Notes on the theory of granular lattices

In this paper, the following two open problems in the theory of granular lattices are studied. 1. In a nondistributive complete lattice, is it true that any component of an element with finite width is also of finite width? 2. Give the characterization of the functions which preserve the granule-decomposable property or granule property. It is shown through constructing a counterexample that there is a nondistributive complete lattice in which a component of some element with finite width is not of finite width. Furthermore, it is showed that an isomorphism between complete lattices can preserve the granule-decomposable property and granule property, and some counterexamples are given out to show that the condition of isomorphism is indispensable. So the two open problems list above are completely solved.