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:
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".