On the countable orthogonal lifting property

Abstract

We proceed from Conrad’s counterexample to countable orthogonal lifting which disproved a positive result by Topping when working over the real field. It is shown that Conrad’s work applies more generally, beginning with an abelian group which is a lower semi-lattice that also contains an element greater than 0. Conrad’s example was of uncountable dimension. It is shown that Conrad-type vector space examples of countable dimension exist when working with a field whose elements are linearly ordered, in particular with the real field. An example shows the failure of the orthogonal lifting property in a finite dimensional vector lattice over the two-element field.