Prove every interval of a lattice is a sublattice.

Question

To prove this I think I will have to prove that every interval of a lattice (which is a subset of that lattice) is closed under both meet and join operation. How can I do this?

Answer 1

Suppose that your interval is $I=[a,b]=\{x\in L:a\le x\le b\}$, where $L$ is the original lattice. Clearly $\langle I,\le\rangle$ is a poset, so as you say, all that remains is to show that if $x,y\in I$, then $x$ and $y$ have both a meet and a join in $\langle I,\le\rangle$. You know that they have a meet and join in $L$. Show that if $u=x\land y$ and $v=x\lor y$ in $L$, then $u\in I$, $v\in I$, and moreover $u$ is the greatest lower bound and $v$ the least upper bound of $x$ and $y$ in $I$ as well as in $L$. To show that $u,v\in I$ you’ll want to use the fact that $a$ is a lower and $b$ an upper bound for $x$ and $y$.