Siberian Federal University

For any formal commutator R of a free group F , we constructively prove the existence of a logical formula ER with the following properties. First, if we apply the collection process to a positive word W of the group F , then the structure of ER is determined by R, and the logical values of ER are determined by W and the arrangement of the collected commutators. Second, if the commutator R was collected during the collection process, then its exponent is equal to the number of elements of the set D(R) that satisfy ER, where D(R) is determined by R. We provide examples of ER for some commutators R and, as a consequence, calculate their exponents for different positive words of F . In particular, an explicit collection formula is obtained for the word (a1 . . . an)m, n, m > 1, in a group with the Abelian commutator subgroup. Also, we consider the dependence of the exponent of a commutator on the arrangement of the commutators collected during the collection process