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:

• ${\displaystyle x_{1}\dots x_{n}=y_{1}\dots y_{n}}$ for each ${\displaystyle x_{i},y_{i}\in S}$, where ${\displaystyle n}$ 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 ${\displaystyle I_{n}=[a,n]}$ a bounded interval of positive real numbers. For x, y belonging to I, define ${\displaystyle x\star _{n}y}$ as ${\displaystyle \min(x+y,n)}$. We now show that ${\displaystyle \langle I,\star _{n}\rangle }$ is a nilsemigroup whose zero is n. For each natural number k, kx is equal to ${\displaystyle \min(kx,n)}$. For k at least equal to ${\displaystyle \left\lceil {\frac {n-x}{x}}\right\rceil }$, 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 ${\displaystyle S=\prod _{i\in \mathbb {N} }\langle I_{n},\star _{n}\rangle }$, where ${\displaystyle \langle I_{n},\star _{n}\rangle }$ 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 ${\displaystyle x^{\omega }y=x^{\omega }=yx^{\omega }}$.

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.