Vector spaces of delta-plurisubharmonic functions and extensions of the complex Monge-Ampère operator

In this paper we shall consider two types of vector ordering on the vector space of differences of negative plurisubharmonic functions, and the problem whether it is possible to construct supremum and infimum. Then we consider two different approaches to define the complex Monge-Ampère operator on these vector spaces, and we solve some Dirichlet problems. We end this paper by stating and discussing some open problems.