Null semigroup

In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero. If every element of a semigroup is a left zero then the semigroup is called a left zero semigroup; a right zero semigroup is defined analogously.
According to Clifford and Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."

Null semigroup
Let S be a semigroup with zero element 0. Then S is called a null semigroup if for all x and y in S we have xy = 0.

Cayley table for a null semigroup

Let S = { 0, a, b, c } be a null semigroup. Then the Cayley table for S is as given below:

Left zero semigroup

A semigroup in which every element is a left zero element is called a left zero semigroup. Thus a semigroup S is a left zero semigroup if for all x and y in S we have xy = x.

Cayley table for a left zero semigroup

Let S = { a, b, c } be a left zero semigroup. Then the Cayley table for S is as given below:

Right zero semigroup

A semigroup in which every element is a right zero element is called a right zero semigroup. Thus a semigroup S is a right zero semigroup if for all x and y in S we have xy = y.

Cayley table for a right zero semigroup

Let S = { a, b, c } be a right zero semigroup. Then the Cayley table for S is as given below:

Properties
A non trivial null (left/right zero) semigroup does not contain an identity element. It follows that the only null(left/right zero) monoid is the trivial monoid.

The set of null semigroup is:
closed under taking subsemigroup
closed under taking quotient of subsemigroup
closed under arbitrary direct product. 

It follows that the set of null (left/right zero) semigroup is a variety of universal algebra, and thus a variety of finite semigroups. The variety of finite null semigroups is defined by the identity ab = cd.

References

Category:Semigroup theory