First order logic without equality on relativized semantics

Let $\alpha \ge 2$ be any ordinal. We consider the class ${\mathsf{D}\mathsf{r}\mathsf{s}}_{\alpha }$ 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 ${\mathsf{D}\mathsf{r}\mathsf{s}}_{\alpha }$ are atomless, and none of these free algebras contains zero-dimensional elements other than zero and top element. The class ${\mathsf{D}\mathsf{r}\mathsf{s}}_{\alpha }$ 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.