FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2000, VOLUME 6, NUMBER 4, PAGES 1229-1238
K. Champagnier
Abstract
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A set of nonzero pairwise distinct elements of a free algebra $F$
is said to be
a primitive system of elements if it is a subset of some set of free
generators of $F$ .
The rank of $U\subset F$
is the smallest number of free generators of $F$ on which
elements of the set $\phi(U)$ depend, where $\phi$
runs through the automorphism
group of $F$ (in other words, it is the smallest
rank of a free factor of $F$ containing $U$ ).
We consider free non-associative algebras, free commutative
non-associative algebras, and free anti-commutative
non-associative algebras.
We construct the algorithm 1 to realize
the rank of a homogeneous element of these free algebras.
The algorithm 2
for the general case is presented. The problem is decomposed
into homogeneous parts. Next, algorithm 3
constructs an automorphism realizing the rank of a system
of elements reducing
it to the case of one element.
Finally, algorithms 4 and 5 deal with a system of primitive elements.
The algorithm 4 presents an automorphism converting it into a part of
a system of
free generators of the algebra. And the algorithm 5
constructs a complement of a primitive system with respect to a free
generating set of the whole free algebra.
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