A set S on which the relation < is defined, satisfying the following properties:

1. (a) given x,y,z in S,

   $$\text{if}x<y\text{and}y<z,\text{then}x<z$$

2. (b) given x,y in S, then exactly one of the following three possibilities is true:

   $$x<y,x=y,\text{or}y<x$$

3. (c) if T is any nonempty subset of S, then there exists an element x in T such that

   $$\begin{array}{l}x=y\text{or}x<y,\\ \text{i.e.}x\le y\text{for}\text{all}y\text{in}T\end{array}$$