Atomic pre-Riesz spaces and order convergent series of atoms in pre-Riesz spaces

A pre-Riesz space X is an ordered vector space that can be order densely embedded into a vector lattice Y. There are three different concepts of order convergences in pre-Riesz spaces. For certain directed nets, we show that three order convergence types coincide. We prove that a series of positive elements in an Archimedean pre-Riesz space X order converges if and only if it order converges in the vector lattice cover Y of X. Moreover, we show the “additivity” of order summable sets that consist of positive disjoint elements in an Archimedean pre-Riesz space X. In case that X is an Archimedean atomic pervasive pre-Riesz space, positive elements in X can be written as an order sum of a series of positive atoms.