A lattice $L$ is said to be modular if $x \lor (y \land z) = (x \lor y) \land z$ for all $x,y,z\in L$ such that $x \leq z$ . In fact it is sufficient to show that $x \lor (y \land z) \ge (x \lor y) \land z$ for all $x,y,z\in L$ such that $x \leq z$ , as the reverse inequality holds in all lattices (see modular inequality).

There are a number of other equivalent conditions for a lattice $L$ to be modular:

• $(x\meet y)\join(x\meet z)=x\meet(y\join(x\meet z))$ for all $x,y,z\in L$ .
• $(x\join y)\meet(x\join z)=x\join(y\meet(x\join z))$ for all $x,y,z\in L$ .
• For all $x,y,z\in L$ , if $x<z$ then either $x\meet y<z\meet y$ or $x\join y<z\join y$ .

The following are examples of modular lattices.

• All distributive lattices.
• The lattice of normal subgroups of any group.
• The lattice of submodules of any module. (See modular law.)

A finite lattice $L$ is modular if and only if it is graded and its rank function $\rho$ satisfies $\rho(x)+\rho(y)=\rho(x\land y)+\rho(x\lor y)$ for all $x,y\in L$ .