null semigroup

A left zero semigroup is a semigroup in which every element is a left zero element. In other words, it is a set $S$ with a product defined as $xy=x$ for all $x,y\in S$.

A right zero semigroup is defined similarly.

Let $S$ be a semigroup. Then $S$ is a null semigroup if it has a zero element and if the product of any two elements is zero. In other words, there is an element $\theta \in S$ such that $xy=\theta$ for all $x,y\in S$.