Orthogonal Additivity of a Product of Powers of Linear Operators

In this note it is established that a finite family of positive linear operators acting from an Archimedean vector lattice into an Archimedean $f$-algebra with unit is disjointness preserving if and only if the polynomial presented in the form of the product of powers of these operators is orthogonally additive. A similar statement is established for the sum of polynomials represented as products of powers of positive operators.