How an interval [a, b] in a Boolean algebra B can be made into a Boolean algebra?

Question

How an interval $[a, b]$ in a Boolean algebra $B$ can be made into a Boolean algebra?

Answer 1

Assuming $a\le b$ (that is, $a\wedge b=a$ or, equivalently, $a\vee b=b$), the operations are exactly the same as in $B$, except for complementation; the minimum is $a$, the maximum is $b$ and the complement is $$ x^*=(x'\wedge b)\vee a=(x'\vee a)\wedge b $$ where $x'$ denotes the complement in the algebra $B$. We're “cutting $x'$ above by $b$ and below by $a$”.

Check the axioms.