Let $A$ and $B$ be subsets of abelian groups $G_A$ and $G_B$ respectively. A Freiman isomorphism of order $s$ is a bijective mapping $f\colon A\to B$ such that \begin{equation*} a_1+a_2+\dotsb+a_s=a'_1+a'_2+\dotsb+a'_s \end{equation*}holds if and only if \begin{equation*} f(a_1)+f(a_2)+\dotsb+f(a_s)=f(a'_1)+f(a'_2)+\dotsb+f(a'_s). \end{equation*} The Freiman isomorphism is a restriction of the conventional notion of a group isomorphism to a limited number of group operations. In particular, a Freiman isomorphism of order $s$ is also a Freiman isomorphism of order $s-1$ and the mapping is a Freiman isomorphism of every order precisely when it is the conventional isomorphism.

Freiman isomorphisms were introduced by Freiman in his monograph [1] to build a general theory of set addition that is independent of the underlying group.

The number of equivalence classes of $n$ element sets of integers under Freiman isomorphisms of order $2$ is $n^{2n(1+o(1))}$ [2].

References

1

Gregory Freiman.
Foundations of Structural Theory of Set Addition, volume 37 of Translations of Mathematical Monographs.
AMS, 1973.
Zbl 0271.10044.

2

Sergei V. Konyagin and Vsevolod F. Lev.
Combinatorics and linear algebra of Freiman's isomorphism.
Mathematika, 47:39-51, 2000.
Available at http://math.haifa.ac.il/~seva/.

3

Melvyn B. Nathanson.
Additive Number Theory: Inverse Problems and Geometry of Sumsets, volume 165 of GTM.
Springer, 1996.
Zbl 0859.11003.