The first two chapters provide the necessary prerequisites. In the third and fourth chapters we demonstrate than an affine Hjelmslev plane (or A. H. plane) is coordinatized by a biternary ring; and that given a biternary ring, one can construct an affine Hjelmslev plane. In the fifth and sixth chapters we introduce the notions of an ordering of an A. H. plane and an ordering of a biternary ring. In the seventh chapter we show that an ordering of an A. H. plane H induces an ordering on the coordinate biternary ring. In the eighth chapter we show that a given ordering of a biternary ring M induces an ordering on the A.H. plane constructed over M. In the remaining chapters we examine the associated ordinary affine plane of an A. H. plane, the case where an A. H. plane is Desarguesian, and give an example of an ordered non-Desarguesian A. H. plane. / Thesis / Master of Science (MSc)