Lesson plan
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. 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 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.
Article summary
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 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.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.