# Empty semigroup

In mathematics, a **semigroup with no elements** (the **empty semigroup**) is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a *non-empty* set together with an associative binary operation.^{[1]}^{[2]} However not all authors insist on the underlying set of a semigroup being non-empty.^{[3]} One can logically define a semigroup in which the underlying set *S* is empty. The binary operation in the semigroup is the empty function from *S* × *S* to *S*. This operation vacuously satisfies the closure and associativity axioms of a semigroup. Not excluding the empty semigroup simplifies certain results on semigroups. For example, the result that the intersection of two subsemigroups of a semigroup *T* is a subsemigroup of *T* becomes valid even when the intersection is empty.

When a semigroup is defined to have additional structure, the issue may not arise. For example, the definition of a monoid requires an identity element, which rules out the empty semigroup as a monoid.

In category theory, the empty semigroup is always admitted. It is the unique initial object of the category of semigroups.

A semigroup with no element is an inverse semigroup, since the necessary condition is vacuously satisfied.

## 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 - ↑ J M Howie (1976).
*An Introduction to Semigroup Theory*. L.M.S.Monographs.**7**. Academic Press. pp. 2–3 - ↑ P A Grillet (1995).
*Semigroups*. CRC Press. ISBN 978-0-8247-9662-4 pp. 3–4