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Let $𝕂$ be the field of real or complex numbers. In this note we characterize all inner product norms $p_1,...,p_m$ on ${𝕂}^n$ for which the norm $x\mapsto max\{p_1\left(x\right),...,p_m\left(x\right)\}$ on ${𝕂}^n$ is monotonic.

Let the spaces ${𝐑}^m$ and ${𝐑}^n$ be ordered by cones $P$ and $Q$ respectively, let $A$ be a nonempty subset of ${𝐑}^m$, and let $f:A⟶{𝐑}^n$ be an order-preserving function. Suppose that $P$ is generating in ${𝐑}^m$, and that $Q$ contains no affine line. Then $f$ is locally bounded on the interior of $A$, and continuous almost everywhere with respect to the Lebesgue measure on ${𝐑}^m$. If in addition $P$ is a closed halfspace and if $A$ is connected, then $f$ is continuous if and only if the range $f\left(A\right)$ is connected.

Let $R$ be an Archimedean partially ordered ring in which the square of every element is positive, and $N\left(R\right)$ the set of all nilpotent elements of $R$. It is shown that $N\left(R\right)$ is the unique nil radical of $R$, and that $N\left(R\right)$ is locally nilpotent and even nilpotent with exponent at most $3$ when $R$ is 2-torsion-free. $R$ is without non-zero nilpotents if and only if it is 2-torsion-free and has zero annihilator. The results are applied on partially ordered rings in which every element $a$ is expressed as $a=a_1-a_2$ with positive $a_1$,...

It is shown that a monotone function acting between euclidean spaces $R^n$ and $R^m$ is continuous almost everywhere with respect to the Lebesgue measure on $R^n$.