ORDERING RELATION

ordering relation: translation

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 ofx<y,x=y, andy<xholds (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 nox? m in the set such thatx< 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.