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Several of Levi's commutator theorems in group theory (see page 99 in [1]) are generalized to cancellative semigroups. The theorems involve associativity, distributivity, and class two nilpotence of commutator expressions. First we rewrite the concept of 2-nilpotency as a semigroup law [2] and then the new semigroup theorems are proved within the equational theory of semigroups with cancellation. The proofs, found by automated deduction, support the CS-conjecture that if a certain type of statement is provable in group theory, then it is also provable in cancellative semigroups without the facility of having the unary inverse and the nullary identity element [3]. Yang Zhang and I are currently working on similar ideas generalizing certain commutativity theorems from fields or skew fields to semirings.

1. A. G. Kurosh, The Theory of Groups, Volume 1. Chelsea, New York, 1956.
2. B.H. Neumann, Some Semigroup Laws in Groups, Canad. Math. Bull. Vol. 44 (1), 2001 pp. 93\22696
3. R. Padmanabhan, W. McCune and R. Veroff. Levi's Commutator Theorems for Cancellative Semigroups. Semigroup Forum, 71:152-157, 2005.