Big lattices of hereditary and natural classes of linear modular lattices (Q2238020)

The aim of this paper, as its title shows, is to study big lattices of hereditary and natural classes of linear modular lattices. Its main ingredient is the concept of \textit{linear modular lattice}, introduced by \textit{T. Albu} and \textit{M. Iosif} [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 56(104), No. 1, 33--46 (2013; Zbl 1313.06010)]. The first part of Section 2 contains definitions and results taken from the above mentioned paper of Albu and Iosif [loc. cit.]. The authors denote by \(\mathcal{L}_{\mathcal{M}_c}\) the full subcategory of the category of all linear modular bounded complete lattices whose morphisms are linear morphism consisting of all upper semicontinuous modular complete lattices. Among others, they prove that in \(\mathcal{L}_{\mathcal{M}_c}\), the skeleton \(\mathcal{L}_{\mathrm{nat}}\) of the big lattice of all hereditary classes is precisely the class of natural classes. Finally, they show that \(\mathcal{L}_{\mathrm{nat}}\) is a Boolean big lattice. Reviewer's comments: The authors do not mention at all in their paper the monograph [\textit{J. Dauns} and \textit{Y. Zhou}, Classes of modules. Boca Raton, FL: Chapman \& Hall/CRC (2006; Zbl 1108.16001)], where some of their main definitions and results in the module case were introduced and investigated. Notice that what the authors study is a simple lattice counterpart of these module theoretical facts.