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ABSTRACT. The diagonal of the product of two triangular matrices is the product of the diagonals of each matrix. This idea is used to characterize partially ordered linear algebras which have order properties similar to an algebra of real triangular matrices.

We show that the range of a positive derivation on a Dedekind o-complete partially ordered linear algebra with an order unit is a set of generalized nilpotents. With additional assumptions on the algebra, we show that the algebra has an important property similar to a property of the algebra of upper triangular matrices.

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The diagonal of the product of two triangular matrices is the product of the diagonals of each matrix. This idea is used to characterize Dedekind o-complete lattice ordered linear algebras which admit isotone functions with familiar functional and order properties as possessed by the real-valued logarithm or root function.