First order logic without equality on relativized semantics

Let $alphageq 2$ be any ordinal. We consider the class $mathsf{Drs}_{alpha}$ of relativized diagonal free set algebras of dimension $alpha$. With same technique, we prove several important results concerning this class. Among these results, we prove that almost all free algebras of $mathsf{Drs}_{alpha}$ are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class $mathsf{Drs}_{alpha}$ corresponds to first order logic, without equality symbol, with $alpha$-many variables and on relativized semantics. Hence, in this variation of first order logic, there is no finitely axiomatizable, complete and consistent theory.