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It is proved by an order theoretical and purely algebraic method that any order bounded orthosymmetric bilinear operator $b:E\times E\to F$ where $E$ and $F$ are Archimedean vector lattices is symmetric. This leads to a new and short proof of the commutativity of Archimedean almost $f$-algebras.

The main topic of the first section of this paper is the following theorem: let $A$ be an Archimedean $f$-algebra with unit element $e$, and $TA\to A$ a Riesz homomorphism such that $T^2\left(f\right)=T\left(fT\left(e\right)\right)$ for all $f\in A$. Then every Riesz homomorphism extension $\tilde{T}$ of $T$ from the Dedekind completion $A^{\delta }$ of $A$ into itself satisfies ${\tilde{T}}^2\left(f\right)=\tilde{T}\left(fT\left(e\right)\right)$ for all $f\in A^{\delta }$. In the second section this result is applied in several directions. As a first application it is applied to show a result about extensions of positive projections to the Dedekind completion. A second application...

Let $E$ be a Riesz space, $F$ a Hausdorff topological vector space (t.v.s.). We prove, under a certain separation condition, that any orthosymmetric bilinear map $T:E\times E\to F$ is automatically symmetric. This generalizes in certain way an earlier result by F. Ben Amor [On orthosymmetric bilinear maps, Positivity 14 (2010), 123–134]. As an application, we show that under a certain separation condition, any orthogonally additive homogeneous polynomial $P:E\to F$ is linearly represented. This fits in the type of results by...