Infinite Independent Systems of Identities of an Associative Algebra over an Infinite Field of Characteristic Two

Abstract

Let $$\mathfrak{B}$$ $$\user2{(}\mathfrak{D}\user2{)}$$ be the variety of associative (special Jordan, respectively) algebras over an infinite field of characteristic 2 defined by the identity ((((x 1,x 2),x 3), ((x 4,x 5),x 6)), (x 7,x 8)) = 0 (((x 1 x 2 · x 3)(x 4 x 5 · x 6))(x 7 x 8) = 0, respectively). In this paper, we construct infinite independent systems of identities in the variety $$\mathfrak{B}$$ ( $$\mathfrak{D}$$ , respectively). This implies that the set of distinct nonfinitely based subvarieties of the variety $$\mathfrak{B}$$ $$\user2{(}\mathfrak{D}\user2{)}$$ has the cardinality of the continuum and that there are algebras in $$\mathfrak{B}$$ $$\user2{(}\mathfrak{D}\user2{)}$$ with undecidable word problem.