Show that a interval from a boolean algebra is also a boolean algebra and that a function is surjective

Question

We have an boolean algebra $(B,\lor, \land, ', 0, 1)$ and $b \in B - \{0\}$.

We consider $[0,b] = \{x \in B | 0\le x\le b \} \subset B$, where $\le$ means an order relationship introduced in the lattice $(B, \lor, \land)$

I have to show that :

$(B_b, \lor_b, \land_b, *, 0, b)$ it's also an boolean algebra (they are noted with $_b$ because they are restrictions in the $[0,b]$ interval).

The function $f_b:B \to B_b, f_b(x) = b\land x, \forall x \in B$ is an surjective function.

I mention that I have no experience working with boolean algebra, I'm just studying from a book, I read the theory, but do not know how to aproach the exercises. Any help would be very appreciated...

Thank you !

Answer 1

The interval $[0,b]$ is a sublattice and so it is distributive. Every interval in a complemented modular lattice is complemented. $[0,b]$ is a distributive complemented lattice. What is a Boolean lattice?