# The equality-free first-order theory of antisymmetric relations and of coreflexive relations

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?

Hint: if $$xRy$$, $$xRz$$ and $$yRx$$ hold and $$R$$ is weakly antisymmetric, what can you say about $$yRz$$?