On the finite basis property of abstract $T$-spaces
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 3, pp. 669-700
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Let $F=k\langle x_1,\dots,x_i,\dots\rangle$ 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.
@article{FPM_1995_1_3_a7,
author = {A. V. Grishin},
title = {On the finite basis property of abstract $T$-spaces},
journal = {Fundamentalʹna\^a i prikladna\^a matematika},
pages = {669--700},
publisher = {mathdoc},
volume = {1},
number = {3},
year = {1995},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FPM_1995_1_3_a7/}
}
A. V. Grishin. On the finite basis property of abstract $T$-spaces. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 3, pp. 669-700. http://geodesic.mathdoc.fr/item/FPM_1995_1_3_a7/