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In mathematics, a semigroup with no elements 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.
References
- ↑ A. H. Clifford; G. B. Preston (1964). The Algebraic Theory of Semigroups. Vol. I (2nd ed.). American Mathematical Society. ISBN 978-0-8218-0272-4.
- ↑ P. A. Grillet (1995). Semigroups. CRC Press. pp. 3–4. ISBN 978-0-8247-9662-4.
- ↑ Howie, J. M. (1976). An Introduction to Semigroup Theory. LMS Monographs. Vol. 7. Academic Press. pp. 2–3.