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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The fact that e.g. $A={3,5}\subset\Bbb N$ does not have a least element with respect to the partial order "divides" (i.e., there does not exists an element $m\in A$ such that $m\mid a$ for all $a\in A$) shows that $\mathbb N$ is not well-ordered by "divides" (in fact, not even totally ordered). – Hagen von Eitzen Oct 04 '15 at 14:01
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Example of a well-ordered set: Define an order $<$ such that $0<1<2<\dotsb<\omega<\omega+1<\omega+2<\dotsb$. Then the set $S={0,1,2,\dotsb,\omega,\omega+1,\omega+2,\dotsb}$ is well-ordered. (Note that $\omega-1\notin S$.) It's well-ordered because every (nonempty) subset has a least element. – Akiva Weinberger Oct 04 '15 at 14:08
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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 $s\le a$.
Rishabh Gupta
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BigbearZzz
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@HagenvonEitzen Misread your criticism. You are absolutely right. – Akiva Weinberger Oct 04 '15 at 14:05
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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.
truebaran
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