# Trivial semigroup

In mathematics, a **trivial semigroup** (a **semigroup with one element**) is a semigroup for which the cardinality of the underlying set is one. The number of distinct nonisomorphic semigroups with one element is one. If *S* = { *a* } is a semigroup with one element then the Cayley table of *S* is as given below:

a | |
---|---|

a |
a |

The only element in *S* is the zero element 0 of *S* and is also the identity element 1 of *S*.^{[1]} However not all semigroup theorists consider the unique element in a semigroup with one element as the zero element of the semigroup. They define zero elements only in semigroups having at least two elements.^{[2]}^{[3]}

In spite of its extreme triviality, the semigroup with one element is important in many situations. It is the starting point for understanding the structure of semigroups. It serves as a counterexample in illuminating many situations. For example, the semigroup with one element is the only semigroup in which 0 = 1, that is, the zero element and the identity element are equal.
Further, if *S* is a semigroup with one element, the semigroup obtained by adjoining an identity element to *S* is isomorphic to the semigroup obtained by adjoining a zero element to *S*.

The semigroup with one element is also a group.

In the language of category theory, any semigroup with one element is a terminal object in the category of semigroups.

## See also

## References

- ↑ A H Clifford, G B Preston (1964).
*The Algebraic Theory of Semigroups Vol. I*(Second Edition). American Mathematical Society. ISBN 978-0-8218-0272-4 - ↑ P A Grillet (1995).
*Semigroups*. CRC Press. ISBN 978-0-8247-9662-4 pp.3-4 - ↑ J M Howie (1976).
*An Introduction to Semigroup Theory*. L.M.S.Monographs.**7**. Academic Press. pp.2-3