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In mathematics, and more precisely in semigroup theory, a nilsemigroup or nilpotent semigroup is a semigroup whose every element is nilpotent.
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (April 2018) |
Formally, a semigroup S is a nilsemigroup if:
Equivalent definitions exists for finite semigroup. A finite semigroup S is nilpotent if, equivalently:
The trivial semigroup of a single element is trivially a nilsemigroup.
The set of strictly upper triangular matrix, with matrix multiplication is nilpotent.
Let a bounded interval of positive real numbers. For x, y belonging to I, define as . We now show that is a nilsemigroup whose zero is n. For each natural number k, kx is equal to . For k at least equal to , kx equals n. This example generalize for any bounded interval of an Archimedean ordered semigroup.
A non-trivial nilsemigroup does not contain an identity element. It follows that the only nilpotent monoid is the trivial monoid.
The class of nilsemigroups is:
It follows that the class of nilsemigroups is not a variety of universal algebra. However, the set of finite nilsemigroups is a variety of finite semigroups. The variety of finite nilsemigroups is defined by the profinite equalities .
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