  An algebra $\mathcal{A}$ of real or complex valued functions defined on a set
$\mathbf{T}$ shall be called \textit{homotonic} if $\mathcal{A}$ is closed
under forming of absolute values, and for all $f$ and $g$ in $\mathcal{A}$, the
product $f\times g$ satisfies $|f\times g|\le|f|\times|g|$. Our main purpose in
this paper is two-fold: To show that the above definition is equivalent to an
earlier definition of homotonicity, and to provide a simple inequality which
characterizes sub-multiplicativity and strong stability for weighted sup norms
on homotonic algebras.
