A class of join-completions of partially ordered semigroups

We introduce the notions of the join-completions of a partially ordered semigroup S and the weakly consistent nuclei on the power-set P(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {P}(S)$$\end{document}, and prove that the join-completions of a partially ordered semigroup S up to isomorphism are completely determined by the weakly consistent nuclei on P(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {P}(S)$$\end{document}. Then we provide the largest join-completion and show that the least join-completion does not exist in general. Finally, the universal property of the join-completions is investigated.