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 $\mathrm{\Lambda }$ in $\mathbb{C}$ is a subgroup of $\mathbb{C}$ generated by elements $0\ne {\omega }_1,{\omega }_2$ such that ${\omega }_1,{\omega }_2$ form a $\mathbb{R}$-basis of $\mathbb{C}$.