  Problems related to projections on closed convex cones are frequently
encountered in optimization theory and related fields. To study these problems,
various unifying ideas have been introduced, including asymmetric vector-valued
norms and certain generalized lattice-like operations. We propose a new
perspective on these studies by describing how the problem of cone projection
can be formulated using an order-theoretic formalism developed in this paper.
The underlying mathematical structure is a partially ordered vector space that
generalizes the notion of a vector lattice by using two partial orderings and
having certain lattice-type properties with respect to these orderings. In this
note we introduce a generalization of these so-called mixed lattice spaces, and
we show how such structures arise quite naturally in some of the applications
mentioned above.
