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:
*$x_1\dots x_n=y_1\dots y_n$ for each $x_i,y_i\in S$, where $n$ 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 I_n= ,n/math> a bounded interval of positive real numbers. For ''x'', ''y'' belonging to ''I'', define $x\star_n y$ as $\min(x+y,n)$. We now show that $\langle I,\star_n\rangle$ is a nilsemigroup whose zero is ''n''. For each natural number ''k'', ''kx'' is equal to $\min(kx,n)$. For ''k'' at least equal to $\left\lceil\frac\right\rceil$, ''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. Indeed, take the semigroup $S=\prod_\langle I_n,\star_n\rangle$, where $\langle I_n,\star_n\rangle$ 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 $x^\omega y=x^\omega=yx^\omega$.

References

*
*{{cite book, last1=Grillet, first1=P A, title=Semigroups, date=1995, publisher=CRC Press, isbn=978-0-8247-9662-4, page=110

Semigroup theory