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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 concerningthis class. Among these results, we prove that almost all free algebras of$\mathsf{Drs}_{\alpha}$ are atomless, and none of these free algebras containszero-dimensional elements other than zero and top element. The class$\mathsf{Drs}_{\alpha}$ corresponds to first order logic, without equalitysymbol, with $\alpha$-many variables and on relativized semantics. Hence, inthis variation of first order logic, there is no finitely axiomatizable,complete and consistent theory.

First order logic without equality on relativized semantics