Least and greatest element of the $(\mathbb{N}, |)$

Question

Consider the relation | on $\mathbb{N}$, where $\mathbb{N} = \{0,1,2,... \}$ and $n|m$ means $n$ divides $m$. I know that the pair $(\mathbb{N}, |)$ is a partial order, :

(1) Find the least and greatest element.

In this case the only least element would be $1$. Also there are no biggest element.

Am I right or wrong?

Thanks,

Answer 1

In a partial order:

• $a$ is minimal means $b\leq a$ implies $a=b$
• $a$ is maximal means $a\leq b$ implies $a=b$
• $a$ is the maximum means $b\leq a$ for all $b$
• $a$ is the minimum means $a\leq b$ for all $b$

If there is a unique maximal (minimal) element, then it is the maximum (minimum). Note that the definitions are different. Also, maximum (minimum) and greatest (least) are synonyms.

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In your example:

$1$ is a minimal element, since $n\mid1$ implies $0<n\leq1$, hence $n=1$.

$0$ is a maximal element, since $0\mid n$ means $0\cdot m=n$ for some $m\in\mathbb{N}$, hence $n=0$.

$1$ is the least element, since $n\cdot 1=n$ shows that $1\mid n$ for all $n$.

$0$ is the greatest element, since $n\cdot 0=0$ shows that $n\mid 0$ for all $n$.