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.

与闭凸锥上的投影相关的问题在优化理论及相关领域中经常遇到。为了研究这些问题，引入了各种统一思想，包括不对称向量值范数和某些广义的类格运算。我们通过描述如何使用本文中开发的序论形式来表述圆锥投影问题，提出了这些研究的新视角。底层数学结构是偏序向量空间，它通过使用两个偏序并具有相对于这些排序的某些晶格类型属性来概括向量格的概念。在这篇文章中，我们介绍了这些所谓的混合晶格空间的概括，并且我们展示了这种结构如何在上述的一些应用中非常自然地出现。