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

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