Weak inverse

In the theory of semigroups, a weak inverse of an element x in a semigroup {{nowrap|(S, •)}} is an element y such that {{nowrap|1=y • x • y = y}}. If every element has a weak inverse, the semigroup is called an E-inversive or E-dense semigroup. An E-inversive semigroup may equivalently be defined by requiring that for every element {{nowrap|x ∈ S}}, there exists {{nowrap|y ∈ S}} such that {{nowrap|x • y}} and {{nowrap|y • x}} are idempotents.{{cite book|editor=Gracinda M. S. Gomes|title=Semigroups, Algorithms, Automata and Languages|chapter-url=https://books.google.com/books?id=IL58mAsfXOgC&pg=PA167|year=2002|publisher=World Scientific|isbn=978-981-277-688-4|pages=167–168|author=John Fountain|chapter=An introduction to covers for semigroups}} [http://www-users.york.ac.uk/~jbf1/coimbra2.pdf preprint]

An element x of S for which there is an element y of S such that {{nowrap|1=x • y • x = x}} is called regular. A regular semigroup is a semigroup in which every element is regular. This is a stronger notion than weak inverse. Every regular semigroup is E-inversive, but not vice versa.

If every element x in S has a unique inverse y in S in the sense that {{nowrap|1=x • y • x = x}} and {{nowrap|1=y • x • y = y}} then S is called an inverse semigroup.

In category theory, a weak inverse of an object A in a monoidal category C with monoidal product ⊗ and unit object {{mvar|I}} is an object B such that both {{math|A ⊗ B}} and {{math|B ⊗ A}} are isomorphic to the unit object {{mvar|I}} of C. A monoidal category in which every morphism is invertible and every object has a weak inverse is called a 2-group.

• Generalized inverse
• Von Neumann regular ring

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Category:Monoidal categories

Category:Semigroup theory

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