Semitransitive subsemigroups of the singular part of the finite symmetric inverse semigroup

We prove that the minimal cardinality of a semitransitive subsemigroup in the singular part \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal{I}_{n}\setminus \mathcal{S}_{n}$ \end{document} of the symmetric inverse semigroup \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal{I}_{n}$ \end{document} is 2n−p+1, where p is the greatest proper divisor of n, and classify all semitransitive subsemigroups of this minimal cardinality.