I Was preparing for my exam and wanted to write Formal Prove, Question:
Let $D_{m}$ be the divisors of m. Defins inf{a,b} = gcd{a, b}, sup{a, b} = lcm{a, b}, and $¬\,d=\frac{m}{d}$ . Investgate when $D_{m}$ is a Boolean algebra.
Here is my idea:
first I need to show that $a$ ∧ $b =$ gcd{a,b} , $a$ ∨ $b =$ lcm{a,b} for any $ a , b \,\,\epsilon\,\,D_{m}$ and the complement is the division so $D_{m}$ is lattice by definition
second, all of the boolean algebra conditions are applied except the complement so $a\,\,\,∧\,\,\,a^{`} = 0$ and $a\,\,\,∨\,\,\,a^{`} = 1$
so if m is product distinct primes then the complement of any element $¬\,d=\frac{m}{d}$ for $d\,\,\epsilon\,\,D_{m}$ so it's boolean algebra
and if m is a product of non-distinct primes so there will be primes with no complement so it's not a boolean algebra,
I want to order my think process and write consistant mathematical prove.