In Wikipedia, well-order is defined as a strict total order on a set $S$ with the property that every non-empty subset of $S$ has a least element in this ordering.
But then later, well-order is defined as a total order on $S$ with the property that every non-empty subset of $S$ has a least element in this ordering.
As far as I know, a total order and a strict total order are different. One is not the other. So I was wondering if well-order is defined for total order or strict total order or both? If for both, are they equivalent in the sense that if a total order is well-order, then its corresponding strict total order is also well-order? Vice versa?
- At the same Wikipedia page, it also says "a well-ordering is a well-founded strict total order". As I clicked into the definition of Well-founded_relation, it says "a binary relation, $R$, is well-founded (or wellfounded) on a class $X$ if and only if every non-empty subset of $X$ has a minimal element with respect to $R$". As minimal element is defined for partial order not for strict total order, is it true that well-founded order is a partial order and not a strict total order? So the aforementioned "a well-ordering is a well-founded strict total order" is not well-stated?
Thanks and regards!