Are A. Malcev's conditions first order?

Question

There is a Russian paper by A. Malcev written in 1939 that give infinitely many jointly necessary and sufficient conditions for a (not necessarily commutative) monoid to be embeddable. Are those conditions first-order? If yes, this would answer my question of whether group-embeddable monoids are an elementary class.

Answer 1

The answer appears to be yes -- see here. For example, the first three conditions are:

1. $(\forall a,b,c,c')(ac=ac' \Rightarrow bc'=bc')$.
2. $(\forall a,a',b,c)(ab=a'b \Rightarrow ac=a'c)$.
3. $(\forall a,a',b,b',c,c',d,d')(ab=cd \land ab'=cd'\land a'b'=c'd' \Rightarrow a'b=c'd)$.