The lateral order on Riesz spaces and orthogonally additive operators
Let \(E\) be a Riesz space and \(F\) a real vector space. For two elements \(x,y\in E\) we say that \(x\) is a fragment of \(y\) (in symbols \(x\sqsubseteq y\)) if \(x\perp (y-x)\). A function \(T:E\longrightarrow F\) is said to be an orthogonally additive operator if \(T(x+y)=T(x)+T(y)\) for every pair of disjoint elements \(x,y\in E\). The main idea of the present paper is to give a systematic study of the lateral partial order \(\sqsubseteq \) on a Riesz space, especially for the study of orthogonally additive operators. The authors study some notions such as lateral fields, lateral ideals, lateral bands and consistent subsets, showing the importance of these concepts for the theory of orthogonally additive operators. Also, the authors present a positive solution of a problem posed in [\textit{A.~V. Gumenchuk} et al., Mat. Stud. 41, No.~2, 214--219 (2014; Zbl 1321.47115)] concerning the orthogonally additive extension.
scientific article; zbMATH DE number 7345571
scientific article