Nilsemigroup Explained

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...} x _{n=y} _{1...} 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 (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\inN} \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

• Book: Pin. Jean-Éric. Mathematical Foundations of Automata Theory. 2018-06-15. 198.
• Book: Grillet. P A. Semigroups. 1995. CRC Press. 978-0-8247-9662-4. 110.