  Let $\alpha\geq 2$ 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.
