What does it mean by binary operation ( taking gcd and lcm )?

Question

I'm less than a rookie so it might seem like an easy question but I want a head start to figure things later on my own. The question wants me to prove that the set $D(m) = \{ x \in \mathbb N - \{0\} : x\mid m \}$ where $m$ is a positive number, is a lattice under two binary operations of taking : gcd and lcm. So I figured out that first I have to prove that it's a partially ordered set ( reflexive - anti Symmetric - transitive), but the problem here I can't understand what does it mean by these two operations. I need an example on the reflexive part so I could understand. or at least any example. and Sorry for this low Question.

Answer 1

It is reflexive because each number divides itself, so $a$ divides $a$ for every non-zero $a$, it is antisymmetric, because if $a$ divides $b$ and $b$ divides $a$, then $a=\pm b$, but $a$ and $b$ being positive, it must be that $a=b$. Surely you can prove that if $a$ divides $b$ and $b$ divides $c$ then $a$ divides $c$, which is transitivity.

To show that it is a lattice you have to prove that for every two numbers $a$ and $b$ there exist $\inf\{a,b\}$ and $\sup\{a,b\}$. Recall that $\inf\{a,b\}$ must divide both $a$ and $b$ and if $c$ divides both $a$ and $b$, then $c$ must divide $\inf\{a,b\}$. Dually, both $a$ and $b$ divide $\sup\{a,b\}$ and if both $a$ and $b$ divide $c$, then $\sup\{a,b\}$ must divide $c$. Hence you are left to prove that $\gcd(a,b)$ and $\operatorname{lcm}(a,b)$ play the role of $\inf\{a,b\}$ and $\sup\{a,b\}$respectively.

Answer 2

It seems that most of the people are missing PO's question, which is how to use $\operatorname{lcm}$ and $\gcd$.

You need to prove that any subset $\left\{a,b\right\}\subseteq D(m)$ has a least upper bound $\operatorname{LUB}(\left\{a,b\right\})$ and a greatest lower bound $\operatorname{GLB}(\left\{a,b\right\})$ defined by you.

By defining $\operatorname{LUB}(\left\{a,b\right\}) = \operatorname{lcm}(a,b)$ and $\operatorname{GLB}(\left\{a,b\right\})=\gcd(a,b)$, you need to prove that for any $\left\{a,b\right\}\subseteq D(m)$, the $\operatorname{LUB}(\left\{a,b\right\})\in D(m)$ and $\operatorname{GLB}(\left\{a,b\right\})\in D(m)$.

This is easy to prove as if $\left\{a,b\right\}\subseteq D(m)$, then $a\mid m$ and $b\mid m$, hence $\gcd(a,b)\mid m$, so $\operatorname{GLB}(\left\{a,b\right\})\in D(m)$. You can work on the second part. Hint: The expression $ab=\gcd(a,b)\times \operatorname{lcm}(a,b)$ might be useful. Try to think what happens if $\operatorname{LUB}(\left\{a,b\right\})\not\in D(m)$.