First-order axiomatization of rough mereology

In this paper, I will look into a possible first-order axiomatization of rough mereology and then show that if the domain is finite, such an axiomatization is in fact equivalent to a theory which is axiomatizable by some mereological axioms that can be found in the literature. From such a result it can be seen that any model of the theory considered forms a finite Boolean algebra. Besides, based on the same result, I will also make some observations about the conditions on a rough inclusion function which can be used to define rough mereology.