The lateral order on Riesz spaces and orthogonally additive operators
type: Item
instance of: scholarly article
title: The lateral order on Riesz spaces and orthogonally additive operators
zbMATH Open document ID: 1476.46002
author: M. M. Popov
author: V. V. Mykhaylyuk
author: M. Pliev
published in: Positivity
publication date: 2021-05-07T00:00:00Z
review text: 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.
reviewed by: Elhadj Dahia
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 47H99
zbMATH DE Number: 7345571
zbMATH Keywords: Riesz spaces
zbMATH Keywords: fragments
zbMATH Keywords: orthogonally additive operators
zbMATH Keywords: laterally continuous operators
MaRDI profile type: Publication
full work available at URL: s11117-020-00761-x
OpenAlex ID: W3025026935
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DOI: 10.1007/S11117-020-00761-X
description: scientific article; zbMATH DE number 7345571