A topology on the Fremlin tensor product of locally convex-solid vector lattices

Suppose that E and F are Banach lattices. It is known that there are several norms on the Fremlin tensor product E circle times<overline>F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E{\overline{\otimes }} F$$\end{document} that turn it into a normed lattice; in particular, the projective norm |pi|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\pi |$$\end{document} (known as the Fremlin projective norm) and the injective norm |& varepsilon;|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\epsilon |$$\end{document} (known as the Wittstock injective norm). Now, assume that E and F are locally convex-solid vector lattices. Although we have a suitable vector lattice structure for the tensor product E and F (known as the Fremlin tensor product and denoted by E circle times<overline>F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E{\overline{\otimes }}F$$\end{document}), there is a lack of topological structure on E circle times<overline>F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E{\overline{\otimes }}F$$\end{document}, in general. In this note, we consider a linear topology on E circle times<overline>F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E{\overline{\otimes }}F$$\end{document} that makes it into a locally convex-solid vector lattice, as well; this approach can be taken as a generalization of the projective norm of the Fremlin tensor product between Banach lattices.