The paper contains a systematic study of the lateral partial order \(\sqsubseteq \) in a Riesz space (the relation \(x \sqsubseteq y\) means that x is a fragment of y) with applications to nonlinear analysis of Riesz spaces. We introduce and study lateral fields, lateral ideals, lateral bands and consistent subsets and show the importance of these notions to the theory of orthogonally additive operators, like ideals and bands are important for linear operators. We prove the existence of a lateral band projection, provide an elegant formula for it and prove some properties of this orthogonally additive operator. One of our main results (Theorem 7.5) asserts that, if D is a lateral field in a Riesz space E with the intersection property, X a vector space and \(T_0:D\rightarrow X\) an orthogonally additive operator, then there exists an orthogonally additive extension \(T:E\rightarrow X\) of \(T_0\). The intersection property of E means that every two-point subset of E has an infimum with respect to the lateral order. In particular, the principal projection property implies the intersection property.

本文对Riesz空间中的横向偏序\（\ sqsubseteq \）进行了系统的研究（关系\（x \ sqsubseteq y \）表示x是y的一个片段），并应用于Riesz空间的非线性分析。我们介绍并研究了横向场，横向理想，横向带和一致子集，并显示了这些概念对正交加法算子理论的重要性，例如理想和带对于线性算子很重要。我们证明了边带投影的存在，为其提供了一个优雅的公式，并证明了这个正交加法算子的一些性质。我们的主要结果之一（定理7.5）断言，如果D是Riesz空间中的横向场，Ë与交叉点属性，X向量空间和\（T_0：d \ RIGHTARROW X \）的正交添加剂操作者，则存在一个正交添加剂扩展\（T：电子\ RIGHTARROW X \）的\（T_0 \） 。的交点属性Ë意味着每两个点的子集Ê具有相对于横向顺序的下确界。特别地，主投影特性暗示相交特性。