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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. 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.".
- Empty_semigroup wikiPageID "23004136".
- Empty_semigroup wikiPageLength "2025".
- Empty_semigroup wikiPageOutDegree "17".
- Empty_semigroup wikiPageRevisionID "650822092".
- Empty_semigroup wikiPageWikiLink Algebraic_structure.
- Empty_semigroup wikiPageWikiLink Category:Algebraic_structures.
- Empty_semigroup wikiPageWikiLink Category:Nothing.
- Empty_semigroup wikiPageWikiLink Category:Semigroup_theory.
- Empty_semigroup wikiPageWikiLink Category_theory.
- Empty_semigroup wikiPageWikiLink Empty_function.
- Empty_semigroup wikiPageWikiLink Empty_set.
- Empty_semigroup wikiPageWikiLink Field_with_one_element.
- Empty_semigroup wikiPageWikiLink Identity_element.
- Empty_semigroup wikiPageWikiLink Initial_and_terminal_objects.
- Empty_semigroup wikiPageWikiLink Initial_object.
- Empty_semigroup wikiPageWikiLink Inverse_semigroup.
- Empty_semigroup wikiPageWikiLink Mathematics.
- Empty_semigroup wikiPageWikiLink Monoid.
- Empty_semigroup wikiPageWikiLink Semigroup.
- Empty_semigroup wikiPageWikiLink Semigroup_with_one_element.
- Empty_semigroup wikiPageWikiLink Semigroup_with_two_elements.
- Empty_semigroup wikiPageWikiLink Trivial_semigroup.
- Empty_semigroup wikiPageWikiLink Underlying_set.
- Empty_semigroup wikiPageWikiLink Vacuous_truth.
- Empty_semigroup wikiPageWikiLink Vacuously.
- Empty_semigroup wikiPageWikiLinkText "Empty semigroup".
- Empty_semigroup wikiPageWikiLinkText "empty semigroup".
- Empty_semigroup hasPhotoCollection Empty_semigroup.
- Empty_semigroup wikiPageUsesTemplate Template:Reflist.
- Empty_semigroup subject Category:Algebraic_structures.
- Empty_semigroup subject Category:Nothing.
- Empty_semigroup subject Category:Semigroup_theory.
- Empty_semigroup hypernym Semigroup.
- Empty_semigroup type Integer.
- 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 wasDerivedFrom Empty_semigroup?oldid=650822092.
- Empty_semigroup isPrimaryTopicOf Empty_semigroup.