Nilsemigroup

In mathematics, and more precisely in semigroup theory, a nilsemigroup or nilpotent semigroup is a semigroup whose every element is nilpotent.

Definitions

Formally, a semigroup S is a nilsemigroup if:

S contains 0 and
for each element a∈''S, there exists a positive integer k'' such that a^k=0.

Finite nilsemigroups

Equivalent definitions exists for finite semigroup. A finite semigroup S is nilpotent if, equivalently:

for each, where is the cardinality of S.
The zero is the only idempotent of S.

Examples

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.

Properties

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:

closed under taking subsemigroups
closed under taking quotients
closed under finite products
but is not closed under arbitrary Direct product#Direct [product in universal algebra|direct product]. Indeed, take the semigroup, where is defined as above. The semigroup S is a direct product of nilsemigroups, however its contains no nilpotent element.

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.