Weak inverse

In mathematics, the term weak inverse is used with several meanings.

Theory of semigroups

In the theory of semigroups, a weak inverse of an element x in a semigroup (S, •) is an element y such that 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 x ∈ S, there exists y ∈ S such that x • y and y • x are idempotents.[1]

An element x of S for which there is an element y of S such that 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.[1]

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

Category theory

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

See also

• Generalized inverse
• Von Neumann regular ring

References

1. John Fountain (2002). "An introduction to covers for semigroups". In Gracinda M. S. Gomes (ed.). Semigroups, Algorithms, Automata and Languages. World Scientific. pp. 167–168. ISBN 978-981-277-688-4. preprint