{"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}
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}
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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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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