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I know it has to be a partial order relation in order for it to be a well order relation or total order relation, but what are the differences between them.

SAR
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1 Answers1

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From Wikipedia:

In mathematics, a well-order (or well-ordering or well-order relation) on a set $S$ is a total order on $S$ with the property that every non-empty subset of $S$ has a least element in this ordering.

In particular, a well-order is a total order (which itself is a partial order).

But the other implication is not true: for instance, take the (natural) order on $\mathbb{Z}$. It is a total order, yet not a well-order — since in particular $\mathbb{Z}$ has no least element.

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Clement C.
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  • What would this diagram look like if it included circles for all six of strict well-orders, non-strict well-orders, strict total orders, non-strict total orders, strict partial orders, and non-strict partial orders? – Joe Stephen Jun 03 '18 at 18:26