An AE axiomatization of groups

Question

Let $L=\{*\}$. The usual axiomatization of groups in this language has the EA axiom $\exists{e}\forall{x}$ $ e*x = x$. But the union of a chain of groups is also a group. This means that the theory of groups has an AE axiomatization. My question is, what is a system of AE axioms for groups.

I want to replace the above axiom with $\exists{y}$ ${y^2=y}$. But I have had trouble showing that the axiom system leads to the theory of groups.

Answer 1

Say you add the axiom $\exists e\,\, e^2=e$ and also cancellation laws $ab=ac\implies b=c$ and $ba=ca\implies b=c$. Then $xe^2=xe$, which implies $xe=x$ by cancellation.

Answer 2

Another cute axiomatization is

• $\exists x\, x = x$ (just to rule out the empty structure, if you allow it)
• $\forall x\,\forall y\,\forall z\,(xy)z = x(yz)$
• $\forall x\,\forall y\,\exists z\, xz = y$
• $\forall x\,\forall y\,\exists z\, zx = y$

It's a bit more complicated to show that every model of these axioms is a group, see this question.