  The article associates two fundamental lattice constructions with each
regular unital real ordered Banach space (function system). These are used to
establish certain results in the theory of operator algebras, specifically
relating the injective envelope of a separable C*-algebra with its enveloping
von Neumann algebra in a given faithful separable representation. The last
section investigates on lattices of projections arising in injective
C*-algebras and von Neumann algebras and certain nonlinear maps sending
projections to projections which are essentially determined by their values on
positive projections.
