This is yet another of my questions regarding first-order logic without equality. It is actually two questions. First, some definitions: An antisymmetric relation on a set $S$ is a binary relation $R$ that satisfies the sentence $\forall x \forall y ((xRy \land yRx) \rightarrow x=y)$. Also, a coreflexive relation on a set $S$ is a binary relation $R$ that satisfies the sentence $\forall x \forall y (xRy \rightarrow x=y)$. My question is, what is an axiomatization of the equality-free first-order theory of both these classes of relations? I conjecture that the equality-free theory of the class of antisymmetric relations can be axiomatized by the sentence $\forall x \forall y ((xRy \land yRx) \rightarrow xRx)$. I also conjecture that the equality-free theory of coreflexive relations can be axiomatized by the two sentences $\forall x \forall y(xRy \rightarrow yRx)$ and $\forall x \forall y \forall z ((xRy \land yRz) \rightarrow xRz)$. Are either or both of these conjecture true?
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Hint: if $xRy$, $xRz$ and $yRx$ hold and $R$ is weakly antisymmetric, what can you say about $yRz$?
