Isomorphisms and strong finite projec- tive classes of commutative semigroups

In “Sverdlovsk notebook” (Sverdlovsk, 1969), I proposed a question: Are any too first-order equivalent finitely generated commutative semigroups isomorphic? In 1970, B.I.Zilber answered the question negatively. A question arises: In what language, any equivalent over the language finitely generated commutative semigroups are isomorphic? In the note, we propose such a language. Moreover, we prove that there is an algorithm which for a given finite set of generators, a given finite set of defining relations of a commutative semigroup for the generators, and a closed formula of the language decides whether the formula holds in the semigroup. 1