Department of Mathematics
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In a seminal paper, A. I. Malcev showed that the elementary theory of (not necessarily associative) rings and the elementary theory of a certain class of metabelian groups are mutually interpretable. From the proofs, it follows that the commutator [x,y], viewed purely as a binary operation, turns out to be associative in such groups. This was also observed independently by Levi and a complete proof was given by A. G. Kurosh in his volume on group theory. Of course, the commutator [x, y] = x'y'xy is not directly expressible in the language of semigroups. However, we can express it in the form an identity in an enriched language. Earlier, B. H. Neumann has shown that metabelian property can be expressed in the language of semigroups. Here we prove the validity of Levy-Malcev theorem for cancellative semigroups: A cancellative semigroup is metabelian if and only if the binary operation of commutator is associative. We will give both human proofs and automated proofs. It is immediately clear that this kind of theorem generates several open problems. We will mention a few such problems.

1. Kurosh, A. G., Theory of Groups, Volume 1, Chelsea, New York, 1956. Mal'cev, A. I., Some correspondences between rings and groups. (Russian) Mat. Sb. (N.S.) 50 (92) 1960 257–266.
2. Neumann, B. H.; Taylor, Tekla; Subsemigroups of nilpotent groups. Proc. Roy. Soc. Ser. A 274 1963, 1–4.
3. Padmanabhan, R.; McCune, W.; Veroff, R. Levi's commutator theorems for cancellative semigroups. Semigroup Forum 71 (2005), no. 1, 152–157.