Domination problem for orthogonally additive operators in vector lattices

‎The ``Up-and-down theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result in operator theory‎. ‎An analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice was proven‎. ‎This result is used to prove a theorem of domination for $AM$-compact positive abstract Uryson operators from a Dedekind complete vector lattice $E$ to a Banach lattice $F$ with an order continuous norm‎.