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Minimal variety of associative algebras with unsolvable word problem. (Russian) Zbl 0694.16014
One says the word problem is solvable in a variety V of universal algebras if in any finitely defined (in V) algebra from V the word problem is solvable.
In this work, associative algebras with the signature unit over a finitely generated field of characteristic zero are considered. Let S be the algebra presented by generators A, B, q, a and relations \(xA=Bx=q^ 2=qaq=[q,a,a]=0\), where x is any generator.
The main result of the work is the following Theorem. The word problem is unsolvable in any overvariety of the variety var S and it is solvable in any proper subvariety of var S. The variety var S has the following basis of identities:
(i) \([x_ 1,x_ 2][x_ 3,x_ 4][x_ 5,x_ 6][x_ 7,x_ 8]=0;\)
(ii) \([[x_ 1,x_ 2][x_ 3,x_ 4][x_ 5,x_ 6],x_ 7]=0;\)
(iii) \([x_ 1,x_ 2][x_ 3,x_ 4,x_ 5][x_ 6,x_ 7]=0;\)
(iv) \([[x_ 1,x_ 2,x_ 3][x_ 4,x_ 5,x_ 6],x_ 7]=0;\)
(v) \([x_ 1,x_ 2,x_ 3][x_ 4,x_ 5,x_ 6]-[x_ 1,x_ 2,x_ 6][x_ 4,x_ 5,x_ 3]-[x_ 1,x_ 2][x_ 3,x_ 6][x_ 4,x_ 5]+[x_ 1,x_ 2][x_ 4,x_ 5][x_ 3,x_ 6]-[x_ 3,x_ 6][x_ 1,x_ 2][x_ 4,x_ 5]=0;\)
(vi) \([x_ 1,x_ 2][x_ 3,x_ 4][x_ 5,x_ 6]+[x_ 5,x_ 6][x_ 3,x_ 4][x_ 1,x_ 2]+[x_ 1,x_ 6][x_ 3,x_ 4][x_ 2,x_ 5]+[x_ 2,x_ 5][x_ 3,x_ 4][x_ 1,x_ 6]+[x_ 2,x_ 6][x_ 3,x_ 4][x_ 5,x_ 1]+[x_ 5,x_ 1][x_ 3,x_ 4][x_ 2,x_ 6]=0.\)
Further, it is shown that the word problem is solvable in any variety with the identity \([x_ 1,...,x_ n][y_ 1,y_ 2][z_ 1,...,z_ m]=0\).
Reviewer: L.M.Martynov

MSC:
16Rxx Rings with polynomial identity
08B15 Lattices of varieties
08A50 Word problems (aspects of algebraic structures)
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