Formations of orthodox semigroups

Taking formations of groups and of inverse semigroups as the starting point, formations of orthodox semigroups are defined, as well as the wider class of i-formations (i standing for idempotent-separating). The relation between the nature of a class of inverse semigroups F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {F}$$\end{document} [of groups G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}$$\end{document}] and that of certain classes of orthodox semigroups with associated inverse semigroups in F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {F}$$\end{document} [groups in G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}$$\end{document}] is discussed. The product of formations of orthodox semigroups, in particular of R-unipotent semigroups, is considered, and a product like the Gaschütz product known for groups is presented for i-formations. The paper concludes with a list of questions.