"V. Gubarev","doi":"10.1142/9789811215476_0007","DOIUrl":"https://doi.org/10.1142/9789811215476_0007","url":null,"abstract":"Applying Groebner-Shirshov technique, we prove that any post-Lie algebra injectively embeds into its universal enveloping postassociative algebra.","PeriodicalId":106509,"journal":{"name":"New Trends in Algebras and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132371440","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 : 2018-07-23DOI: 10.1142/9789811215476_0016
P. Kolesnikov
We present an approach to the computation of confluent systems of defining relations in associative conformal algebras based on the similar technique for modules over ordinary associative algebras.
基于普通结合代数上的模的类似技术，给出了结合共形代数中定义关系的合流系统的计算方法。
{"title":"Gröbner–Shirshov bases for associative conformal algebras with arbitrary locality function","authors":"P. Kolesnikov","doi":"10.1142/9789811215476_0016","DOIUrl":"https://doi.org/10.1142/9789811215476_0016","url":null,"abstract":"We present an approach to the computation of confluent systems of defining relations in associative conformal algebras based on the similar technique for modules over ordinary associative algebras.","PeriodicalId":106509,"journal":{"name":"New Trends in Algebras and Combinatorics","volume":"28 14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126505170","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 : 2018-07-01DOI: 10.1142/9789811215476_0024
H. P. Sankappanavar
The variety DMSH of semi-Heyting algebras with a De Morgan negation was introduced in [12] and an increasing sequence DMSHn of level n, n being a natural number, of its subvarieties was investigated in the series [12], [13], [14], [15], [16], and [17], of which the present paper is a sequel. In this paper, we prove two main results: Firstly, we prove that DMSH1-algebras of level 1 satisfy Stone identity, generalizing an earlier result that regular DMSH1-algebras of level 1 satisfy Stone identity. Secondly, we prove that the variety of DmsStSH of dually ms, Stone semi-Heyting algebras is at level 2. As an application, it is derived that the variety of De Morgan semi-Heyting algebras is also at level 2. It is also shown that these results are sharp.
{"title":"De Morgan Semi-Heyting and Heyting Algebras","authors":"H. P. Sankappanavar","doi":"10.1142/9789811215476_0024","DOIUrl":"https://doi.org/10.1142/9789811215476_0024","url":null,"abstract":"The variety DMSH of semi-Heyting algebras with a De Morgan negation was introduced in [12] and an increasing sequence DMSHn of level n, n being a natural number, of its subvarieties was investigated in the series [12], [13], [14], [15], [16], and [17], of which the present paper is a sequel. In this paper, we prove two main results: Firstly, we prove that DMSH1-algebras of level 1 satisfy Stone identity, generalizing an earlier result that regular DMSH1-algebras of level 1 satisfy Stone identity. Secondly, we prove that the variety of DmsStSH of dually ms, Stone semi-Heyting algebras is at level 2. As an application, it is derived that the variety of De Morgan semi-Heyting algebras is also at level 2. It is also shown that these results are sharp.","PeriodicalId":106509,"journal":{"name":"New Trends in Algebras and Combinatorics","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128566007","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 : 2017-12-18DOI: 10.1142/9789811215476_0020
D. Rumynin
In this expository paper we review some recent results about representations of Kac-Moody groups. We sketch the construction of these groups. If practical, we present the ideas behind the proofs of theorems. At the end we pose open questions.
{"title":"Kac-Moody Groups and Their Representations","authors":"D. Rumynin","doi":"10.1142/9789811215476_0020","DOIUrl":"https://doi.org/10.1142/9789811215476_0020","url":null,"abstract":"In this expository paper we review some recent results about representations of Kac-Moody groups. We sketch the construction of these groups. If practical, we present the ideas behind the proofs of theorems. At the end we pose open questions.","PeriodicalId":106509,"journal":{"name":"New Trends in Algebras and Combinatorics","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134146146","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 : 2017-08-12DOI: 10.1142/9789811215476_0023
Yuqun Chen, Lili Ni
We construct free modules over an associative conformal algebra. We establish Composition-Diamond lemma for associative conformal modules. As applications, Gr"obner-Shirshov bases of the Virasoro conformal module and module over the semidirect product of Virasoro conformal algebra and current algebra are given respectively.
{"title":"Gröbner-Shirshov bases for associative conformal modules","authors":"Yuqun Chen, Lili Ni","doi":"10.1142/9789811215476_0023","DOIUrl":"https://doi.org/10.1142/9789811215476_0023","url":null,"abstract":"We construct free modules over an associative conformal algebra. We establish Composition-Diamond lemma for associative conformal modules. As applications, Gr\"obner-Shirshov bases of the Virasoro conformal module and module over the semidirect product of Virasoro conformal algebra and current algebra are given respectively.","PeriodicalId":106509,"journal":{"name":"New Trends in Algebras and Combinatorics","volume":"135 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116604210","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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