FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2007, VOLUME 13, NUMBER 1, PAGES 135-159

Limit T-spaces

E. A. Kireeva

Abstract

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Let $F$ be a field of prime characteristic $p$ and let $$V_p be the variety of associative algebras over $F$ without unity defined by the identities $[[x,y],z]\; =\; 0$ and $x4=\; 0$ if $p\; =\; 2$ and by the identities $[[x,y],z]\; =\; 0$ and $xp=\; 0$ if $p\; >\; 2$ (here $[x,y]\; =\; xy$- yx). Let $A/V$_p be the free algebra of countable rank of the variety $$V_p and let $S$ be the T-space in $A/V$_p generated by $x$₁²x₂² ... x_k² + V₂, where $k$Î N if $p\; =\; 2$ and by $x$₁^α₁x₂^α₂ [x₁,x₂] ... x_2k−1^α_2k−1 x_2k^α_2k [x_2k−1,x_2k] + V_p, where $k$Î N and $$a₁,..., a_2k Î {0, p - 1} if $p\; >\; 2$. As is known, $S$ is not finitely generated as a T-space. In the present paper, we prove that $S$ is a limit T-space, i.e., a maximal nonfinitely generated T-space. As a corollary, we have constructed a limit T-space in the free associative $F$-algebra without unity of countable rank.

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