Can the class of equality relations be axiomatized by just one elementary sentence?

Question

This is a follow-up to my previous question, here: Can equivalence relations be axiomatized using just one elementary sentence?. Referring back to that question, I define an elementary sentence to be an atomic sentence, a negated atomic sentence, or finite disjunctions of the previous two. Now for the question proper. Suppose we are working in first-order logic with equality, along with a binary relation symbol $R$. The class $C$ of equality relations can be axiomatized by the biconditional $xRy \leftrightarrow x=y$. However, that is not an elementary sentence. We can break it up into the two sentences $xRy \rightarrow x=y$ and $x=y \rightarrow xRy$ and then rewrite them as $\neg xRy \vee x=y$ and $x \neq y \vee xRy$, which are elementary sentences. So, that shows that you need at most two elementary sentences to axiomatize the class of equality relations. My question is, do you need at least two elementary sentences? Or can you get away with just one elementary sentence to axiomatize the class of equality relations?

Answer 1

Minor quibble: when you define "elementary sentence," stricty speaking you are conflating a formula with its universal closure. This is a fairly common abuse of terminology, so I'm just going to follow it in the rest of this answer, but it's worth being aware of.

Any single elementary sentence $\varphi$ in the language $\{R\}$ (with equality) will either be satisfied in every structure where $R$ is interpreted as the empty relation or will be satisfied in every structure where $R$ is interpreted as the complement of the empty relation: consider whether $\varphi$ contains a non-negated instance of $R$ as a disjunct. So two such sentences are indeed necessary.