Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4120190101Dominated orthogonally additive operators in lattice-normed spaces2512646667610.15352/aot.1804-1354ENNariman AbasovMoscow Aviation Institute, RussiaMarat PlievSouthern Mathematical Institute of the Russian Academy of Sciences, RussiaJournal Article20180427In this paper we introduce a new class of operators in lattice-normed spaces. We say that an orthogonally additive operator $T$ from a lattice-normed space $(V,E)$ to a lattice-normed space $(W,F)$ is dominated if there exists a positive orthogonally additive operator $S$ from $E$ to $F$ such that $ls Tx rsleq Sls<br /> xrs$ for any element $x$ of $(V,E)$. We show that under some mild conditions, a dominated orthogonally additive operator has an exact dominant and obtain formulas for calculating the exact dominant of a dominated orthogonally additive operator. In the last part of the paper we consider laterally-to-order continuous operators. We prove that a dominated orthogonally additive operator is laterally-to-order continuous if and only if the same is its exact dominant.http://www.aot-math.org/article_66676_5dcc9862c75677d3f4f27bea043fd952.pdf