Empty semigroup

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title: "Empty semigroup" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["algebraic-structures", "semigroup-theory"] description: "Semigroup containing no elements" topic_path: "general/algebraic-structures" source: "https://en.wikipedia.org/wiki/Empty_semigroup" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Semigroup containing no elements ::

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. However not all authors insist on the underlying set of a semigroup being non-empty. 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 elements is an inverse semigroup, since the necessary condition is vacuously satisfied.

References

References

1. [[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
2. Howie, J. M.. (1976). "An Introduction to Semigroup Theory". Academic Press.
3. P. A. Grillet (1995). ''Semigroups''. [[CRC Press]]. {{ISBN. 978-0-8247-9662-4 pp. 3–4

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