(FUNDAMENTAL AND APPLIED MATHEMATICS)

1995, VOLUME 1, NUMBER 3, PAGES 669-700

## On the finite basis property of abstract T-spaces

A.V.Grishin

Let $F = k \langle x_1,\ldots, x_i,\ldots \rangle$ be the free countably generated algebra over a field k of the characteristic 0. A vector subspace V of the algebra F is called a T-space of F if it is closed under substitutions. It is clear that an ideal I of F is a T-ideal if and only if I is a T-space of F. The aim of this paper is to introduce the definition of the abstract T-space and to prove the finite basis property for some large class of T-spaces.

The main result of this paper is the following

Theorem. Let I be a T-ideal of algebra F which contains a Capelly polynomial. Then every T-space of F/I is finitely based.

The statement of this theorem allows us to give a positive answer to the local Specht's problem (A.Kemer gave a positive answer to Specht's problem using another approach) and to the representability problem.

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