An elementary example of an order bound dual space that is not directed (Q850558)

Let \(E\) be a directed partially ordered vector space. A linear functional from \(E\) into \(\mathbb{R}\) is called order bounded if it maps order bounded sets to order bounded sets; it is called positive if it is positive on the positive cone. In the present paper, the author gives a simple example of an order bounded linear functional on a directed partially ordered vector space which is not equal to the difference of two positive linear functionals.