Homotonic Algebras

An algebra $\mathcal{A}$ of real or complex valued functions defined on a set $\mathbf{T}$ shall be called $\mathit{\text{homotonic}}$ 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 $\mathrm{\mid }f\times g\mathrm{\mid }\le \mathrm{\mid }f\mathrm{\mid }\times \mathrm{\mid }g\mathrm{\mid }$. 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.