Some remarks on orthogonally additive operators on Riesz spaces

TL;DR

This paper investigates properties of orthogonally additive operators on Riesz spaces, providing conditions for boundedness, lattice operations, and disjointness preservation without assuming Dedekind completeness.

Contribution

It introduces new necessary and sufficient conditions for boundedness and lattice operations of orthogonally additive operators on Riesz spaces without Dedekind completeness.

Findings

Characterizes when orthogonally additive operators are laterally-to-order bounded.

Provides conditions for the existence of lattice operations like supremum and infimum.

Establishes an analogue of Meyer's theorem for disjointness preserving operators.

Abstract

We study orthogonally additive operators between Riesz spaces without the Dedekind completeness assumption on the range space. Our first result gives necessary and sufficient conditions on a pair of Riesz spaces $(E,F)$ for which every orthogonally additive operator from $E$ to $F$ is laterally-to-order bounded. Second result provides sufficient conditions on a pair of orthogonally additive operators $S$ and $T$ to have $S \vee T$, as well as to have $S \wedge T$, and consequently, for an orthogonally additive operator $T$ to have $T^+$, $T^-$ or $|T|$ without any assumption on the domain and range spaces. Finally we prove an analogue of Meyer's theorem on the existence of modules of disjointness preserving operator for the setting of orthogonally additive operators.