On a Theorem of Šutov

Sava Krstić

This note deals with formulas occurring in Mal'cev's and Šutov's axiomatizations of the class of semigroups embeddable in a group. Assuming $\alpha$ and $\beta$ are schemes as defined by Mal'cev and $T(\alpha)$, $T(\beta)$ corresponding Mal'cev quasi-identities and $T(\beta,x)$ the Šutov quasi-identity arising from $T(\beta)$ it is proved that there exists a semigroup on which $T(\beta,x)$ is true and $T(\alpha)$ is not whenever $\alpha$ is irreducible and $|\alpha| > |\beta|/2+2$.