Module Arens regularity for semigroup algebras

We extend the concept of Arens regularity of a Banach algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}$\end{document} to the case that there is an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{O}$\end{document} -module structure on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}$\end{document} , and show that when S is an inverse semigroup with totally ordered subsemigroup E of idempotents, then A=ℓ1(S) is module Arens regular if and only if an appropriate group homomorphic image of S is finite. When S is a discrete group, this is just Young’s theorem which asserts that ℓ1(S) is Arens regular if and only if S is finite.