# 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.^{[1]} If every element of the semigroup is a left zero then the semigroup is called a **left zero semigroup**; a **right zero semigroup** is defined analogously.^{[2]}
According to Clifford and Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."^{[1]}

## 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:

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 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:

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 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:

a |
b |
c | |
---|---|---|---|

a |
a |
b |
c |

b |
a |
b |
c |

c |
a |
b |
c |

## References

- 1 2 A H Clifford; G B Preston (1964).
*The algebraic theory of semigroups Vol I*. mathematical Surveys.**1**(2 ed.). American Mathematical Society. pp. 3–4. ISBN 978-0-8218-0272-4. - ↑ M. Kilp, U. Knauer, A.V. Mikhalev,
*Monoids, Acts and Categories with Applications to Wreath Products and Graphs*, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p. 19