N. Martins-Ferreira, A. Montoli, A. Patchkoria, M. Sobral
{"title":"The third cohomology group of a monoid and admissible abstract kernels","authors":"N. Martins-Ferreira, A. Montoli, A. Patchkoria, M. Sobral","doi":"10.1142/s0218196722500436","DOIUrl":null,"url":null,"abstract":"We define the product of admissible abstract kernels of the form [Formula: see text], where [Formula: see text] is a monoid, [Formula: see text] is a group and [Formula: see text] is a monoid homomorphism. Identifying [Formula: see text]-equivalent abstract kernels, where [Formula: see text] is the center of [Formula: see text], we obtain that the set [Formula: see text] of [Formula: see text]-equivalence classes of admissible abstract kernels inducing the same action of [Formula: see text] on [Formula: see text] is a commutative monoid. Considering the submonoid [Formula: see text] of abstract kernels that are induced by special Schreier extensions, we prove that the factor monoid [Formula: see text] is an abelian group. Moreover, we show that this abelian group is isomorphic to the third cohomology group [Formula: see text].","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":"20 1","pages":"1009-1041"},"PeriodicalIF":0.0000,"publicationDate":"2022-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Int. J. Algebra Comput.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0218196722500436","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We define the product of admissible abstract kernels of the form [Formula: see text], where [Formula: see text] is a monoid, [Formula: see text] is a group and [Formula: see text] is a monoid homomorphism. Identifying [Formula: see text]-equivalent abstract kernels, where [Formula: see text] is the center of [Formula: see text], we obtain that the set [Formula: see text] of [Formula: see text]-equivalence classes of admissible abstract kernels inducing the same action of [Formula: see text] on [Formula: see text] is a commutative monoid. Considering the submonoid [Formula: see text] of abstract kernels that are induced by special Schreier extensions, we prove that the factor monoid [Formula: see text] is an abelian group. Moreover, we show that this abelian group is isomorphic to the third cohomology group [Formula: see text].