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

K. Champagnier

Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 4, pp. 1229-1238

Résumé

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.