Fragments of orthogonally additive operators and the nonlinear Dodds-Fremlin's theorem

Abstract

In the first part of the paper we describe the structure of the Boolean algebra $F_T$ of all fragments of a positive orthogonally additive operator $T:E\to F$, generalizing classical results of Aliprantis, Burkinshaw, de Pagter to the setting of orthogonally additive (in general nonlinear) operators on Banach lattices. In the second part of the article we present the nonlinear version of the well known Dodds-Fremlin's theorem.