In first order logic without equality, is the theory of equality the same as equivalence relations?

Question

In first order logic without equality with a single binary relation R, is the theory of equality relations the same as the theory of equivalence relations? An equality relation is just equality restricted to a certain set.

Answer 1

Yes. To see this, let $(A,\sim)$ be an equivalence relation. Prove that the map $f\colon (A,\sim)\to (A/\!\sim,=), a\mapsto [a]_\sim$ is elementary (in the logic that does not contain equality). This gives you that $(A,\sim)$ and $(A/\!\sim,=)$ are elementarily equivalent in this logic. Therefore, if $\mathcal K$ is the class of equality relations and $\mathcal K'$ is the class of equivalence relations, then $\operatorname{Th}(\mathcal K)\subseteq\operatorname{Th}(\mathcal K')$. The other inclusion is trivial since $\mathcal K\subseteq\mathcal K'$.