Stone Representation Theorem

Question

Given two Boolean algebras $A$ and $B$ such that $A$ is a subalgebra of $B$. What is the relation between the Stone space of $A$ and the Stone space of $B$.

The question maybe silly but I am getting confused !

Answer 1

If $A$ is a subalgebra of $B$, the inclusion map $i\colon A\hookrightarrow B$ induces a continuous surjection between their Stone spaces, call it $S(i)\colon S(B)\to S(A)$ (notice the arrow-reversal). Since Stone spaces are (in particular) compact Hausdorff, it follows that $S(i)$ is a quotient map. So if $A$ is a subalgebra of $B$, the Stone space of $A$ is a quotient of the Stone space of $B$.

As a side note, when $A$ runs over all subalgebras of $B$ you do not get all the possible quotients of $S(B)$, but only the quotients whose associated equivalence relation has the property that for any two different equivalence classes there is a saturated clopen set containing one class and not the other.