Isomorphic intervals in lattice

Question

Let $(L,\leq)$ be a modular lattice and let $a,b \in L$ be disctinct. How does one prove that the intervals $[a \wedge b,b]$ and $[a, a \vee b]$ are isomorphic sublattices?

Answer 1

Hint: Consider the following isomorphism: $$\varphi :[a\wedge b,b]\longrightarrow [a,a\vee b],$$ defined as $\varphi (x)=a\vee x,$ notice that this is well defined because $a\leq a\vee x$ and $a\vee x\leq a\vee b$ becase $x\leq b.$

We have to show that this is homomorphism, so $\varphi (x\vee y)=\varphi(x)\vee \varphi (y)$ why?

and $\varphi (x\wedge y)=\varphi (x)\wedge \varphi(y)$ meaning $a\vee (x\wedge y)=(a\vee x)\wedge (a\vee y).$ Notice that modularity implies that lhs is $(a\vee x)\wedge y=(a\vee x)\wedge (a\vee y)$ why?

Notice that the inverse looks like $\phi (x)=x\wedge b.$

Show that they are inverses, modularity is very handy. This is called the Diamond isomorphism.