In mathematics a '''null semigroup''' (also called a '''zero semigroup''') is a semigroup with a zero element in which the product of any two elements is zero

## Zero element in a semigroup

An element

*z* in a

semigroup *S* is called a

**zero element** of

*S* if for all

*a* in

*S* we have

*az* =

*za* =

*z* Zero element in a

semigroup if it exists is unique and it is usually denoted by 0.

*The zero element* of a

semigroup *S* if it exists is also called

*the zero* of

*S*If a

semigroup *S* has no zero element we can construct a

semigroup *S*^{0} with zero as follows: Let

*S*^{0} = S ∪ { 0 } and define a binary operation

* in *S

^{0} as follows:*For all *a* *b* in *S^{0} set a* . *b* = *ab* if *a* ≠ 0, *b* ≠ 0 and *a* . *b* = 0 otherwise*

S

^{0} with this binary operation is a semigroup with zero element the zero element being 0.

## Null semigroup

Let S

* be a semigroup with zero element 0. Then *S

* is called a *null semigroup

* if the following condition is satisfied:
**
**For all *x* *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:*

Cayley table for a null semigroup | 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 element and right zero element in a semigroup

An element *z* in a semigroup *S* is called a left zero element** of ***S* if for all *a* in *S* we have *za* = *z* Left zero element in a semigroup if it exists need not be unique A *left zero element* of a semigroup *S* if it exists is also called a *left zero* of *S*

An element *z* in a semigroup *S* is called a right zero element** of ***S* if for all *a* in *S* we have *za* = *z* Right zero element in a semigroup if it exists need not be unique A *right zero element* of a semigroup *S* if it exists is also called a *right zero* of *S*

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:

Cayley table for a left zero semigroup | *a* | *b* | *c* |
---|

a | *a* | *a* | *a* |

b | *b* | *b* | *b* |

c | *c* | *c* | *c* |

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:

Cayley table for a right zero semigroup | *a* | *b* | *c* |
---|

a | *a* | *b* | *c* |

b | *a* | *b* | *c* |

c | *a* | *b* | *c* |

## End note

"In spite of their triviality these semigroups arise naturally in a number of investigations"

## References