(related to Definition: Well-order, Well-ordered Set)
In literature, the above two different versions of well-ordered sets can be found, one for partial orders and one for strict orders. Because partial orders are reflexive, and because strict orders are irreflexive, the alternative definitions require the existence of either a minimum or a minimal element in the subset $S.$ Both definitions have their advantages and disadvantages, but only the latter has the advantage that it is compatible with the definition of well-founded relations, which are very useful in introducing the ordinal numbers, which we learn more about later.
For the time being, we will use the poset version, unless otherwise stated.