FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

**Limit T-spaces**

E. A. Kireeva

Abstract

View as HTML
View as gif image

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 $x$_{1}^{2}x_{2}^{2} ... x_{k}^{2} + V_{2},
where $k$Î **N** if
$p\; =\; 2$ and by
$x$_{1}^{α1}x_{2}^{α2}
[x_{1},x_{2}] ...
x_{2k−1}^{α2k−1}
x_{2k}^{α2k}
[x_{2k−1},x_{2k}] + V_{p}, where $k$Î
**N** and $$a_{1},...,
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.

Location: http://mech.math.msu.su/~fpm/eng/k07/k071/k07108h.htm

Last modified: December 21, 2006