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A monomial in finite collection of homogeneous polynomials is a point-wise product of these polynomials each raised to some power. It was established in [14] and [15] that, given a finite set of positive linear operators acting from an Archimedean vector lattice into an f-algebra with multiplicative unit, the corresponding monomial is orthogonally additive polynomial if and only if the collection of these operators is a disjointness preserving set. It is shown in this paper that a similar result is also true in the case when the target space is a uniformly complete vector lattice. Moreover, the result remains valid if we replace the linear operators with homogeneous orthogonally additive polynomials. The correct definition of monomials in homogeneous polynomials is guaranteed by the homogeneous functional calculus and the concavification procedure. [ABSTRACT FROM AUTHOR]

The so-called pseudo-integral multilinear and polynomial operators, defined by random measures, are introduced, and their lattice properties are studied. Radon–Nikodým type theorems for multilinear and polynomial pseudo-integral operators are proved. They generalize a similar result obtained earlier by Grobler, de Pagter, and Rambane for linear pseudo-integral operators. [ABSTRACT FROM AUTHOR]