Tight representations of semilattices and inverse semigroups

By a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{S}$\end{document} in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of Booleanness present in the semilattice of idempotents of  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{S}$\end{document} . After observing that the Vagner–Preston representation is not tight, we exhibit a canonical tight representation for any inverse semigroup with zero, called the regular tight representation. We then tackle the question as to whether this representation is faithful, but it turns out that the answer is often negative. The lack of faithfulness is however completely understood as long as we restrict to continuous inverse semigroups, a class generalizing the E*-unitaries.