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 ak=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 $$x^\omega y=x^\omega=yx^\omega$$.