Strongly fully invariant-extending modular lattices

This paper is a natural continuation of our previous joint paper [Albu, Kara, Tercan, Fully invariant-extending modular lattices, and applications (I), J. Algebra517 (2019), 207–222], where we introduced and investigated the notion of a fully invariant-extending lattice, the latticial counterpart of a fully invariant-extending module. In this paper we introduce and investigate the latticial counter-part of the concept of a strongly FI-extending module defined by Birkenmeier, Park, Rizvi (2002) as a module M having the property that every fully invariant submodule of M is essential in a fully invariant direct summand of M. Our main tool in doing so, is again the very useful concept of a linear morphism of lattices introduced in the literature by Albu and Iosif (2013).