The join of varieties with associative-commutative intersection of bounded index.
type: Item
instance of: scholarly article
title: The join of varieties with associative-commutative intersection of bounded index.
zbMATH Open document ID: 1332.08007
DOI: 10.1134/S0037446615030192
published in: Siberian Mathematical Journal
publication date: 2015-09-28T00:00:00Z
review text: In the present paper, by an algebra it is meant a linear algebra over a field of characteristic 0. The join of finitely-based varieties of algebras need not be finitely-based. The main result of the paper is the following theorem: Theorem 1. Let \(\mathcal A\) and \(\mathcal B\) be finitely-based varieties of algebras. Suppose that \(\mathcal A\cap\mathcal B\) is an associative commutative variety of algebras of index \(n\geq 3\). Then \(\mathcal A+\mathcal B\) is a finitely-based variety of algebras. -- Next, three open problems are proposed.
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 17D10
zbMATH DE Number: 6485710
zbMATH Keywords: varieties of algebras
zbMATH Keywords: joins of varieties
zbMATH Keywords: finite bases of systems of identities
zbMATH Keywords: bases of identities
zbMATH Keywords: unions of varieties
zbMATH Keywords: PI-algebras
zbMATH Keywords: finitely based T-ideals
zbMATH Keywords: multilinear polynomials
zbMATH Keywords: Lie algebras
zbMATH Keywords: Mal'cev algebras
author: Oleg Vladimirovich Shashkov
reviewed by: Danica Jakubiková-Studenovská
MaRDI profile type: Publication
cites work: Properties of the join of varieties of algebras
cites work: Q4210624
cites work: Finite basis property of identities of associative algebras
cites work: Q4761662
cites work: Finiteness of basis of identities of a finitely generated alternative PI- algebra over a field of characteristic zero
cites work: The Specht property of the ideals of identities of certain simple nonassociative algebras
cites work: Counterexamples to the Specht problem
cites work: Q4761675
cites work: Varieties of generalized standard and generalized accessible algebras
cites work: The join of varieties of algebras
cites work: The varieties of standard and accessible algebras
cites work: Q3947798
cites work: ON SOME VARIETIES OF ASSOCIATIVE ALGEBRAS
cites work: Malcev Algebras
cites work: The finite basis property for the union of some varieties of algebras
description: scientific article; zbMATH DE number 6485710