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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).

Strongly fully invariant-extending modular lattices