- ordering relation
- A partial ordering on a set is a relation < that is transitive and reflexive and antisymmetric. That is, (i)
*x*<*y*&*y*<*z*→*x*<*z*; (ii)*x*<*x*; (iii)*x*<*y*&*y*<*x*→*x*=*y*. If we add (iv) that at least one of*x*<*y*,*x*=*y*, and*y*<*x*holds (the relation is connected, or, all elements of the set are comparable), then the ordering is a total ordering (intuitively, the elements can be arranged along a straight line); otherwise it is a partial ordering. A well-ordering is an ordering such that every non-empty subset of the set contains a minimal element, that is, some element m such that there is no*x*? m in the set such that*x*< m. A well-ordering on a set A is a linear ordering with the property that every nonempty subset of A has a minimal element.

*Philosophy dictionary.
Academic.
2011.*