A relation for general and inverse semigroups

The relation in the title is S defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$a\mathcal{S}b \Leftrightarrow a^2 = ab = ba$$ \end{document} on an arbitrary semigroup. We investigate antisymmetry of S by means of a (minimal) family Ϝ whose members can not appear as subsemigroups. Transitivity of S is characterized similarly by means of the family Ϝ and homomorphic images of a certain semigroup. We study the transfer of certain properties of a monoid T and the Bruck semigroup B(T,α) over T. The paper concludes with a consideration of certain properties of the relation S on inverse semigroups.