I've been studying the possibility of collapsing axioms and I'm stuck trying to prove something (by collapsing two axioms i mean find an axiom equivalent to them).
Take the following characterization (where the ' is the result of applying skolemization to the regularity of the semigroup):
$\begin{align} x(yz)&=(xy)z \tag{A1}\\ xx'x&=x \tag{A2}\\ xx'&=x'x \tag{A3} \end{align}$
I've been trying to prove that (A3) and (A1) cannot be collapsed into a single identity (which I believe to be true since I haven't been able to collapse for a while now) but I'm having a hard time finding any literature on the subject.
So my question is: Do you know any book on the subject of the collapsing of axioms or know a proof of the impossibility of collapsing them (or an identity equivalent to both of them, if my statement is false)?
An example of an axiom that can be collapsed with associativity is $x''=x$.
One can prove that $\begin{align} x(yz)&=(xy)z \tag{B1}\\ xx'x&=x \tag{B2}\\ x''&=x \tag{B3}\\ x'x&=xx' \tag{B4} \end{align}$ Is equivalent to
$\begin{align} x(yz)''&=(xy)z \tag{E1}\\ xx'x&=x \tag{E2}\\ xx'&=x'x \tag{E3}\\ \end{align}$
What I've been trying to do thus far is to find two identities equivalent to (A1) and (A3) that have the same left (/right) member. So something of the sort:
$\begin{align} P&=Q\\ P&=R \end{align}$
I think (although I haven't seen a proof of it) that if such identities exist than you cannot collapse the two initial axioms.
Thanks in advance.
Ps: If you search for "On Single Equational-Axiom Systems for Abelian Groups" you find a proof that abelian groups are representable by a single identity.
Edit(15/03): Is this argument valid?
Any identity expressing both associativity and comutativity of pseudo-inverses can't have x and x' together (like $\textbf{(xx')(yz) = ((xx')y)z}$ ), since one could make a non associative model where xx'=1, so the identity would be true.