  The connection between classical model theoretical types (MT-types) and
logically-geometrical types (LG-types) introduced by B. Plotkin is considered.
It is proved that MT-types of two $n$-tuples in two universal algebras coincide
if and only if their LG-types coincide. An algebra $H$ is called logically
perfect if for every two $n$-tuples in $H$ whose types coincide, one can be
sent to another by means of an automorphism of this algebra. Some sufficient
condition for logically perfectness of free finitely generated algebras is
given which helps to prove that finitely generated free Abelian groups,
finitely generated free nilpotent groups and finitely generated free semigroups
are logically perfect. It is proved that if two Abelian groups have the same
type and one of them is finitely generated and free then these groups are
isomorphic.

Logic;Group Theory