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- Empty_semigroup abstract "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. With a more restrictively defined semigroup 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.".
- Empty_semigroup wikiPageID "23004136".
- Empty_semigroup wikiPageRevisionID "574717313".
- Empty_semigroup hasPhotoCollection Empty_semigroup.
- Empty_semigroup subject Category:Algebraic_structures.
- Empty_semigroup subject Category:Semigroup_theory.
- Empty_semigroup subject Category:Zero.
- Empty_semigroup type AlgebraicStructures.
- Empty_semigroup type Artifact100021939.
- Empty_semigroup type Object100002684.
- Empty_semigroup type PhysicalEntity100001930.
- Empty_semigroup type Structure104341686.
- Empty_semigroup type Whole100003553.
- Empty_semigroup type YagoGeoEntity.
- Empty_semigroup type YagoPermanentlyLocatedEntity.
- Empty_semigroup comment "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.".
- Empty_semigroup label "Empty semigroup".
- Empty_semigroup sameAs m.064pyfh.
- Empty_semigroup sameAs Q5374737.
- Empty_semigroup sameAs Q5374737.
- Empty_semigroup sameAs Empty_semigroup.
- Empty_semigroup wasDerivedFrom Empty_semigroup?oldid=574717313.
- Empty_semigroup isPrimaryTopicOf Empty_semigroup.