Null semigroup
In mathematics, a null 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 A. H. Clifford and G. B. 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 xy = 0 for all x and y in S.Cayley table for a null semigroup
Let S = be a null semigroup. Then the Cayley table for S is as given below:| 0 | a | b | c | |
| 0 | 0 | 0 | 0 | 0 |
| a | 0 | 0 | 0 | 0 |
| b | 0 | 0 | 0 | 0 |
| c | 0 | 0 | 0 | 0 |
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 = be a left zero semigroup. Then the Cayley table for S is as given below:| a | b | c | |
| a | a | a | a |
| b | b | b | b |
| c | c | c | c |
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 = be a right zero semigroup. Then the Cayley table for S is as given below:| a | b | c | |
| a | a | b | c |
| b | a | b | c |
| c | a | b | c |
Properties
A non-trivial null semigroup does not contain an identity element. It follows that the only null monoid is the trivial monoid. On the other hand, a null semigroup with an identity adjoined is called a find-unique monoid.The class of null semigroups is:
- closed under taking subsemigroups
- closed under taking quotient of subsemigroup
- closed under arbitrary direct products.