On Laterally Continuous Orthogonally Additive Operators

The aim of this article is to consider some problems of the theory of orthogonally additive operators in vector lattices. Order bounded orthogonally additive operators acting between vector lattices were introduced and studied in 1990 by Maz$\acute{o}$n and Segura de Le$\acute{o}$n. Recently, a new class of orthogonally additive operators in vector lattices where the condition of order boundness of an operator is replaced with a much weaker property was investigated by the author of these notes and Ramdane. It is worth to note that today the theory of orthogonally additive operators is an area of the intense study. Let $E$ be a vector lattice and $F$ a real linear space. An operator $T : E \rightarrow F$ is said to be orthogonally additive if $T \left ( x+y \right ) = T \left ( x \right ) + T \left ( y \right )$ whenever $x,y \in E$ are disjoint. Evidently from the definition that $T\left ( 0 \right ) = 0$. It is clear that the set of all orthogonally additive operators is a real vector space with respect to the natural linear operations. Let $E$ and $F$ be vector lattices. We say that an orthogonally additive operator $T:E \rightarrow F$ is positive if $T\left ( E \right ) \subset F_{+}$ and we say that an orthogonally additive operator $T:E \rightarrow F$ is regular if $T = S_{1} - S_{2}$ for some positive orthogonally additive operators $S_{i} : E \rightarrow F, i \in 1,2$. In this paper we investigate the band of laterally continuous operators in the vector lattice of all regular orthogonally additive operators between vector lattices $E$ and $F$. We show that the band which is disjoint to the band generated by all singular orthogonally additive operators coincides with the band of all laterally continuous orthogonally additive operators.
Keywords: Orthogonally additive operator, regular operator, laterally continuous operator, singular operator, orthogonally additive map, vector lattice, lateral ideal.