Freiman isomorphism

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$f:A\to B$ such that

$$a_1+a_2+\mathrm{\cdots }+a_s=a_1^{\prime }+a_2^{\prime }+\mathrm{\cdots }+a_s^{\prime }$$

holds if and only if

$$f(a_1)+f(a_2)+\mathrm{\cdots }+f(a_s)=f(a_1^{\prime })+f(a_2^{\prime })+\mathrm{\cdots }+f(a_s^{\prime }).$$

The Freiman isomorphism is a restriction of the conventional notion ofa group isomorphism to a limited number of group operations. Inparticular, a Freiman isomorphism of order $s$ is also a Freimanisomorphism of order $s-1$, and the mapping is a Freiman isomorphism ofevery order precisely when it is the conventional isomorphism.

Freiman isomorphisms were introduced by Freiman in his monograph[1] to build a general theory ofset addition (http://planetmath.org/Sumset) that is independent of theunderlying 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].