Freiman s-homomorphism (Freiman $s$-homomorphism)

A map $\varphi :A\to B$ is a Freiman $s$-homomorphism whenimplies the corresponding equality between the $\varphi (a_i)$. It is a Freiman $s$-isomorphism when it is bijective and the converse implication also holds.

A Freiman $s$-isomorphism is a bijection whose forward and inverse maps are both Freiman s-homomorphisms.

The Ruzsa modelling lemma says that a finite set with bounded doubling has a large subset Freiman $s$-isomorphic to a dense subset of a finite cyclic group, whose order is bounded by a constant depending only on the doubling constant and $s$ times the original set size.