CORRECTION

In the article by L. N. Shevrin, “On semi-isomorphisms and structural isomorphisms of semigroups with the cancellation law” (DAN, vol. 171, no. 2, pp. 296–298), through an oversight an erroneous version of the first paragraph, including Theorem 1, was printed. The correct text is as follows:

As in (¹), a one-to-one mapping $\varphi$ of a semigroup $\Gamma$ into some semigroup will be called a semi-isomorphism if, for any $x, y \in \Gamma$, at least one of the equalities $\varphi(xy)=\varphi(x)\varphi(y)$, $\varphi(xy)=\varphi(y)\varphi(x)$ holds. In the work cited, Scott proved that every semi-isomorphism of a semigroup with the cancellation law $\Gamma$ into a semigroup with the cancellation law $\Gamma'$ is an isomorphism or an anti-isomorphism. In the proof, the validity of the cancellation law in the semigroup $\Gamma'$ was used essentially; therefore, naturally, the question arose whether it is possible to dispense with the latter requirement. There are examples showing that, generally speaking, this cannot be done. However, if one considers semi-isomorphisms onto a semigroup, then the fulfillment of the cancellation law in the semi-isomorphic image is immaterial, and the following stronger assertion is true than the one obtained from Scott’s result mentioned above by replacing in it the word “into” by the word “onto”:

Theorem 1. Every semi-isomorphism of a semigroup with the cancellation law onto an arbitrary semigroup is an isomorphism or an anti-isomorphism.