Consider the class of groups in the signature {\*}. Is the quasi-equational theory of that class axiomatized by the associative law and the left and right cancellative laws?



Is the quasi-equational theory of groups the same as cancellative semigroups?
No.


Every group satisfies 


$$
(xy\approx x'y')\wedge (zy\approx z'y')\wedge (zw\approx z'w')\to (xw\approx x'w')
$$


but this quasi-identity is not derivable from associativity + cancellativity. You can find this in Maltsev's papers *Uber die Einbettug von assoziativen Systemen in Gruppen* parts I and II. In fact, Maltsev proves that the quasi-equational theory of groups in the signature $\{\*\}$ is not finitely axiomatizable relative to the theory axiomatized by the associative law and the cancellation laws.


