Let $(A,\leq)$ be a lattice. Show that $A$ is modular if and only if for all $x,y,z\in A$, $(x\wedge z)\vee (y\wedge z)=[(x\wedge z)\vee y]\wedge z$.

Question

Let $(A,\leq)$ be a lattice. Show that $A$ is modular if and only if for all $x,y,z\in A$, $(x\wedge z)\vee (y\wedge z)=[(x\wedge z)\vee y]\wedge z$.

How can I start this problem?

Answer 1

I'm using the following definition of modularity of a lattice: For all $x,y,z$, $$x \leq z \;\Rightarrow\; x \vee ( y \wedge z ) = ( x \vee y ) \wedge z.$$

Hint: In one direction note that $x \wedge z \leq z$. In the opposite direction note that $x \leq z$ is equivalent to $x \wedge z = x$