Lattice in complex numbers

Definition

Symbol-free definition

• A lattice in $\mathbb {C}$ is a discrete closed subgroup isomorphic to a free Abelian group on two generators
• A lattice in $\mathbb {C}$ is the Abelian subgroup generated by two nonzero complex numbers, that are not real multiples of each other

Definition with symbols

Let $\mathbb {C}$ denote the complex numbers. A lattice $\Lambda$ in $\mathbb {C}$ is a subgroup of $\mathbb {C}$ generated by elements $0\neq \omega _{1},\omega _{2}$ such that $\omega _{1},\omega _{2}$ form a $\mathbb {R}$-basis of $\mathbb {C}$.