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 $I_{n}=[a,n]$ 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 $\left\lceil {\frac {n-x}{x}}\right\rceil$ , 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 $S=\prod _{i\in \mathbb {N} }\langle I_{n},\star _{n}\rangle$ , 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

Pin, Jean-Éric (2018-06-15). Mathematical Foundations of Automata Theory (PDF). p. 198.
Grillet, P A (1995). Semigroups. CRC Press. p. 110. ISBN 978-0-8247-9662-4.

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