How to find all minima, maxima, largest and smallest elements of in a partially ordered set $S=\mathbb{N}$?

Question

As the title suggests; how do I find the all minima, maxima, largest and smallest elements of in a partially ordered set $$S=\mathbb{N}$$ with the "Divisibility relation" $$a\preceq b: \Leftrightarrow a|b$$

And how can I prove that they are correct?

Answer 1

A minimal element $m$ is one such that $a\preceq m$ implies $a=m$. Therefore $1$ is the only minimal element.

A maximal element $m$ is one such that $m\preceq b$ implies $m=b$. As we can pick $b=2m$, there is no maximal element.

A largest element $m$ is one such that $a\preceq m$ for all $a$. Again, trying $a=2m$, we see than there is no largest element.

A smallest element $m$ is one such that $m\preceq b$ for all $b$. Therefore $1$ is the only smallest element.

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The problem gets a bit more interesting if we consider $\Bbb N\setminus\{1\}$ instead of $\Bbb N$