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.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}} {{cite book|first = J. M.|last=Howie|authorlink = John Mackintosh Howie|title=An Introduction to Semigroup Theory|publisher=Academic Press|date=1976|series=L.M.S.Monographs|volume=7}} pp. 2–3 However not all authors insist on the underlying set of a semigroup being non-empty.P. A. Grillet (1995). Semigroups. CRC Press. {{ISBN|978-0-8247-9662-4}} pp. 3–4 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 {{nowrap|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.