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In 1977, Colville, Davis, and Keimel [Positive derivations on f-rings, J. Aust. Math. Soc. Ser. A 23 1977, 3, 371–375] proved that a positive derivation on an Archimedean f-algebra A has its range in the set of nilpotent elements of A. The main objective of this paper is to obtain a generalization of the above Colville, Davis and Keimel result to general derivations. Moreover, we give a new version of the Singer–Wermer conjecture for the class of second-order derivations acting on uniformly complete almost f-algebras. [ABSTRACT FROM AUTHOR]

Orthogonality of homogeneous orthogonally additive polynomials and orthosymmetry of multilinear maps are in general independent properties. In this paper, we show, for some important vector lattices, that the vector space of all orthogonally additive scalar-valued polynomials and the vector space of orthosymmetric scalar-valued multilinear maps are isomorphic. [ABSTRACT FROM AUTHOR]