It is proved by an order theoretical and purely algebraic method that any order bounded orthosymmetric bilinear operator where and are Archimedean vector lattices is symmetric. This leads to a new and short proof of the commutativity of Archimedean almost -algebras.
We prove that on the basis of ZF the ultrafilter theorem and the theorem of Artin-Schreier are equivalent. The latter says that every formally real field admits a total order.
The notions of a preordering and an ordering of a ring R with involution are investigated. An algebraic condition for the existence of an ordering of R is given. Also, a condition for enlarging an ordering of R to an overring is given. As for the case of a field, any preordering of R can be extended to some ordering. Finally, we investigate the class of archimedean ordered rings with involution.
Let be a Riesz space, a Hausdorff topological vector space (t.v.s.). We prove, under a certain separation condition, that any orthosymmetric bilinear map 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 is linearly represented. This fits in the type of results by...