World Library  
Flag as Inappropriate
Email this Article

Empty semigroup

Article Id: WHEBN0023004136
Reproduction Date:

Title: Empty semigroup  
Author: World Heritage Encyclopedia
Language: English
Subject: Semigroup, Zero object (algebra), Special classes of semigroups, Semigroup with two elements, Trivial semigroup
Publisher: World Heritage Encyclopedia

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.

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.

See also


This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.