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Regular Hom-algebras admitting a multiplicative basis

Regular Hom-algebras admitting a multiplicative basis AbstractLet (ℌ,μ,α){({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field 𝔽{{\mathbb{F}}}.A basis ℬ={ei}i∈I{{\mathcal{B}}=\{e_{i}\}_{i\in I}} of ℌ{{\mathfrak{H}}} is called multiplicative if for any i,j∈I{i,j\in I}, we have that μ⁢(ei,ej)∈𝔽⁢ek{\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and α⁢(ei)∈𝔽⁢ep{\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some k,p∈I{k,p\in I}.We show that if ℌ{{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum ℌ=⊕rℑr{{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis.Also, the minimality of ℌ{{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case ℬ{{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

Regular Hom-algebras admitting a multiplicative basis

Georgian Mathematical Journal , Volume 28 (4): 11 – Aug 1, 2021

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Publisher
de Gruyter
Copyright
© 2020 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1572-9176
eISSN
1572-9176
DOI
10.1515/gmj-2020-2068
Publisher site
See Article on Publisher Site

Abstract

AbstractLet (ℌ,μ,α){({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field 𝔽{{\mathbb{F}}}.A basis ℬ={ei}i∈I{{\mathcal{B}}=\{e_{i}\}_{i\in I}} of ℌ{{\mathfrak{H}}} is called multiplicative if for any i,j∈I{i,j\in I}, we have that μ⁢(ei,ej)∈𝔽⁢ek{\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and α⁢(ei)∈𝔽⁢ep{\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some k,p∈I{k,p\in I}.We show that if ℌ{{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum ℌ=⊕rℑr{{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis.Also, the minimality of ℌ{{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case ℬ{{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

Journal

Georgian Mathematical Journalde Gruyter

Published: Aug 1, 2021

Keywords: Hom-algebra; multiplicative basis; infinite dimensional algebra; structure; 17A60; 17A30

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