On positively complemented subspaces of c0

It has been proved, see [1], that a closed infinite dimensional subspace of c0 is isomorphic to c0 if and only if it is the range of a bounded linear projection. In [6] we proved half of the order-theoretic analogue of this result. In fact we showed that an infinite dimensional subspace of c0 which is the range of a positive projection is order-isomorphic to c0. We left open the question whether the converse holds also true. In this paper we answer this question negatively by providing an example in Section 4. In Section 3 we give necessary and sufficient conditions in order that an ordered-subspace of c0 be the range of a positive projection.