| Abstract:
Apart from serving as a measure of non-commutativity, commutators in group
theory play an important role in the actual description of nilpotent groups,
which are closest to being abelian. In view of this, there are several
results in group theory which describe group varieties in the language of
commutators. Thus, F. W. Levi proved that a group is of nilpotent class 2
if and only if the commutator viewed as a binary law of composition is
associative i.e. the group satisfies the law
\( [[x, y], z] = [x, [y, z]] \).
Here we generalize such results for semigroups with cancellation.
Specifically, we prove that for cancellation semigroups admitting
commutators, the following statements are equivalent: nilpotency of class
2, associativity of the commutators, distributivity of the commutator over
the semigroup multiplication and the semigroup law
\( x*y*z*y*x = y*x*z*x*y \).
We also characterize such semigroups using only the language of conjugates,
which incidentally, gives a new characterization for nilpotent groups as
well. For example, we prove that a cancellative semigroup admitting
conjugates is nilpotent class 2 if and only if the semigroup satisfies the
conjugate law
\( x^{(y^{z})}=x^{y} \).
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