Null semigroup

Let S be a semigroup with zero element 0. Then S is called a null semigroup if xy = 0 for all x and y in S. Cayley table for a null semigroup Let S = {0, a, b, c} be (the underlying set of) 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 xy = x for all x and y in S. 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 xy = y for all x and y in S. 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. On the other hand, a null (left/right zero) semigroup with an identity adjoined is called a find-unique (find-first/find-last) monoid. The class of null semigroups is: • closed under taking subsemigroups • closed under taking quotient of subsemigroup • closed under arbitrary direct products. It follows that the class of null (left/right zero) semigroups 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. ==See also==