Linear algebra in lattices, the Fitting lemma

Lattice theoretical generalizations of some classical linear algebra results are formulated. A vector space is replaced by its subspace lattice and a linear map is replaced by the induced lattice map. This map is a complete join homomorphism and satisfies some additional natural conditions. The object of our study is a pair consisting of a complete lattice and one of its complete join endomorphisms. We prove the Fitting lemma for such a pair.