Rigid extensions of algebraic frames

An extension G ≤ H of lattice-ordered groups is said to be a rigid extension if for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h \in H}$$\end{document} there exists a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${g \in G}$$\end{document} such that h⊥⊥ = g⊥⊥. In this paper, we will define rigid extensions and some other generalizations in the context of algebraic frames satisfying the FIP. One of the main results is a characterization of rigid extensions using d-elements of the frame. We also show that a rigid extension between two algebraic frames satisfying the FIP will induce a homeomorphism between their corresponding minimal prime spaces with respect to both the hull-kernel topology and the inverse topology. Moreover, basic open sets map to basic open sets.