Elementary group equivalence with the integral length function

The paper determines criteria of elementary equivalence for some classes of free groups with operators and free products with the length function. The case of a group with operators admitting rational coordinatizati0n with a finite basis is completely analyzed. They are polycyclic, solvable groups of finite rank without torsion, and Chernikov groups. The concept of t0-isomorphism of groups intermediate between elementary equivalence and isomorphism is important for the aspects of elementary equivalence of groups with operators and free product. It is proved that to-isomorphism of arbitrary groups of operators is followed by the elementary equivalence of the respective free operator groups (free products) with length function.