Generalized arithmetic progression

A generalized arithmetic progression of rank $r$ is a setIt is proper when every parameter tuple in the displayed box represents a different element.

An abelian progression is a generalized arithmetic progression inside an abelian group. It is the image of an integer box under a group homomorphism.

A coset progression is a sumset $H+P$, where $H$ is a finite subgroup of an abelian group and $P$ is a generalized arithmetic progression.