A property of meets in slim semimodular lattices and its application to retracts

Slim semimodular lattices were introduced by G. Grätzer and E. Knapp in 2007, and they have intensively been studied since then. These lattices can be given by C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{C}_1$$\end{document}-diagrams, defined by the author in 2017. We prove that if x and y are incomparable elements in such a lattice L, then their meet has the property that the interval [x∧y,x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[x \wedge y, x]$$\end{document} is a chain, this chain is of a normal slope in every C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{C}_1$$\end{document}-diagram of L, and except possibly for x, the elements of this chain are meet-reducible.