FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2000, VOLUME 6, NUMBER 4, PAGES 1229-1238

**Algorithms to realize the rank and primitivity of systems of
elements in free non-associative algebras**

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 $U$Ì F is the smallest
number of free generators of $F$ on which elements of
the set $$j (U) depend, where
$$j 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.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k00/k004/k00418h.htm

Last modified: February 13, 2001