Necessary and sufficient conditions for the embeddability of a semigroup in a group

Question

According to wikipedia,

The first set of necessary and sufficient conditions for the embeddability of a semigroup in a group were given in (Malcev 1939).[5] Though theoretically important, the conditions are countably infinite in number and no finite subset will suffice, as shown in (Malcev 1940).[6]

Although very important, Malcev's two papers share a fundamental flaw; they're both written in Russian, and alas, I do not speak Russian.

Question 1. What was Malcev's countably infinite set of (necessary and sufficient) conditions?

Question 2. And what other characterizations have been discovered since 1940?

Answer 1

You can read the Malcev's theory in 2nd vol. of Algebraic theory of semigroups by Clifford and Preston. You will also find there the more late results of Lambeck.

Addendum:

N. Bouleau, Classes de semigroupes immersible dans groupes, Semigroup Forum,6(1973), N.2, pp.100-112.

S.I. Adyan, Defining relations and algorithmic problems for groups and semigroups. Proc. Steklov Inst. Math. 85(1966), 152 p.; translation from Tr. Mat. Inst. Steklov 85(1966), 123 p.