Null semigroup

[2] According to A. H. Clifford and G. B. Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations.

Then S is called a null semigroup if xy = 0 for all x and y in S. Let S = {0, a, b, c} be (the underlying set of) a null semigroup.

Then the Cayley table for S is as given below: A semigroup in which every element is a left zero element is called a left zero semigroup.

Then the Cayley table for S is as given below: A non-trivial null (left/right zero) semigroup does not contain an identity element.

The variety of finite null semigroups is defined by the identity ab = cd.