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:

• [math]\displaystyle{ x_1\dots x_n=y_1\dots y_n }[/math] for each [math]\displaystyle{ x_i,y_i\in S }[/math], where [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ I_n=[a,n] }[/math] a bounded interval of positive real numbers. For x, y belonging to I, define [math]\displaystyle{ x\star_n y }[/math] as [math]\displaystyle{ \min(x+y,n) }[/math]. We now show that [math]\displaystyle{ \langle I,\star_n\rangle }[/math] is a nilsemigroup whose zero is n. For each natural number k, kx is equal to [math]\displaystyle{ \min(kx,n) }[/math]. For k at least equal to [math]\displaystyle{ \left\lceil\frac{n-x}{x}\right\rceil }[/math], 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 [math]\displaystyle{ S=\prod_{i\in\mathbb N}\langle I_n,\star_n\rangle }[/math], where [math]\displaystyle{ \langle I_n,\star_n\rangle }[/math] 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 [math]\displaystyle{ x^\omega y=x^\omega=yx^\omega }[/math].

References

• Pin, Jean-Éric (2018-06-15). Mathematical Foundations of Automata Theory. p. 198. http://www.liafa.jussieu.fr/~jep/PDF/MPRI/MPRI.pdf. 
• Grillet, P A (1995). Semigroups. CRC Press. p. 110. ISBN 978-0-8247-9662-4.