I am unable to get the difference between a well ordered set and a totally ordered set ,I have gone through book , it says that if some non-empty subset of a poset has a least element then it is a well-ordered set but this least element can only be found in the relation less than equal to , we can't find it in a relation like "x divides y ", so then what is the significance of the term least here ?
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$\begingroup$In a totally ordered set $P$, every pair of elements is comparable i.e. if we have $a,b\in P$, then $a\le b$ or $b\le a$ holds.
In contrast, a well-ordered set is a totally ordered set with an additional property that every subset $W$ of $P$ contains a smallest element $s\in W$ in the sense that for any $a\in W$ we have $s\le a$.
$\endgroup$ 5 $\begingroup$Each well ordered set is totally ordered (apply the definition of well order to the two point set $\{a,b\}$) but the converse is not true: for example consider the reals $\mathbb{R}$ with the standard ordering: then $(\mathbb{R},\leq)$ is totally ordered but is not well ordered since $(0,1)$ has no smallest element. The theorem of Zermelo states that each set can be well ordered but this theorem is equivalent to the axiom of choice: therefore most well orderings are quite exotic (at least for uncountable sets) and it is impossible to construct it concretely.
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