On Compact Orthogonally Additive Operators

In this article we explore orthogonally additive (nonlinear) operators in vector lattices. First we investigate the lateral order on vector lattices and show that with every element $$e$$ of a $$C$$ -complete vector lattice $$E$$ is associated a lateral-to-order continuous orthogonally additive projection $$\mathfrak{p}_{e}\colon E\to\mathcal{F}_{e}$$ . Then we prove that for an order bounded positive $$AM$$ -compact orthogonally additive operator $$S\colon E\to F$$ defined on a $$C$$ -complete vector lattice $$E$$ and taking values in a Dedekind complete vector lattice $$F$$ all elements of the order interval $$[0,S]$$ are $$AM$$ -compact operators as well.