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:
Finite nilsemigroups.
Equivalent definitions exists for finite semigroup. A finite semigroup "S" is nilpotent if, equivalently:
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 formula_4 a bounded interval of positive real numbers. For "x", "y" belonging to "I", define formula_5 as formula_6. We now show that formula_7 is a nilsemigroup whose zero is "n". For each natural number "k", "kx" is equal to formula_8. For "k" at least equal to formula_9, "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:
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 formula_12.